# Improved well-posedness for the limit flow of differentiation of roots of polynomials

Charles Bertucci, Valentin Pesce

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Charles Bertucci, Valentin Pesce. Improved well-posedness for the limit flow of differentiation of roots of polynomials. 2026. hal-05497629

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
IMPROVED WELL-POSEDNESS FOR THE LIMIT FLOW OF
DIFFERENTIATION OF ROOTS OF POLYNOMIALS

CHARLES BERTUCCI AND VALENTIN PESCE

ABSTRACT. In this paper, we study the partial differential equation on the circle that was heuristically obtained by Steinerberg [32] on the real line and which represents the evolution of the density of the roots of polynomials under differentiation. After integrating the partial differential equation in question, we observe that it can be treated with the theory of viscosity solutions. This equation at hand is a non linear parabolic integro-differential equation which involves the elliptic operator called the half-Laplacian. Due to the singularity of the equation, we restrict our study to strictly positive initial condition. We obtain a comparison principle for solutions of the primitive equation which yields uniqueness, existence, continuity with respect to initial condition. We also present heuristics to justify that the system of particles indeed approximates the solution of the equation.

CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. General introduction</td><td>2</td></tr><tr><td>1.2. Overview of the results of the paper</td><td>3</td></tr><tr><td>1.3. Structure of the paper</td><td>4</td></tr><tr><td>2. A brief overview of the problem</td><td>4</td></tr><tr><td>2.1. Location of the roots of the derivative of a polynomial</td><td>4</td></tr><tr><td>2.2. Known results about random polynomials</td><td>5</td></tr><tr><td>2.3. Heuristic of the PDE</td><td>7</td></tr><tr><td>3. Study of the PDE and primitive equation</td><td>9</td></tr><tr><td>3.1. Notation</td><td>9</td></tr><tr><td>3.2. Primitive equation</td><td>10</td></tr><tr><td>3.3. Viscosity solutions of the truncated PDEs</td><td>12</td></tr><tr><td>3.4. Comparison principles</td><td>12</td></tr><tr><td>3.5. Consequences of comparison principles for truncated PDEs</td><td>15</td></tr><tr><td>3.6. Existence, uniqueness, continuity in the initial condition of viscosity solutions of the original equation</td><td>19</td></tr><tr><td>4. System of particles</td><td>20</td></tr><tr><td>4.1. Reminders and notation for trigonometric polynomials</td><td>20</td></tr><tr><td>4.2. Comparison principles on the particles</td><td>20</td></tr><tr><td>4.3. An heuristic of the derivation of the viscosity solutions of the PDE</td><td>22</td></tr><tr><td>Appendix A. Proof of Proposition 4.3 in the particular case s<sub>N</sub> = 1/(2N)</td><td>27</td></tr><tr><td>A.1. Notation</td><td>27</td></tr><tr><td>A.2. Near part</td><td>27</td></tr><tr><td>A.3. Far away part</td><td>28</td></tr><tr><td>A.4. Very near part</td><td>28</td></tr><tr><td>A.5. Proof of Proposition A.3</td><td>28</td></tr><tr><td>Acknowledgments</td><td>35</td></tr><tr><td>References</td><td>35</td></tr></table>

Date: Winter 2026, compiled February 6, 2026.

Key words and phrases. Random polynomials, Partial differential equation, Viscosity solutions, Mean field limit.
# 1. INTRODUCTION

## 1.1. General introduction.
In this paper we study the limit partial differential equation (PDE) that was heuristically obtained by Steinerberg [32] in a dynamical approach to characterize the flow of differentiation. The heuristic is quite difficult to set going but Steinerberg points in his paper some explicit solutions of the PDE which come from orthogonal polynomials families. This motivates the studying of the limit PDE to make rigorous the heuristic of the convergence of particles. Moreover this problem has been surprisingly connected to free probability, which motivated several works in this direction [19, 25, 21, 22].

The problem was first studied for random roots of trigonometric polynomials because there is not a loss of roots after differentiation (and so at the level of the PDE in the trigonometric case the mass of a solution is preserved). In this case a connection with free probability was also studied [24]. In [27], Kiselev and Tan study the limit PDE on the circle and they prove the existence and uniqueness of smooth solutions of the PDE starting from a strictly positive smooth initial condition. They also obtain a notion of convergence of the system of particles towards the strong solution of the PDE under assumptions of smoothness.

We first briefly state already known results about the location of roots of polynomials after differentiation both for the sake completeness and to motivate the problem. We recall the heuristic of the proof of Steinerberg [32] to formally obtain the PDE that shall be studied.

Then, the main goal of this paper is to introduce a new kind of solutions of this PDE in the trigonometric case and to prove existence and uniqueness for this new notion of solutions. Thanks to this approach we could study the PDE for less regular initial conditions in comparison with what was done in [27]. We note that if we integrate the PDE, which means we look at the PDE satisfied by the cumulative distribution function of a solution of the PDE, we obtain a new PDE that can be studied using the theory of viscosity solutions. First, this PDE is invariant by translation in space and so the integration of the initial PDE does not depend on the angle of reference we chose to primitive a measure on the circle. Then, this primitive equation is a parabolic integro-differential partial differential equation. The integro-differential operator that appears is known as the half Laplacian on the circle and satisfies an elliptic property which hints to look at viscosity solutions of this equation. However, this PDE has a singularity which makes difficult to state properly a comparison principle as we usually do in viscosity theory. To overcome this issue we change the PDE by truncating the singularity. At the level of the original PDE it physically says that we are looking at solution starting from a strictly positive initial condition as in [27]. For these truncated primitive equations we get a comparison principle and so uniqueness of the solution of the PDE by using standard arguments in viscosity theory solutions ([10] for fully non linear PDE of second order and [2, 3, 4] for PDE with an integro-differential term). Furthermore we can show that starting from an appropriate initial condition, we can guarantee that truncated PDEs and the original PDE are the same.

Later we go back to the system of particles that are the roots of trigonometric polynomials. We try to understand the solutions that we just defined as a limit of the system of particles as it was done in a similar approach to the Dyson equation for the eigenvalues of random matrices [6, 8]. We first prove a discrete comparison principle for the particles. Finally, even if we were not able to provide a complete proof, we explain heuristically how to obtain the viscosity solutions of the PDE as the limit of the system of particles.
1.2. **Overview of the results of the paper.** The goal of this paper is to study the PDE:

$$ \partial_t u + \frac{1}{\pi} \partial_\theta \left( \arctan \left( \frac{H[u]}{u} \right) \right) = 0 \text{ in } (0, +\infty) \times \mathbb{T}, \quad (1.1) $$

where $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$ and $H[u]$ is the periodic Hilbert transform. In [27], it was proved that if the initial condition of this PDE is smooth (in $H^s$ for $s > 3/2$) and strictly positive there exists a unique strong solution of this PDE and some properties as regularization, long time behaviour of these solutions have been studied.

The PDE (1.1) is very singular and having a "good" notion of solutions to obtain uniqueness or existence is not clear at all. Moreover a regularizing property of (1.1) is not expected in general. Indeed, as we shall mention from the heuristic derivation of the PDE it is expected that if the initial condition of (1.1) is the Dirac mass $\delta_0$ then the solution of the PDE will contain a Dirac mass for $t \in [0, 1]$ at least.

We introduce a new notion of solutions to study the PDE (1.1) using the theory of viscosity solutions. Thanks to this approach one can obtain existence and uniqueness of solutions of (1.1) without assumptions of regularity of the initial condition but still with hypothesis of strict positivity. We integrate the PDE (1.1) to obtain a PDE satisfied by the primitive of a solution of (1.1):

$$ \partial_t F(t, \theta) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[F(t, \cdot)](\theta)}{(\partial_\theta F(t, \theta))_+} \right) + \frac{\pi}{2} \right) = 0 \text{ in } (0, +\infty) \times \mathbb{R}, \quad (1.2) $$

where $A_0$ is the so-called periodic half Laplacian. This PDE is well settled in the viscosity solutions theory thanks to the elliptic property of the half Laplacian. Thanks to this notion of solution we can now study solutions of (1.1) starting for instance from $\delta_0(dx)$. However, the singularity of the PDE makes difficult to properly establish comparison principles for this PDE. So we consider truncated modification of (1.2) for which the comparison principle holds:

$$ \partial_t F(t, \theta) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[F(t, \cdot)](\theta)}{\max((\partial_\theta F(t, \theta))_+, m)} \right) + \frac{\pi}{2} \right) = 0 \text{ in } (0, +\infty) \times \mathbb{R}, \quad (E_m) $$

where $m > 0$ is a fixed real number.

The consequence of the truncation of the PDE (1.2) is that we have to consider solutions of (1.1) starting from strictly positive initial condition (or equivalently solutions of (1.2) with strictly non-decreasing initial condition). For a suitable notion of viscosity solutions introduced in Section 3.2 we prove existence and uniqueness of strictly non-decreasing solutions.

**Theorem 1.1.** Let $\mu \in \mathcal{P}(\mathbb{T})$ such that $F_\mu$, the cumulative distribution function of $\mu$, satisfies a condition of strict monotony ($H_m$) for a $m > 0$. Then there exists a viscosity solution of (1.2) with initial condition $F_\mu$ satisfying ($H_m$) for all time.

**Theorem 1.2.** Let $\mu \in \mathcal{P}(\mathbb{T})$ such that $F_\mu$ satisfies ($H_m$) for a $m > 0$. Then there exists at most one viscosity solution of (1.2) which satisfies ($H_{m'}$) for $0 < m' \le m$ with initial condition $F_\mu$.

We also prove a continuity of this notion of solutions with respect to the initial condition.

**Theorem 1.3.** Fix $m > 0$. Let $(\mu_n)_{n \in \mathbb{N}} \in \mathcal{P}(\mathbb{T})^N$ be a family of probabilities measures on $\mathbb{T}$ such that for all $n \ge 0$ $F_{\mu_n}$, the cumulative distribution function of $\mu_n$, satisfies ($H_m$). Assume that $(\mu_n)_{n \in \mathbb{N}} \in \mathcal{P}(\mathbb{T})^N$ converges in law towards $\mu \in \mathcal{P}(\mathbb{T})$ and that $\mu$ is absolutely continuous with respect to Lebesgue measure. Let $(F_n(t, x))_{t \ge 0, x \in \mathbb{R}}$ (resp.
($F_\infty(t, x)_{t \ge 0, x \in \mathbb{R}}$) be a viscosity solution of (3.5) with initial condition $F_{\mu_n}$ (resp. $F_\mu$). Then we have the following convergence

$$ \sup_{t \ge 0} \|F_n(t, \cdot) - F_\infty(t, \cdot)\|_{L^\infty(\mathbb{R})} \xrightarrow{n \to +\infty} 0. $$

**1.3. Structure of the paper.** First, in Section 2 we recall some well known results about random polynomials that motivate the problem and the study of the PDE (1.1).

Then, in Section 3 we study the PDE (1.1) by integrating it and introducing a notion of viscosity solutions for this primitive equation.

Finally, in Section 4 we investigate how viscosity solutions of (1.2) can be constructed as limits of the system of particles heuristically introduced by Steinerberger in [32]. Even if we were not able to completely derive the convergence, we present the partial proof that we were able to find in order to justify this heuristic at the level of viscosity solutions of (1.2).

## 2. A BRIEF OVERVIEW OF THE PROBLEM

**2.1. Location of the roots of the derivative of a polynomial.** The problem of the location of the roots of the derivative of a polynomial knowing the roots of this polynomial have been a very well studied problem in the literature. For instance, see the non exhaustive references [11, 34, 35] and references therein.

The most important result on this topic is the so-called Gauss-Lucas theorem. This theorem can be viewed as a generalisation in $\mathbb{C}$ of the Rolle theorem for polynomials. We quickly recall proofs of this result since similar arguments shall be used later on. For $P \in \mathbb{C}[X]$, let $\mathcal{Z}(P)$ be the set of the roots of $P$. We write $\text{Conv}(A)$ the convex hull of $A \subset \mathbb{C}$ which is the smallest convex of $\mathbb{C}$ that contains $A$.

**Theorem 2.1 (Gauss-Lucas theorem).** Let $P \in \mathbb{C}[X]$, then $\mathcal{Z}(P') \subset \text{Conv}(\mathcal{Z}(P))$.

We give two proofs of this result because these two proofs use arguments that shall be used later on. The first one is based on the fundamental identity:

$$ \frac{P'(X)}{P(X)} = \sum_{i=1}^{n} \frac{1}{X - x_j}, \qquad (2.1) $$

for $P \in \mathbb{C}_n[X]$ (where $\mathbb{C}_n[X]$ is the set of polynomial of $\mathbb{C}[X]$ of degree at most $n$) whose roots are $(x_i)_{i=1}^n$ with multiplicity. This relation is the key ingredient in what shall follow to obtain the partial differential equation.

The second proof gives an electrostatic point of view for the Theorem 2.1. It makes a link between the roots of complex polynomials and potential theory. This point of view was a key ingredient to obtain results about random polynomials as we will mention in Section 2.2.

*Proof.* • First proof. Let $x_1, \dots, x_k$ be the roots of $P$ and $z \in \mathcal{Z}(P')$. If $z$ is a multiple root of $P$ the result is obvious. Now assume that $z \in \mathcal{Z}(P') \setminus \mathcal{Z}(P)$. Evaluating

$$ \frac{P'(X)}{P(X)} = \sum_{i=1}^{k} \frac{1}{X - x_j} $$

in $z$ yields:

$$ 0 = \sum_{i=1}^{k} \frac{1}{z - x_j}. $$

To express $z$ as a convex combination of the $z_j$ we write:

$$ 0 = \sum_{j=1}^{k} \frac{1}{z - x_j} = \sum_{j=1}^{k} \frac{\bar{z} - \bar{x}_j}{|z - x_j|^2} = \sum_{j=1}^{k} \frac{z - x_j}{|z - x_j|^2} $$
It gives:

$$z = \frac{\sum_{j=1}^{k} \frac{x_j}{|z - x_j|^2}}{\sum_{j=1}^{k} \frac{1}{|z - x_j|^2}} \in \text{Conv}(\mathcal{Z}(P)).$$

• Second proof. Let $x_1, ..., x_k$ be the roots of $P$ and assume that $P$ is monic. Consider the $x_i$ as positive electrical charges in $\mathbb{C} = \mathbb{R}^2$. The electric charge density created by these charges is $\rho = \sum_{i=1}^k \delta_{x_i}$. We can compute the electrical field $\vec{E}$ created by the charges using the Maxwell-Gauss law. Indeed, we have:

$$\text{div}(\vec{E}) = \frac{\rho}{\varepsilon_0} \\ \vec{E} = -\nabla V,$$

where $V$ is the electric potential associated to $\vec{E}$ and $\varepsilon_0$ is the vacuum permittivity. It yields the so-called Laplace equation

$$-\Delta V = \frac{\rho}{\varepsilon_0}.$$

We recall that the fundamental solution of the Laplacian in $\mathbb{R}^2$ is $g(x, y) = -\frac{1}{2\pi} \ln(|(x, y)|)$ (with $|(x, y)|$ the canonic norm in $\mathbb{R}^2$ of $(x, y)$). More precisely we have that for all $a \in \mathbb{R}^2$ in the sense of distributions:

$$-\Delta [-\ln(|(x, y) - a|)] = 2\pi\delta_a.$$

Hence the solution of the Laplace equation is

$$V(z) = -\frac{1}{2\pi\varepsilon_0} \sum_{i=1}^{k} \ln|z - x_i| = -\frac{\ln|P(z)|}{2\pi\varepsilon_0}, z \in \mathbb{C}.$$

So the electric field $\vec{E} = -\nabla V$ vanishes exactly where $P'$ vanishes. However since the charges are all positive, the electric field can not vanish outside the convex hull of the charges proving the result. □

## 2.2. Known results about random polynomials.

Notation. We introduce the notation of measure theory that shall be used in this section.

• For $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ we write $\mathcal{P}(\mathbb{K})$ the set of probability measures supported on $\mathbb{K}$.

• For $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ and $\mu \in \mathcal{P}(\mathbb{K})$ and $f$ a reasonable function we write $\mu(f) = \int_{\mathbb{K}} f d\mu$.

• Let $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$. We recall that we say that a sequence of probability measures $\mu_N$ on $\mathcal{P}(\mathbb{K})$ converges weakly (or in law) towards $\mu \in \mathcal{P}(\mathbb{K})$ if and only if for all $f: \mathbb{K} \to \mathbb{R}$ continuous and bounded functions we have $\mu_N(f) \xrightarrow{N \to +\infty} \mu(f)$. In this case, we write

$$\mu_N \xrightarrow{\mathcal{L}} \mu.$$

There are two natural ways to generate a random polynomial. We can sample its roots or its coefficients. These two constructions are linked by the Vieta formulas.

A question that really emerged during the 20th century was: given a polynomial in $\mathbb{R}[X]$ whose coefficient are independent and identically distributed, could we estimate the average number of real roots this random polynomial has? Up to the knowledge of the authors Bloch and Pólya were the first one to try to answer this question in [9]. Littlewood and Offord [28] dealt with this question in particular cases as when the coefficients are uniform on $[0, 1]$ or Bernoulli random variables. Then, an astonishing result was obtained
by Kac by giving an exact formula to compute the expectation of real roots of such polynomials and computing asymptotically the expectation when the degree of the polynomial goes to +∞ and when the coefficients are Gaussian [26]. Actually, the result obtained by Kac is quite "universal" in the sense that for reasonable distributions this asymptotic still holds when the degree increases [33] and similar computations were also done for random trigonometric polynomials [14, 17] using the Kac approach. This result was improved by obtaining an expansion of this expectation for instance using a geometric interpretation of the Kac formula obtained by Edelman and Kostlan [16]. We refer to [12, 13] for recent works on this topic and references therein. We also refer to [20] which is a general textbook about the study of the roots of standard models of random polynomials.

In this paper, we shall focus on the problem of random polynomials when the roots are sample independently and following the same distribution. This question was more recently asked by Pemantle and Rivin in [31]. More precisely, let $\mu \in \mathcal{P}(\mathbb{C})$ be a probability measure on $\mathbb{C}$ and $(X_i)_{i \in \mathbb{N}}$ a sequence of independent and identically distributed random variables with law $\mu$, we define for all $n \in \mathbb{N}$ the random polynomial $P_n(X) = \prod_{i=1}^n (X - X_i)$ and the empirical measure of the roots of $P'_n$ as

$$ \nu_n := \frac{1}{n-1} \sum_{z \in \mathbb{C}, P'_n(z)=0} \delta_z. $$

By the law of large numbers the empirical measure of the roots of $P_n$ almost surely converges weakly towards $\mu$. The question raised by Pemantle and Rivin was: how does the empirical measure of the roots of $P'_n$ behave? Thanks to their simulations Pemantle and Rivin conjectured that $(\nu_n)_{n \in \mathbb{N}}$ also almost surely converges weakly towards $\mu$. They proved the result under a technical assumption that must be satisfied by the measure $\mu$. However, this assumption fails for certain measures such as the uniform measure on the unit circle. Then, a universal result was obtained by Kabluchko in [23]. He proved that for all $\mu \in \mathcal{P}(\mathbb{C})$ the empirical mean of the roots of $P'_n$ converges in probability towards $\mu$. This proof is based on the electrostatic interpretation of the zeros of polynomials as it was mentioned in the second proof of Theorem 2.1. He used that for $f$ a non zero holomorphic function on $\mathbb{C}$ we have that $\Delta \ln |f(z)| = 2\pi \sum_{z \in \mathbb{C}, f(z)=0} \delta_z$ in the sense of distributions. Using this formula yields that for all $\phi$ smooth and compactly supported functions

$$ \frac{1}{2\pi n} \int_{\mathbb{C}} \ln(|L_n(z)|) \Delta\phi(z) d\text{Leb}(z) = \frac{1}{n} \sum_{z \in \mathbb{C}, f'(z)=0} \phi(z) - \frac{1}{n} \sum_{z \in \mathbb{C}, f(z)=0} \phi(z), \quad (2.2) $$

with $L_n(z) := \frac{P'_n(z)}{P_n(z)}$. The technical part of the proof consists in proving that the left hand side of (2.2) goes to 0 in probability which implies the result obtained by Kabluchko. The key argument is the use of potential theory tools as the Poisson-Jensen formula. This result represents an important step in the proof of the conjecture of Pemantle and Rivin but the convergence obtained by Kabluchko is in probability and not almost surely. Finally, the conjecture was solved by Angst & al. in [1]. The proof is based on the same idea as Kabluchko but they combined it with an anti-concentration inequality [18]. Using the same arguments, this result was extended by proving that the result holds when we look at the empirical measure of the roots of the $k^{th}$ derivatives of $P_n$ [29, 30] (also when $k$ depends on $n$ but slow enough).

To finish this section and for the sake of completeness we give a counterpart of the proof of the Pemantle and Rivin conjecture in the case of zeros of trigonometric polynomials since we shall work on this case later on. This proof is nearly similar with the proof of the Pemantle and Rivin conjecture in the particular case when $\mu \in \mathcal{P}(\mathbb{R})$.

**Proposition 2.2.** Let $\mu \in \mathcal{P}(\mathbb{T})$ and $(X_i)_{i \in \mathbb{N}}$ be a family of independent and identically distributed random variables of law $\mu$. For $N \ge 1$, let us define $P_N(X) = \prod_{i=1}^{2N} \sin(\frac{X - X_i}{2})$
the trigonometric polynomial whose roots are the $(X_i)_{1\le i\le 2N}$. Let $\nu_N = \frac{1}{2N} \sum_{\theta \in \mathbb{T}}, P'_N(\theta)=0$ $\delta_\theta$ be the empirical measure of the roots of $P'_N$. Then almost surely

$$ \nu_N \xrightarrow[N \to \infty]{\mathcal{L}} \mu. $$

*Proof.* For all $N > 0$ we relabel the roots of $P_N$ as follows. We denote $k(N)$ the (random) number of distinct roots of $P_N$ and we introduce $(\alpha_i(N))_{1\le i\le k(N)}$ the (random) multiplicity of the respective distinct roots of $P_N$. We first order the roots of $P_N$

$$ 0 \le X_{1,1} = \dots = X_{1,\alpha_1(N)} < X_{2,1} = \dots = X_{2,\alpha_2(N)} < \dots < X_{k(N),1} = \dots = X_{k(N),\alpha_{k(N)}(N)} < 2\pi. $$

We can now relabel the roots of $P_N$ such that $\overline{X_1} := X_{1,1}$, ..., $\overline{X_{\alpha_1(N)}} := X_{1,\alpha_1(N)}$, $\overline{X_{\alpha_1(N)+1}} := X_{2,1}$, ..., $\overline{X_{\alpha_1(N)+\alpha_2(N)}} := X_{2,\alpha_2(N)}$, ..., $\overline{X_{2N}} := X_{k(N),\alpha_{k(N)}(N)}$.

By the Rolle theorem we know that there are two kinds of roots of $P'_N$: the same roots as $P_N$ but with multiplicity $\alpha_i(N) - 1$ for all $1 \le i \le k(N)$ and roots (that are simple) obtained by using the Rolle theorem between $X_{i,\alpha_i(N)} < X_{i+1,1}$ for all $1 \le i \le k(N) - 1$ and between $X_{k(n),\alpha_{k(N)}(N)} < X_{1,1} + 2\pi$. For all $N > 0$, let $(Y_i^N)_{1\le i\le 2N}$ be the $2N$ roots of $P'_N$ such that $Y_i^N$ is either $\overline{X_i}$ in the case of a multiple root or the unique root of $P'_N$ between $\overline{X_i}$ and $\overline{X_{i+1}}$ otherwise. Let $f$ be a bounded Lipschitz function with Lipschitz constant $K$. Almost surely, for all $N$ we have:

$$ \begin{aligned} \left| \frac{1}{2N} \sum_{i=1}^{2N} f(\overline{X_i}) - \frac{1}{2N} \sum_{i=1}^{2N} f(Y_i^N) \right| &\le \frac{1}{2N} \sum_{i=1}^{2N} \left| f(\overline{X_i}) - f(Y_i^N) \right| \\ &\le \frac{K}{2N} \sum_{i=1}^{2N} \left| \overline{X_i} - Y_i^N \right| \qquad (2.3) \\ &\le \frac{K}{2N} \sum_{i=1}^{2N} \left[ \overline{X_{i+1}} - \overline{X_i} \right], \end{aligned} $$

where $\overline{X_{2N+1}} = \overline{X_1} + 2\pi$ by convention.

So we deduce that almost surely, for all $N$

$$ \left| \frac{1}{2N} \sum_{i=1}^{2N} f(\overline{X_i}) - \frac{1}{2N} \sum_{i=1}^{2N} f(Y_i^N) \right| \le \frac{K}{2N} (\overline{X_{N+1}} - \overline{X_1}) = \frac{\pi K}{N}. $$

By the strong law of large numbers for the $X_i$ we deduce that almost surely for all $f$ bounded Lipschitz function

$$ \frac{1}{2N} \sum_{i=1}^{2N} f(Y_i) \xrightarrow[N \to +\infty]{\mathcal{L}} \int_{\mathbb{T}} f d\mu $$

which proves the result.

□

2.3. **Heuristic of the PDE.** In this section we recall how the PDE was formally obtained in [32]. We give the heuristic of the proof for polynomials and not trigonometric polynomials to focus on the main ideas. The heuristic of the proof for trigonometric polynomials is quite similar and can be found in [27].

Let $(x_i)_{i \in \mathbb{N}} \in \mathbb{R}^N$ be all distinct and write $p_N(X) = \prod_{i=1}^N (X - x_i)$. Let $u \in \mathcal{P}(\mathbb{R})$ be a distribution that approximates the $x_i$ in the sense that $\sum_{i=1}^N \frac{\delta_{x_i}}{N} \sim u$. Let $y_i$ be the unique root of $p'_N$ in $(x_i, x_{i+1})$. The goal is to understand the flow created by the derivation and more precisely to look at $y_i - x_i$. Fix $m \in \mathbb{N}$. As for the Gauss-Lucas theorem, we start from the identity:

$$ 0 = \sum_{i=1}^{N} \frac{1}{y_m - x_i}. $$
Since we look at a mean field limit we naturally renormalise the sum by starting from the identity:

$$0 = \frac{1}{N} \sum_{i=1}^{N} \frac{1}{y_m - x_i}.$$

We separate the sum in two parts associating to a near interaction and a far away interaction:

$$0 = \frac{1}{N} \sum_{|x_i - y_m| \le N^{-1/2}} \frac{1}{y_m - x_i} + \frac{1}{N} \sum_{|x_i - y_m| > N^{-1/2}} \frac{1}{y_m - x_i} =: S_{1,N} + S_{2,N}. \quad (2.4)$$

Since the distribution $u$ approximates the $x_i$ we formally have:

$$\frac{1}{N} \sum_{|x_i - y_m| > N^{-1/2}} \frac{1}{y_m - x_i} \sim \lim_{\epsilon \to 0} \int_{|x - y_m| > \epsilon} \frac{1}{y_m - x} u(dx) = \pi H[u](y_m), \quad (2.5)$$

where $H[u] := \frac{1}{\pi} P.V.(1/x) * u$ is the real Hilbert transform of $u$ with P.V. the principal value.

About $S_{1,N}$, the idea introduced by Steinerberger is to approximate the $x_i$ near $y_m$ by equally distributed points around $x_m$. We want to construct points $\tilde{x}_i$ such that

$$\int_{x_m}^{\tilde{x}_i} u(x) dx = \frac{i-m}{N}.$$

Formally, approximating $\int_{x_m}^{\tilde{x}_i} u(x) dx \sim u(x_m)(\tilde{x}_i - x_m)$, we set $\tilde{x}_i = x_m + \frac{i-m}{Nu(x_m)}$.

Formally, we rewrite $S_{1,N}$ by changing the $x_i$ by these new points:

$$\begin{align}
S_{1,N} = \frac{1}{N} \sum_{|x_i - y_m| < N^{-1/2}} \frac{1}{y_m - x_i} &\sim \frac{1}{N} \sum_{|\tilde{x}_i - y_m| < N^{-1/2}} \frac{1}{y_m - \tilde{x}_i} \nonumber \\
&\sim \frac{1}{N} \sum_{|k| < N^{1/2}} \frac{1}{y_m - x_m - \frac{k}{Nu(x_m)}} \tag{2.6} \\
&\sim u(x_m) \sum_{k \in \mathbb{Z}} \frac{1}{(y_m - x_m)Nu(x_m) + k}. \nonumber
\end{align}$$

We recall the Euler identity for the function cotan := $\frac{\cos}{\sin}$:

$$\cotan(z) = \sum_{k \in \mathbb{Z}} \frac{1}{z + k\pi}.$$

Using this identity and (2.6), we get:

$$S_{1,N} \sim \pi u(x_m) \cotan(N\pi u(x_m)(y_m - x_m)) \quad (2.7)$$

Passing to the limit in (2.4) formally yields:

$$0 = \pi H[u](y_m) + \pi u(x_m) \cotan(N\pi u(x_m)(y_m - x_m)). \quad (2.8)$$

Since $u$ is a density that approximates the $x_i$, we have

$$u(x_m)(x_{m+1} - x_m) \approx \int_{x_m}^{x_{m+1}} u(y) dy = \frac{1}{N}$$

which gives that

$$0 \le y_m - x_m \le x_{m+1} - x_m \le \frac{1}{Nu(x_m)},$$
and so

$$0 \leq N\pi u(x_m)(y_m - x_m) \leq \pi.$$

The formula (2.8) yields a flow created by the derivation:

$$y_m - x_m = \frac{1}{\pi N u(x_m)} \arctan \left( - \frac{H[u](x_m)}{u(y_m)} \right),$$

where $\arctan$ is the reciprocal function of $\cotan: (0, \pi) \to \mathbb{R}$. Using the identity $\arctan(-\theta) = \arctan(\theta) + \pi/2$ for all $\theta \in \mathbb{R}$, the flow creating by the derivation is

$$y_m - x_m = \frac{1}{\pi N u(x_m)} \left( \arctan \left( \frac{H[u](y_m)}{u(x_m)} \right) + \frac{\pi}{2} \right). \quad (2.9)$$

Taking $\Delta t = 1/N$ as time scale, (2.9) gives that the derivation creates a macroscopic flux at speed:

$$v(x_m) = \frac{1}{\pi u(x_m)} \left( \arctan \left( \frac{H[u](y_m)}{u(x_m)} \right) + \frac{\pi}{2} \right).$$

It gives at a macroscopic level the following conservative equation:

$$\partial_t u + \partial_x(uv) = \partial_t u + \frac{1}{\pi} \partial_x \left( \arctan \left( \frac{H[u]}{u} \right) \right) = 0. \quad (2.10)$$

Let us also mention that in [32] Steinerberger also finds explicit solutions for the PDE (2.10) that can be constructed using well known results on orthogonal polynomials. Another example starting from the degenerate condition $x_i = 0$ for all $i \in \mathbb{N}$ gives that a solution of the PDE (2.10) starting from $\delta_0(dx)$ should be $u(t, dx) = (1 - t)\delta_0(dx)$.

### 3. STUDY OF THE PDE AND PRIMITIVE EQUATION

We shall study the following PDE which is the analogous of (2.10) for trigonometric polynomials:

$$\partial_t u + \frac{1}{\pi} \partial_\theta \left( \arctan \left( \frac{H[u]}{u} \right) \right) = 0 \text{ in } (0, +\infty) \times \mathbb{T}, \quad (3.1)$$

where $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$ and $H[u] := \frac{1}{2\pi}P.V. \cotan(\frac{\pi}{2}) * u$ is the periodic Hilbert transform with $P.V.$ the principal value. If $u$ is smooth enough we can more explicitly write $H[u](y) = \frac{1}{2\pi}P.V. \int_{\mathbb{T}} \cotan(\frac{y-x}{2})(u(x) - u(y))dx$. Since the mass of a solution of (3.1) is preserved, we shall look at solution $u$ of (3.1) such that $u(t, \cdot) \in \mathcal{P}(\mathbb{T})$ for every $t \ge 0$. The existence, regularity and long time behaviour of solutions of (3.1) starting from a smooth initial condition have been studied in [27]. The main idea of this section is to integrate the equation (3.1) to obtain what we call the "primitive" equation of (3.1). This new equation shall be treated with the theory of viscosity solutions [10, 4, 2, 3] to define a new kind of solutions of (3.1). This allows us to improve the existing results of existence and uniqueness of solutions of (3.1) namely by allowing us to consider more general initial conditions.

#### 3.1. Notation.
We introduce the notation for the functional spaces that shall be used in the following parts.

* A function $f: \mathbb{R} \to \mathbb{R}$ is an element of $C^k(\mathbb{T})$ for $k \in [0, \infty]$ (resp. $L^p(\mathbb{T})$ for $p \in [1, \infty]$) if $f$ is a $2\pi$ periodic function of class $C^k$ (resp. if $f$ is a $2\pi$ periodic function in $L^p([0, 2\pi])$). We write $\mathcal{F}(\mathbb{T})$ for the space of functions on the circle that means the functions from $\mathbb{R}$ to $\mathbb{R}$ that are $2\pi$ periodic.

* Let $\mathcal{F}_{2\pi}$ (resp. $\mathcal{F}_{t,2\pi}$) be the space of functions $F$ defined on $\mathbb{R}$ (resp. $\mathbb{R}^+ \times \mathbb{R}$) for which there exists $a \in \mathbb{R}$ such that for all $x \in \mathbb{R}: F(x + 2\pi) = F(x) + a$ (resp. for all $t \ge 0$, for all $x \in \mathbb{R}$, we have $F(t, x + 2\pi) = F(t, x) + a$).
• We say that $F \in \mathcal{F}_{2\pi}$ (resp. $\mathcal{F}_{t,2\pi}$) is bounded if there exists $C > 0$ such that $F$ is bounded by $C$ on $[0, 2\pi]$ (resp. for all $t \ge 0$, $F(t, \cdot)$ is bounded by $C$ on $[0, 2\pi]$).

• Let $C_{x}^{1,1}(\mathbb{R}^{+} \times \mathbb{R})$ be the space of functions $f(t, x)$ defined from $\mathbb{R}^{+} \times \mathbb{R}$ with values in $\mathbb{R}$ such that $f$ is a $C^1$ function and $\partial_x f$ is Lipschitz.

• Let $C_{2\pi}^{1,1}$ be the space of functions $F \in C^1(\mathbb{R}, \mathbb{R})$ such that $F'$ is a Lipschitz function and for which there exists $a \in \mathbb{R}$ such that for all $x \in \mathbb{R}$, we have $F(x + 2\pi) = F(x) + a$.

• Let $C_{t,2\pi}^{1,1}$ be the space of functions of $F \in C_{x}^{1,1}(\mathbb{R}^{+} \times \mathbb{R})$, for which there exists $a \in \mathbb{R}$ such that for all $t \ge 0$, for all $x \in \mathbb{R}$, we have $F(t, x + 2\pi) = F(t, x) + a$.

• For a locally bounded function $F$ on a subset $\Omega$ of $\mathbb{R}^d$, we define for all $x \in \Omega$

$$F_*(x) = \liminf_{y \to x} F(y), \quad F^*(x) = \limsup_{y \to x} F(y).$$

**3.2. Primitive equation.** In [6, 8], authors studied what is called the Dyson equation on $\mathbb{R}$ or $\mathcal{T}$ namely

$$\partial_t u + \partial_x(uH[u]) = 0. \qquad (3.2)$$

These PDE appear as limits of the empirical mean of the eigenvalues of continuous in time models of large random matrices introduced by Dyson in 1962 [15]. The main idea in [6, 8] is to integrate the PDE that is to look at the PDE satisfied by the cumulative distribution function of a solution of (3.2). This PDE can be treated with the theory of viscosity solutions. We follow this idea to deal with (3.1).

A priori there is no "good" definition of the cumulative distribution function of a measure $\mu \in \mathcal{P}(\mathcal{T})$, since there is no good point to start integrating from.

Given $\mu \in \mathcal{P}(\mathcal{T})$ we define its cumulative distribution function as $F_{\mu}(x) = \mu([0, x])$ if $x \ge 0$ and $F_{\mu}(x) = -\mu((x, 0))$ if $x < 0$. This define a function $F$ on $\mathbb{R}$ that is non-decreasing, right continuous and which satisfied $F_{\mu}(x + 2\pi) = F_{\mu}(x) + 1$ for every $x \in \mathbb{R}$ since $\mu \in \mathcal{P}(\mathcal{T})$.

Formally, if $F(t, \theta) := \int_{0}^{\theta} \mu(t, \theta')d\theta'$ and $\mu$ is a solution of (3.1), integrating (3.1) with respect to $\theta$ gives:

$$\partial_t F(t, \theta) + \frac{1}{\pi} \left( \arctan \left( \frac{H[\partial_\theta F(t, \cdot)](\theta)}{(\partial_\theta F(t, \theta))_+} \right) + \frac{\pi}{2} \right) = 0 \quad \text{in } (0, +\infty) \times \mathbb{R}. \qquad (3.3)$$

We put $(\partial_{\theta}F(t, \theta))_+ \text{ instead of } \partial_{\theta}F(t, \theta)$ to have a non negative term in the denominator since we shall look at solutions of (3.3) which are non-decreasing in the space variable as primitives of measures. This way the PDE (3.3) is elliptic as we shall explain. We choose the constant of integration with respect to what was obtained in the heuristic of the proof in (2.9).

We introduce the operator $A_0$ defined on $C_{2\pi}^{1,1}$ and called the half Laplacian defined as:

$$A_0[f](\theta) = \frac{1}{4\pi} P.V. \int_{-\pi}^{\pi} \frac{f(\theta) - f(\theta - \theta')}{\sin^2\left(\frac{\theta'}{2}\right)} d\theta' = \frac{1}{8\pi} \int_{\mathcal{T}} \frac{2f(\theta) - f(\theta - \theta') - f(\theta + \theta')}{\sin^2\left(\frac{\theta'}{2}\right)} d\theta'. \qquad (3.4)$$

We notice that $A_0[f] = H[f']$ for $f \in C_{2\pi}^{1,1}$.

Hence, we call primitive equation of (3.1) the following PDE:

$$\partial_t F(t, \theta) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[F(t, \cdot)](\theta)}{(\partial_\theta F(t, \theta))_+} \right) + \frac{\pi}{2} \right) = 0 \quad \text{in } (0, +\infty) \times \mathbb{R}. \qquad (3.5)$$

**Remark 3.1.** Let us notice that this equation is invariant by translation meaning that if $F$ is solution of (3.5) then $G(t, \theta) = F(t, \theta + a) + b$ is also a solution for all $a, b \in \mathbb{R}$. So if instead of considering the cumulative distribution function of $\mu \in \mathcal{P}(\mathcal{T})$ with respect to $0$ as an angle of reference we had chosen another angle $\theta' \in \mathbb{R}$ then the cumulative distribution function with respect to this angle would also satisfied the PDE (3.5).
The operator $A_0$ satisfies a maximum principle: if $F \in C_{2\pi}^{1,1}$ has a maximum in $\theta_0$, then $A_0[F](\theta_0) \ge 0$. This motivates our approach by the viscosity solutions theory as in [6, 7, 8].

**Definition 3.2.**

* An upper semi continuous (usc) function $F \in \mathcal{F}_{t,2\pi}$ is said to be a viscosity subsolution of (3.5) if for any function $\phi \in C_{t,2\pi}^{1,1}$, $(t_0, \theta_0) \in (0, \infty) \times \mathbb{R}$ point of strict maximum of $F - \phi$ such that $\partial_\theta \phi(t_0, \theta_0) \ne 0$ the following holds:

$$ \partial_t \phi(t_0, \theta_0) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[\phi(t_0, \cdot)](\theta_0)}{(\partial_\theta \phi(t_0, \theta_0))_+} \right) + \frac{\pi}{2} \right) \le 0 \quad (3.6) $$

* A lower semi continuous (lsc) function $F \in \mathcal{F}_{t,2\pi}$ is said to be a viscosity supersolution of (3.5) if for any function $\phi \in C_{t,2\pi}^{1,1}$, $(t_0, \theta_0) \in (0, \infty) \times \mathbb{R}$ point of strict minimum of $F - \phi$ such that $\partial_\theta \phi(t_0, \theta_0) \ne 0$ the following holds:

$$ \partial_t \phi(t_0, \theta_0) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[\phi(t_0, \cdot)](\theta_0)}{(\partial_\theta \phi(t_0, \theta_0))_+} \right) + \frac{\pi}{2} \right) \ge 0 \quad (3.7) $$

* A viscosity solution $F$ of (3.5) is a locally bounded function such that $F$ is subsolution and $F_*$ is a supersolution.

* We say that the initial condition of $F$ is $F_0$ if $F^*(0, \cdot) \le F_0^*$ and $F_*(0, \cdot) \ge (F_0)_*$.

* Finally $\mu \in C([0, T], \mathcal{P}(\mathbb{T}))$ is said to be a viscosity solution of (3.1) with initial condition $\mu_0 \in \mathcal{P}(\mathbb{T})$ if $F(t, x) := F_{\mu(t, \cdot)}(x)$ is a viscosity solution of (3.5) with initial condition $F_{\mu_0}$.

**Remark 3.3.** Following the heuristic derivation of the PDE (3.1) a solution of the PDE (3.1) starting from $\delta_0(dx)$ should be in the real case $u(t, x) = (1 - t)\delta_0(dx)$. One can easily check that the cumulative distribution function of $u$ is a viscosity solution of (3.5) in the sense of Definition 3.2.

We have to introduce a weak notion of solutions for (3.1) since we do not expect any regularizing property of the PDE (3.1) for non smooth initial data. Now we can try to prove existence and uniqueness for viscosity solutions of (3.1) starting from non smooth initial data as $\delta_0(dx)$.

However, due to another main difficulty of the PDE which is the singularity, the notion of viscosity solutions introduced in Definition 3.2 is difficult to use in practice. Indeed, we have to impose at a point of maximum that $\partial_\theta \phi \ne 0$ to make sense of the quantities in the PDE. Due to this extra condition it is difficult to state properly a comparison principle for this notion of viscosity solutions and so to obtain uniqueness of the viscosity solutions. To overcome this difficulty we shall restrict our study to solutions of (3.1) that formally satisfies $u(t, x) \ge m$ for all $t \ge 0$, for all $x \in \mathbb{R}$ for a certain $m > 0$. This hypothesis shall be called ($H_m$) in Definition 3.9. Having in mind the so-called monotone principle which states that for a smooth solution of (3.1) the minimum is non-decreasing in time we shall justify (even at the level of viscosity solutions) that it suffices to satisfy the hypothesis ($H_m$) at time 0 to satisfy it for any $t \ge 0$.

To properly state a comparison principle, we introduce a family of PDEs obtained by truncating the singularity. Then, we shall explain how the results of existence, uniqueness and continuity in the initial condition obtained at the level of these truncated PDEs can be used to obtain results for the initial PDE.

For $m > 0$ we consider the following primitive equation that shall be called ($E_m$):

$$ \partial_t F(t, \theta) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[F(t, \cdot)](\theta)}{\max((\partial_\theta F(t, \theta))_+, m)} \right) + \frac{\pi}{2} \right) = 0 \quad \text{in } (0, +\infty) \times \mathbb{R} \quad (E_m) $$
### 3.3. Viscosity solutions of the truncated PDEs.
Equations (E_m) shall be treated with the theory of viscosity solutions [10, 2, 4].

**Definition 3.4.** Fix $m > 0$.

* An upper semi continuous (usc) function $F \in \mathcal{F}_{t,2\pi}$ is said to be a viscosity subsolution of (E_m) if for any function $\phi \in C_{t,2\pi}^{1,1}$, $(t_0, \theta_0) \in (0, \infty) \times \mathbb{R}$ point of strict maximum of $F - \phi$ the following holds:

$$ \partial_t \phi(t_0, \theta_0) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[\phi(t_0, \cdot)](\theta_0)}{\max((\partial_\theta \phi(t_0, \theta_0))_+, m)} \right) + \frac{\pi}{2} \right) \le 0 \quad (3.8) $$

* A lower semi continuous (lsc) function $F \in \mathcal{F}_{t,2\pi}$ is said to be a viscosity supersolution of (E_m) if for any function $\phi \in C_{t,2\pi}^{1,1}$, $(t_0, \theta_0) \in (0, \infty) \times \mathbb{R}$ point of strict minimum of $F - \phi$ the following holds:

$$ \partial_t \phi(t_0, \theta_0) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[\phi(t_0, \cdot)](\theta_0)}{\max((\partial_\theta \phi(t_0, \theta_0))_+, m)} \right) + \frac{\pi}{2} \right) \ge 0 \quad (3.9) $$

* A viscosity solution F of (E_m) is a locally bounded function such that F is subsolution and F_* is a supersolution.

* We say that the initial condition of F is $F_0$ if $F^*(0, \cdot) \le F_0^*$ and $F_*(0, \cdot) \ge (F_0)_*$.

As usual in the theory of viscosity solutions for integro-differential equations, it is more convenient to work with the local reformulation. We define a local and non local operator associated to $A_0$ as in [2, 4, 6]. For $\delta > 0$ let $I_{1,\delta}$ and $I_{2,\delta}$ defined by:

$$ \begin{aligned} I_{1,\delta}[\phi](\theta) &= \int_{\mathbb{T} \cap |\theta'| \le \delta} \frac{2\phi(\theta) - \phi(\theta - \theta') - \phi(\theta + \theta')}{\sin^2(\theta'/2)} \frac{d\theta'}{8\pi} \\ I_{2,\delta}[\phi](\theta) &= \int_{\mathbb{T} \cap |\theta'| > \delta} \frac{2\phi(\theta) - \phi(\theta - \theta') - \phi(\theta + \theta')}{\sin^2(\theta'/2)} \frac{d\theta'}{8\pi}. \end{aligned} $$

This allows us to distinguish between the part of $A_0$ which acts as a differential operator ($I_{1,\delta}$) and the rest ($I_{2,\delta}$).

**Proposition 3.5.** Fix $m > 0$. Let F be a subsolution (resp. supersolution) of (E_m). Then for all $\phi \in C_{t,2\pi}^2$, $\delta > 0$ and $(t_0, \theta_0) \in (0, +\infty) \times \mathbb{R}$ such that $(F - \phi)(t_0, \theta_0) = 0$ and $(F - \phi)(t, \theta) \le 0$ (resp. $\ge 0$) for any $(t, \theta) \in B((t_0, \theta_0), \delta)$, the following holds:

$$ \partial_t \phi(t_0, \theta_0) + \frac{1}{\pi} \left( \arctan \left( \frac{I_{1,\delta}[\phi(t_0, \cdot)](\theta_0) + I_{2,\delta}[F(t_0, \cdot)](\theta_0)}{\max((\partial_\theta \phi(t, \theta))_+, m)} \right) + \frac{\pi}{2} \right) \le 0 \quad (\text{resp. } \ge 0). $$

### 3.4. Comparison principles.
We now state and prove a comparison principle for (E_m). The proof is similar with the proof of the comparison principle obtained in the similar approach to study the eigenvalues of random matrices [6, 8].

**Theorem 3.6.** Fix $m > 0$. Assume that $u \in \mathcal{F}_{t,2\pi}$ and $v \in \mathcal{F}_{t,2\pi}$ are respectively bounded viscosity subsolution and supersolution of (E_m). If $u(0, \cdot) \le v(0, \cdot)$, then for all time t, $u(t, \cdot) \le v(t, \cdot)$.

*Proof.* Let $\gamma > 0$, instead of considering $u$ we could consider $u_\gamma(t, \theta) = u(t, \theta) - \gamma t$ which is a $\gamma$ strict subsolution in the sense that if $\phi \in C_{t,2\pi}^{1,1}$, $\delta > 0$ and $(t_0, \theta_0) \in (0, +\infty) \times \mathbb{R}$ are such that $(u_\gamma - \phi)(t_0, \theta_0) = 0$ and $(u_\gamma - \phi)(t, \theta) \le 0$ then for any $(t, x) \in B((t_0, x_0), \delta)$, the following holds:

$$ \partial_t \phi(t, \theta) + \frac{1}{\pi} \left( \arctan \left( \frac{I_{1,\delta}[\phi(t_0, \cdot)](\theta_0) + I_{2,\delta}[u_\gamma(t_0, \cdot)](\theta_0)}{\max((\partial_\theta \phi(t, \theta))_+, m)} \right) + \frac{\pi}{2} \right) \le -\gamma < 0. $$
Hence if we prove that for all $t, \theta$, $u_\gamma(t, \theta) \le v(t, \theta)$ for all $\gamma > 0$, then by taking the limit $\gamma \to 0$ we shall recover $u \le v$. Hence, we can assume $u$ is a $\gamma$ strict subsolution in the proof and not just a subsolution.

We argue by contradiction and suppose that there exists $t_0 > 0$, $\theta_0 \in \mathbb{R}$ such that $u(t_0, \theta_0) > v(t_0, \theta_0)$. Let $T > t_0$ and use the classical technique of doubling variables. Thanks to the hypothesis, there exists $\alpha > 0$ such that for all $\varepsilon > 0$:

$$ \sup \left\{ u(t, \theta) - v(s, \theta') - \frac{1}{2\varepsilon}(\theta - \theta')^2 - \frac{1}{2\varepsilon}(t - s)^2, t, s \in [0, T], \theta, \theta' \in \mathbb{R}^2 \right\} > \alpha. $$

We shall justify that this supremum is actually a maximum by an usual localisation argument. Indeed, for a bounded $F \in \mathcal{F}_{2\pi}$ we have $a \in \mathbb{R}$ such that $F(t, \theta + 2\pi) = F(t, \theta) + a$ for all $t, \theta$ and so we can deduce that there exists $K > 0$ such that for all $t \ge 0$, $\theta \in \mathbb{R}$, $|F(t, \theta)| \le K(1 + |\theta|)$.

So for $u$ and $v$ in $\mathcal{F}_{2\pi}$ and bounded there exists $L > 0$ such that for all $(t, s) \in \mathbb{R}^+$, for all $(\theta, \theta') \in \mathbb{R}^2$, $u(t, \theta) - v(s, \theta') \le L(1 + |\theta| + |\theta'|)$ (this fact shall be called sublinearity). Now we can consider for $\beta > 0$, $\varepsilon > 0$ the following supremum:

$$ \sup \left\{ u(t, \theta) - v(s, \theta') - \frac{1}{2\varepsilon}(\theta - \theta')^2 - \frac{1}{2\varepsilon}(t - s)^2 - \beta(|\theta|^2 + |\theta'|^2), t, s \in [0, T], \theta, \theta' \in \mathbb{R}^2 \right\}. \quad (3.10) $$

For $\beta > 0$ small enough this supremum is greater than $\alpha/2 > 0$. Moreover thanks to the sublinearity and the fact that $u$ is usc and $v$ is lsc, this supremum is a maximum. Let $t^*, s^*, \theta^*, \theta'^* \in [0, T]^2 \times \mathbb{R}^2$ be a point of maximum of (3.10). We can classically assume that for $\varepsilon$ small enough, $t^*$ and $s^*$ are positive thanks to the fact that $\alpha > 0$. We want to use as tests functions:

$$ \phi_1(t, \theta) = v(s^*, \theta'^*) + \frac{1}{2\varepsilon}(\theta - \theta'^*)^2 + \frac{1}{2\varepsilon}(t - s^*)^2 + \beta(|\theta|^2 + |\theta'^*|^2), $$

for $u$ and:

$$ \phi_2(s, \theta') = u(t^*, \theta^*) - \frac{1}{2\varepsilon}(\theta^* - \theta')^2 - \frac{1}{2\varepsilon}(t^* - s)^2 - \beta(|\theta^*|^2 + |\theta'|^2), $$

for $v$.

The issue is that these functions are not in $C_{t,2\pi}^{1,1}$, we shall modify them a bit.

Take a small $\delta > 0$ that will be specified later on. We can find a function $\tilde{\phi}_1$ such that $\tilde{\phi}_1$ is equal to $\phi_1$ in $B((t^*, \theta^*), \delta)$ and which is in $C_{t,2\pi}^{1,1}$. Indeed for $\delta < \pi/2$, by localisation we can construct $\tilde{\phi}_1$ such that $\tilde{\phi}_1$ is equal to $\phi_1$ in $B((t^*, \theta^*), \delta)$ and impose that $\tilde{\phi}_1(t, \theta^* + \pi) = \tilde{\phi}_1(t, \theta^* - \pi) + 1$ for all $t$ to define $\tilde{\phi}_1(t, \cdot)$ on $[\theta^* - \pi, \theta^* + \pi]$ and then extend periodically $\tilde{\phi}_1(t, \cdot)$ for all $t > 0$. We can do the same for $\phi_2$. We still call $\phi_1$ and $\phi_2$ these modifications of $\phi_1$ and $\phi_2$ that are now in $C_{t,2\pi}^{1,1}$.

Moreover to lighten the computations that shall follow, we can forget the term in $\beta$. Indeed, if we look at $\gamma_\beta(\theta) = \beta(|\theta|^2 + |\theta'^*|^2)$ we have that $\gamma_\beta''$ is uniformly bounded in $\theta$ by $\beta$. Hence, we have that $I_{1,\delta}[\gamma_\beta](\theta'^*)$ converges uniformly in $\delta$ to 0 when $\beta$ converges to 0. So if we let $\beta$ converges to 0 in the subviscosity formulation this part disappears and same for the super viscosity formulation [4]. We still write $\phi_1$ and $\phi_2$ these functions without the term $\beta(|\theta|^2 + |\theta'|^2)$.

By using $\phi_1$ and $\phi_2$ as test functions in the definition of subsolution and supersolution we have that:

$$ \frac{1}{\varepsilon}(t^* - s^*) + \frac{1}{\pi} \left( \arctan \left( \frac{I_{1,\delta}[\phi_1(t^*, \cdot)](\theta^*) + I_{2,\delta}[u(t^*, \cdot)](\theta^*)}{\max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*)_+, m \right)} \right) + \frac{\pi}{2} \right) \le -\gamma $$
and:

$$ \frac{1}{\varepsilon}(t^* - s^*) + \frac{1}{\pi} \left( \arctan \left( \frac{I_{1,\delta}[\phi_2(s^*, .)](\theta'^*) + I_{2,\delta}[v(s^*, .)](\theta'^*)}{\max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*), m \right)} \right) + \frac{\pi}{2} \right) \geq 0. $$

We subtract the first inequality to the second and we have:

$$ \arctan \left( \frac{I_{1,\delta}[\phi_2(s^*, .)](\theta'^*) + I_{2,\delta}[v(s^*, .)](\theta'^*)}{\max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*), m \right)} \right) - \arctan \left( \frac{I_{1,\delta}[\phi_1(t^*, .)](\theta^*) + I_{2,\delta}[u(t^*, .)](\theta^*)}{\max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*), m \right)} \right) \geq \pi\gamma. \quad (3.11) $$

Firstly, we look at the $I_{2,\delta}$ part:

$$ I_{2,\delta}[v(s^*, .)](\theta'^*) - I_{2,\delta}[u(t^*, .)](\theta^*) = \int_{T \cap |z| > \delta} \frac{v(s^*, \theta'^*) - v(s^*, \theta'^* + z) - v(s^*, \theta'^* - z) - (2u(t^*, \theta^*) - u(t^*, \theta^* + z) - u(t^*, \theta^* - z))}{\sin^2(z/2)} \frac{dz}{4}. $$

Using that $(t^*, s^*, \theta^*, \theta'^*)$ is a point of maximum we have that for all $z \in \mathbb{R}$:

$$ \begin{aligned} u(t^*, \theta^*) - v(s^*, \theta'^*) &\geq u(t^*, \theta^* + z) - v(s^*, \theta'^* + z) \\ u(t^*, \theta^*) - v(s^*, \theta'^*) &\geq u(t^*, \theta^* - z) - v(s^*, \theta'^* - z). \end{aligned} $$

Adding these two inequalities we see that the previous numerator is actually non negative. Hence we have:

$$ I_{2,\delta}[v(s^*, .)](\theta'^*) - I_{2,\delta}[u(t^*, .)](\theta^*) \leq 0. \quad (3.12) $$

Then for the terms in $I_{1,\delta}$ we notice that since $\phi_1(t^*,.)$ (resp. $\phi_2(s^*,.)$) is bounded in $C^2$ by $\varepsilon^{-1}$ in $B(\theta'^*, \delta)$ (resp. $B(\theta^*, \delta)$), we have:

$$ I_{1,\delta}[\phi_2(s^*, .)](\theta'^*) - I_{1,\delta}[\phi_1(t^*, .)](\theta^*) \leq \frac{2\delta}{\varepsilon\pi}. \quad (3.13) $$

Using (3.11), (3.12), (3.13), and the fact that arctan is one Lipschitz and non-decreasing:

$$ \begin{aligned} \pi\gamma &\leq \arctan \left( \frac{I_{1,\delta}[\phi_2(s^*, .)](\theta'^*) + I_{2,\delta}[v(s^*, .)](\theta'^*)}{\max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*), m \right)} \right) - \arctan \left( \frac{I_{1,\delta}[\phi_1(t^*, .)](\theta^*) + I_{2,\delta}[u(t^*, .)](\theta^*)}{\max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*), m \right)} \right) \\ &\leq \frac{I_{1,\delta}[\phi_2(s^*, .)](\theta'^*) + I_{2,\delta}[v(s^*, .)](\theta'^*)}{\max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*), m \right)} - \frac{I_{1,\delta}[\phi_1(t^*, .)](\theta^*) + I_{2,\delta}[u(t^*, .)](\theta^*)}{\max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*), m \right)} \\ &\leq \frac{2\delta}{\varepsilon\pi \max \left( \frac{1}{\varepsilon}(\theta^* - \theta'^*), m \right)} \leq \frac{2\delta}{\varepsilon m \pi} \end{aligned} \quad (3.14) $$

Let $\delta = \varepsilon^2$ and let $\varepsilon$ goes to 0 to obtain a contradiction since $\gamma > 0$. $\square$

**Remark 3.7.** This comparison principle was obtained using the parabolicity of the PDE which means at the level of particles that they repeal each others. So, we shall obtain in Section 4.2 a comparison principle at the level of particles.
### 3.5. Consequences of comparison principles for truncated PDEs.
Using the comparison principle, we shall prove results of existence and uniqueness of viscosity solutions of $(E_m)$. We also prove a counterpart of the minimum principle for the viscosity solutions of $(E_m)$.

#### 3.5.1. Uniqueness of viscosity solutions of the truncated PDE.
We can state a uniqueness result for viscosity solutions of $(E_m)$.

**Proposition 3.8.** Given $m > 0$ and $F_0 \in \mathcal{F}_{2\pi}$ a bounded non-decreasing right continuous function, there exists at most one viscosity solution of $(E_m)$ with initial condition $F_0$.

*Proof.* Assume that there exists $F$ and $G$ which are viscosity solutions of $(E_m)$ with the same initial data. Let $\varepsilon > 0$. Since $F$ and $G$ have the same initial condition $F_0$ we get that for all $x \in \mathbb{R}$:

$$F(0, x) = F^*(0, x) \leq (F_0)^*(x) \leq (F_0)_*(x + \varepsilon) \leq G_*(0, x + \varepsilon).$$

Using Theorem 3.6 with $u = F$ as a viscosity subsolution and $v_\varepsilon(t, x) = G_*(t, x + \varepsilon) + \varepsilon$ as a supersolution to $(E_m)$, we get that for all $t \geq 0, \forall x \in \mathbb{R}$:

$$F(t, x) \leq G_*(t, x + \varepsilon).$$

Letting $\varepsilon$ goes to 0 yields that for all $t \geq 0, \forall x \in \mathbb{R}$,

$$F(t, x) \leq G(t, x).$$

By symmetry of $F$ and $G$ we get that $F = G$.

□

#### 3.5.2. Monotone principle.
First let us define what replace the condition $u \geq m$ for a $m > 0$ at the level of a primitive of $u \in \mathcal{P}(\mathbb{T})$.

**Definition 3.9.** Fix $m > 0$. A function $F: \mathbb{R} \to \mathbb{R}$ (resp. $F: \mathbb{R}^+ \times \mathbb{R} \to \mathbb{R}$) is said to satisfied the hypothesis $(H_m)$ if:

$$\begin{gathered} \forall h \geq 0, \forall \theta \in \mathbb{R}, F(\theta + h) - F(\theta) \geq mh \\ (\text{resp. } \forall t \geq 0, \forall h \geq 0, \forall \theta \in \mathbb{R}, F(t, \theta + h) - F(t, \theta) \geq mh). \end{gathered} \qquad (H_m)$$

As explained, for a smooth solution $\mu$ of (3.1) the minimum principle states that $t \mapsto \min_{\mathbb{T}} \mu(t, \cdot)$ is non-decreasing. We state a counterpart of this result at the level of viscosity solutions of $(E_m)$.

**Proposition 3.10.** Fix $m > 0$. Let $F$ be a bounded viscosity solution of $(E_m)$ with initial condition $F_0$. Assume that $F_0 \in \mathcal{F}_{2\pi}$ satisfies $(H_m)$ and is a bounded non-decreasing right continuous function. Then $F$ satisfies $(H_m)$.

*Proof.* Fix $h \geq 0$. By hypothesis on $F(0, \cdot)$, we get that for all $\varepsilon > 0$, for all $x \in \mathbb{R}$:

$$F(0, x) + mh \leq F_0(x) + mh \leq F_0(x + h) \leq (F_0)_*(x + h + \varepsilon) \leq F_*(0, x + h + \varepsilon).$$

Using Proposition 3.5 with $u = F + mh$ as a subsolution and $v(t, x) = F_*(t, x + h + \varepsilon)$ as a supersolution we get that for all $\varepsilon > 0, \forall t \geq 0, \forall x \in \mathbb{R}$:

$$F(t, x) + mh \leq F_*(t, x + h + \varepsilon).$$

Since $F$ is usc, letting $\varepsilon$ goes to 0 gives the result.

□

**Remark 3.11.** Suppose that $F$ satisfies the hypothesis $(H_m)$ for $m > 0$. Then we can replace in the definition of viscosity sub and supersolution $\max((\partial_\theta\phi(t_0, \theta_0))_+, m)$ by $\partial_\theta\phi(t_0, \theta_0)$. Indeed, for any $t_0, h \geq 0$, if $\theta_0$ is a point of maximum of $F(t_0, \cdot) - \phi(t_0, \cdot)$ then:

$$mh + F(t_0, \theta_0) - \phi(t_0, \theta_0 + h) \leq F(t_0, \theta_0 + h) - \phi(t_0, \theta_0 + h) \leq F(t_0, \theta_0) - \phi(t_0, \theta_0)$$

where we first used that $F(t_0, \cdot)$ satisfies $(H_m)$ and then the fact that $\theta_0$ is a point of maximum of $F(t_0, \cdot) - \phi(t_0, \cdot)$. Hence, we deduce that $\partial_\theta\phi(t_0, \theta_0) \geq m$.
This explains why under the hypothesis $(H_m)$ to find viscosity solutions of (3.5) in the sense of Definition (3.2) we can work with solutions of $(E_m)$.

**Remark 3.12.** Assume that $F$ satisfies $(H_m)$ and is a viscosity solution of $(E_m)$ for $m > 0$. Then for all $0 < m' \le m$, $F$ satisfies $(H_{m'})$ and is a viscosity solution of $(E_{m'})$ using Remark 3.11.

**3.5.3. Existence of a viscosity solution.** In this section we prove that there exists a viscosity solution of $(E_m)$ starting from an initial data that satisfies $(H_m)$ for a given $m > 0$. We first prove the result assuming that the initial condition is continuous. In this case we prove that the viscosity solution is continuous for every time. This proof is quite standard in viscosity theory. We then prove the result under the only assumption that the initial condition is the cumulative distribution function of a measure on $\mathbb{T}$. This point is more technical since we have to pay attention to the fact that the initial condition is just usc and not continuous as in the previous case.

We first state a general lemma about viscosity solutions that shall be used in the proof of existence of a viscosity solution of $(E_m)$. We sketch the proof but a detailed proof can be found in [4].

**Lemma 3.13 (Stability result).** Fix $m > 0$. Let $(F_n)_{n \in \mathbb{N}}$ be a bounded sequence of viscosity subsolutions of $(E_m)$ (resp. viscosity supersolutions). The function $F^*(t_0, \theta_0) := \limsup_{n \to +\infty} F_n(t, \theta)$ (resp. $F_*(t_0, \theta_0) := \liminf_{n \to +\infty} F_n(t, \theta)$) is a viscosity subsolution (resp. a visocosity supersolution) of $(E_m)$.

*Proof.* Let $\phi \in C_{t,2\pi}^{1,1}$ and $(t_0, \theta_0)$ a point of strict maximum of $F^* - \phi$. For the moment let us consider a sequence $(t_n, \theta_n)$ such that $F_n - \phi$ has a point of maximum at $(t_n, \theta_n)$ and such that $(t_n, \theta_n) \underset{n \to +\infty}{\to} (t_0, \theta_0)$. As for $n \ge 0$, $F_n$ is a viscosity subsolution we get

$$ \partial_t \phi(t_n, \theta_n) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[\phi(t_n, \cdot)](\theta_n)}{\max((\partial_\theta \phi(t_n, \theta_n))_+, m)} \right) + \frac{\pi}{2} \right) \le 0. \quad (3.15) $$

Since we have

$$ A_0[\phi(t_n, \cdot)](\theta_n) = \frac{1}{4\pi} \int_{-\pi}^{\pi} \frac{\phi(t_n, \theta_n) - \phi(t_n, \theta_n - \theta) - \theta \partial_\theta \phi(t_n, \theta_n)}{\sin^2(\frac{\theta}{2})} d\theta $$

and $\phi$ is in $C_{t,2\pi}^{1,1}$, the dominated convergence theorem yields

$$ A_0[\phi(t_n, \cdot)](\theta_n) \underset{n \to +\infty}{\longrightarrow} A_0[\phi(t_0, \cdot)](\theta_0). $$

Passing to the limit in the inequality (3.15) gives that $F^*$ is a viscosity subsolution. To make this proof complete, we should use the local formulation of viscosity solutions of $(E_m)$ because the points of maximum $(t_n, \theta_n)$ as they are constructed can only be considered in a ball around $(t_0, \theta_0)$ and not globally. The same argument holds for $F_*$. $\square$

**Proposition 3.14.** Fix $m > 0$ and $\mu_0 \in \mathcal{P}(\mathbb{T})$ and write $F_0 := F_{\mu_0}$ the cumulative distribution function of $\mu_0$. Assume that $F_0$ satisfies $(H_m)$.

* If $F_0$ is continuous then there exists a continuous viscosity solution of $(E_m)$ with initial condition $F_0$

* Even if $F_0$ is not continuous there exists a viscosity solution of $(E_m)$ with initial condition $F_0$.

*Proof.* • Assume that $F_0$ is continuous. We start by regularizing the initial data. Let $(\rho_\varepsilon)_{\varepsilon>0}$ be an approximation of the unity on $\mathbb{T}$ and define $\mu_0^\varepsilon := \mu_0 * \rho_\varepsilon$. Fix $\varepsilon > 0$. Since $\mu_0^\varepsilon$ is smooth and satisfies $(H_m)$ we can consider a smooth strong solution $(\mu_t^\varepsilon)_{t \ge 0}$ of (3.1) with initial condition $\mu_0^\varepsilon$ using Theorem 1.1 of [27]. We also have that the cumulative distribution function of $\mu_t^\varepsilon$; $F^\varepsilon(t, \cdot) := F_{\mu^\varepsilon(t, \cdot)}$; is a strong smooth solution of (3.5) and
satisfies ($H_m$) by the monotone principle ($t \mapsto \min_{x \in \mathbb{T}} \mu^\varepsilon(t, x)$ is non-decreasing). In particular, for all $\varepsilon > 0$, $F^\varepsilon$ is a viscosity solution of ($E_m$) that satisfies ($H_m$).

We define $F_1(t_0, x_0) := \limsup_{\varepsilon \to 0, t \to t_0, x \to x_0} F^\varepsilon(t, x)$ and $F_2(t_0, x_0) := \liminf_{\varepsilon \to 0, t \to t_0, x \to x_0} F^\varepsilon(t, x)$ which are finite since $(F^\varepsilon)_{\varepsilon>0}$ is uniformly bounded in $\varepsilon > 0$ and time on compact space of the space variable since $F^\varepsilon(t, \cdot)$ is the cumulative distribution of a probability measure on $\mathbb{T}$.
By Lemma 3.13, $F_1$ is a viscosity subsolution of ($E_m$) and $F_2$ is a viscosity supersolution of ($E_m$). Moreover, they also both satisfy ($H_m$).

Now let us look at the initial condition. Since for all $\varepsilon > 0$, $F^\varepsilon$ is a strong solution of (3.5), we have that for all $t \ge 0$, $x \in \mathbb{R}$,

$$ \partial_t F^\varepsilon(t, x) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[F^\varepsilon(t, \cdot)](x)}{\partial_x F^\varepsilon(t, x)} \right) + \frac{\pi}{2} \right) = 0. $$

From this we deduce that for all $\varepsilon > 0$ for all $t \ge 0$, $x \in \mathbb{R}$

$$ |\partial_t F^\varepsilon(t, x)| \le 1. $$

The mean value theorem gives that for all $\varepsilon > 0$, for all $t \ge 0$, $x \in \mathbb{R}$

$$ |F^\varepsilon(t, x) - F_{\mu_0^\varepsilon}(x)| = |F^\varepsilon(t, x) - F^\varepsilon(0, x)| \le t. $$

Since $F_0$ is continuous, passing to the superior and inferior limit in the previous inequality gives that $x \in \mathbb{R} \ F_1(0, x) = (F_0)^*(x) = F_0(x)$ and $F_2(0, x) = (F_0)_*(x) = F_0(x)$. As $F_1$ is a viscosity subsolution of ($E_m$) and $F_2$ is a viscosity supersolution of ($E_m$), by the comparison principle of Theorem 3.6, we obtain that for every $t \ge 0$ $F_1(t, \cdot) \le F_2(t, \cdot)$ and thus $F := F_1 = F_2$ is a continuous viscosity solution that satisfies $F(0, \cdot) = F_0$.

• We first regularize $F_0$ by considering for all $\varepsilon > 0$ $F_0^\varepsilon := F_0 * \rho_\varepsilon$ with $\rho_\varepsilon$ an approximation of the unity such that $\rho_\varepsilon$ is supported in $[-\varepsilon, \varepsilon]$. We notice that since $F_0$ satisfies ($H_m$) then so is $F_0^\varepsilon$ for all $\varepsilon > 0$. Using Theorem 1.1 of [27] we can consider $(F^\varepsilon(t, x))_{t \ge 0, x \in \mathbb{R}}$ a strong smooth solution of (3.5) starting from $F^\varepsilon(0, \cdot)$. The solution $F^\varepsilon$ satisfies ($H_m$) by the monotone principle ($t \mapsto \min_{x \in \mathbb{T}} \partial_x F^\varepsilon(t, x)$ is non-decreasing). In particular, for all $\varepsilon > 0$, $F^\varepsilon$ is a viscosity solution of ($E_m$) that satisfies ($H_m$).

We define $F_1(t_0, x_0) := \limsup_{\varepsilon \to 0, t \to t_0, x \to x_0} F^\varepsilon(t, x)$ and $F_2(t_0, x_0) := \liminf_{\varepsilon \to 0, t \to t_0, x \to x_0} F^\varepsilon(t, x)$.

We shall prove that $F_1$ is a viscosity solution of ($E_m$) with initial condition $F_0$. By definition of being a viscosity solution of ($E_m$) we have to prove that $F_1$ is a viscosity subsolution of ($E_m$), $(F_1)_*$ is a viscosity supersolution of ($E_m$) and $F_1(0, \cdot) \le F_0$ and $(F_1)_*(0, \cdot) \ge (F_0)_*$.

By Lemma 3.13, $F_1$ is a viscosity subsolution of ($E_m$) and $F_2$ is a viscosity supersolution of ($E_m$). Moreover, they also both satisfy ($H_m$).

Now let us look at the initial condition. Since for all $\varepsilon > 0$, $F^\varepsilon$ is a strong solution of (3.5), we have that for all $t \ge 0$, $x \in \mathbb{R}$,

$$ \partial_t F^\varepsilon(t, x) + \frac{1}{\pi} \left( \arctan \left( \frac{A_0[F^\varepsilon(t, \cdot)](x)}{\partial_x F^\varepsilon(t, x)} \right) + \frac{\pi}{2} \right) = 0. $$

From this we deduce that for all $\varepsilon > 0$ for all $t \ge 0$, $x \in \mathbb{R}$

$$ |\partial_t F^\varepsilon(t, x)| \le 1. $$

The mean value theorem gives that for all $\varepsilon > 0$ for all $t \ge 0$, $x \in \mathbb{R}$

$$ -t + F^\varepsilon(0, x) \le F^\varepsilon(t, x) \le t + F^\varepsilon(0, x). \quad (3.16) $$

We start by noticing that for $x \in \mathbb{R}$

$$ F_1(0, x) = \limsup_{t \to 0, x_0 \to x, \varepsilon \to 0} F^\varepsilon(t, x_0). $$
So using (3.16) yields that for $x \in \mathbb{R}$

$$F_1(0, x) \le \limsup_{t \to 0, x_0 \to x, \varepsilon \to 0} [t + F^\varepsilon(0, x_0)] = \limsup_{x_0 \to x, \varepsilon \to 0} F_0^\varepsilon(x_0).$$

Since $\rho_\varepsilon$ is supported on $[-\varepsilon, \varepsilon]$ we have $F_0^\varepsilon(x) \le \sup_{y \in [x-\varepsilon, x+\varepsilon]} F_0(y)$ and so

$$F_1(0, \cdot) \le (F_0)^*(\cdot).$$

Doing the exact same argument with the liminf, we obtain $(F_0)_* \le F_2(0, \cdot)$.

To conclude about the initial condition it suffices to show that $(F_1)_*(0, \cdot) = F_2(0, \cdot)$. Since $F_1 \ge F_2$ we directly have that $(F_1)_*(0, \cdot) \ge F_2(0, \cdot)$. We prove the reverse. Indeed using (3.16) we get

$$F_1(t, x) \le t + (F_0)^*(x).$$

This implies

$$(F_1)_*(0, x) \le ((F_0)^*)_*(x) = (F_0)_*(x)$$

since $F_0$ is non-decreasing. So $(F_1)_*(0, x) \le (F_0)_*(x) \le F_2(0, x)$.

In summary at time 0 we proved:

$$\left\{ \begin{array}{l} F_1(0, \cdot) \le F_0 \\ (F_1)_*(0, \cdot) \ge (F_0)_* \\ (F_1)_*(0, \cdot) = F_2(0, \cdot) \end{array} \right. \qquad (3.17)$$

It remains to prove that $(F_1)_*$ is a viscosity supersolution of $(E_m)$. To show this we shall prove that $(F_1)_* = F_2$. By definition, since $F_1 \ge F_2$ we have $(F_1)_* \ge (F_2)_* = F_2$.

Using the initial condition (3.17) for all $\gamma > 0$, we have

$$F_1(0, \cdot) \le F_0(\cdot) \le (F_0)_*(\cdot + \gamma) \le (F_2)_*(0, \cdot + \gamma).$$

As $F_1$ is a viscosity subsolution of $(E_m)$ and $F_2(\cdot, \cdot + \gamma)$ is a viscosity supersolution of $(E_m)$ by the comparison principle of Theorem 3.6, we obtain that for every $\gamma > 0$, for every $t \ge 0$ $F_1(t, \cdot) \le F_2(t, \cdot + \gamma)$. Let $\gamma$ goes to 0 to obtain that $(F_1)_* \le (F_2)_* = F_2$.

□

### 3.5.4. Continuity in the initial condition.

**Proposition 3.15.** Fix $m > 0$. Let $(\mu_n)_{n \in \mathbb{N}} \in \mathcal{P}(\mathbb{T})^N$ be a family of probabilities measures on $\mathbb{T}$ such that for all $n \ge 0$ $F_{\mu_n}$, the cumulative distribution function of $\mu_n$, satisfies $(H_m)$.

Assume that $(\mu_n)_{n \in \mathbb{N}} \in \mathcal{P}(\mathbb{T})^N$ converges in law towards $\mu \in \mathcal{P}(\mathbb{T})$ and that $\mu$ is absolutely continuous with respect to Lebesgue measure. Then $F_\mu$ satisfies $(H_m)$.

Moreover, we now denote $(F_n(t, x))_{t \ge 0, x \in \mathbb{R}}$ (resp. $(F_\infty(t, x))_{t \ge 0, x \in \mathbb{R}}$) the unique viscosity solution of $(E_m)$ with initial condition $F_{\mu_n}$ (resp. $F_\mu$). Then we have the following convergence

$$\sup_{t \ge 0} \|F_n(t, \cdot) - F_\infty(t, \cdot)\|_{L^\infty(\mathbb{R})} \xrightarrow{n \to +\infty} 0.$$

*Proof.* Since $\mu$ is absolutely continuous with respect to Lebesgue measure, for all $x \in \mathbb{R}$

$$F_{\mu_n}(x) \xrightarrow{n \to +\infty} F_\mu(x).$$

It directly implies that $F_\mu$ satisfies $(H_m)$. Moreover, by the Dini convergence theorem the convergence of $F_{\mu_n}$ towards $F_\mu$ is also uniform on $[0, 2\pi]$ and so on $\mathbb{R}$ (since for the cumulative distribution of a measure $\nu \in \mathcal{P}(\mathbb{T})$ we have $F_\mu(\cdot + 2\pi) = F_\mu(\cdot) + 1$).

We recall by Proposition 3.14 that $F_\infty$ is continuous and $F_\infty(0, \cdot) = F_\mu(\cdot)$.

Fix $\varepsilon > 0$ and consider an integer $N$ such that for all $n \ge N$ one has

$$\|F_{\mu_n}(\cdot) - F_\mu(\cdot)\|_{L^\infty(\mathbb{R})} \le \varepsilon.$$
To complete the proof we shall prove for $n \ge N$ we have

$$ \sup_{t \ge 0} \|F_n(t, \cdot) - F_\infty(t, \cdot)\|_{L^\infty(\mathbb{R})} \le \varepsilon. $$

For $n \ge N$, we have

$$ F_n(0, \cdot) \le F_{\mu_n}(\cdot) \le F_{\mu}(\cdot) + \varepsilon = F_{\infty}(0, \cdot) + \varepsilon. $$

By the comparison principle we obtain that for every $n \ge N$ and $t \ge 0$

$$ F_n(t, \cdot) \le F_\infty(t, \cdot) + \varepsilon. $$

For the other inequality, for $n \ge N$, we have

$$ -\varepsilon + F_{\mu}(\cdot) \le F_{\mu_n}(\cdot) $$

Passing to the infimum and since $F_\mu$ is continuous one has

$$ -\varepsilon + F_\infty(0, \cdot) = -\varepsilon + F_\mu(\cdot) \le (F_{\mu_n})_*(\cdot) \le (F_n)_*(0, \cdot). $$

Again by the comparison principle we get that for all $n \ge N$, for all $t \ge 0$

$$ -\varepsilon + F_\infty(t, \cdot) \le (F_n)_*(t, \cdot) \le F_n(t, \cdot), $$

which proves the result.

□

**3.6. Existence, uniqueness, continuity in the initial condition of viscosity solutions of the original equation.** Using Remark 3.11 and Remark 3.12, we have that if $F$ satisfies $(H_m)$ for a $m > 0$ then $F$ is a viscosity solution of (3.5) (in the sense of Definition 3.2) if and only if $F$ is a viscosity solution of $(E_m)$. As a corollary of Proposition 3.14 and of the minimum principle we get the following result for the existence of a viscosity solution of (3.5).

**Theorem 3.16.** Let $\mu \in \mathcal{P}(\mathbb{T})$ such that $F_\mu$ satisfies $(H_m)$ for a $m > 0$. Then there exists a viscosity solution of (3.5) with initial condition $F_\mu$.

As a corollary of Proposition 3.8 we get the following result for the uniqueness of a viscosity solution of $(E_m)$.

**Theorem 3.17.** Let $\mu \in \mathcal{P}(\mathbb{T})$ such that $F_\mu$ satisfies $(H_m)$ for a $m > 0$. Then there exists at most one viscosity solution in $(H_{m'})$ for $0 < m' \le m$ of (3.5) with initial condition $F_\mu$.

**Remark 3.18.** In the previous theorem we need the hypothesis of strict monotony of solutions to have the uniqueness of viscosity solutions of (3.5). Indeed, we did not directly obtain a comparison principle for (3.5) because of the singularity in the PDE.

Finally, we can reformulate Proposition (3.15) for viscosity solutions of $(E_m)$.

**Theorem 3.19.** Fix $m > 0$. Let $(\mu_n)_{n \in \mathbb{N}} \in \mathcal{P}(\mathbb{T})^\mathbb{N}$ be a family of probabilities measures on $\mathbb{T}$ such that for all $n \ge 0$ $F_{\mu_n}$, the cumulative distribution function of $\mu_n$, satisfies $(H_m)$. Assume that $(\mu_n)_{n \in \mathbb{N}} \in \mathcal{P}(\mathbb{T})^\mathbb{N}$ converges in law towards $\mu \in \mathcal{P}(\mathbb{T})$ and that $\mu$ is absolutely continuous with respect to Lebesgue measure. Let $(F_n(t, x))_{t \ge 0, x \in \mathbb{R}}$ (resp. $(F_\infty(t, x))_{t \ge 0, x \in \mathbb{R}}$) be a viscosity solution of (3.5) with initial condition $F_{\mu_n}$ (resp. $F_\mu$). Then we have the following convergence

$$ \sup_{t \ge 0} \|F_n(t, \cdot) - F_\infty(t, \cdot)\|_{L^\infty(\mathbb{R})} \xrightarrow{n \to +\infty} 0. $$
## 4. SYSTEM OF PARTICLES

**4.1. Reminders and notation for trigonometric polynomials.** We consider the space of real trigonometric polynomials with $2n$ distinct roots in $(-\pi, \pi]$

$$ \mathbb{R}_{2n,per} = \left\{ p \mid \exists (a_i, b_i)_{i=1}^n \in \mathbb{R}^{2n}, p(x) = \sum_{j=1}^{n} a_j \cos(jx) + b_j \sin(jx), p \text{ has } 2n \text{ distinct roots in } (-\pi, \pi] \right\}. $$

We notice that if $p \in \mathbb{R}_{2n,per}$ then all the derivatives of $p$ are also in $\mathbb{R}_{2n,per}$ by the Rolle theorem. For $p \in \mathbb{R}_{2n,per}$, let $(x_j)_{j=1}^{2n}$ be its $2n$ distinct roots. We recall that for $p \in \mathbb{R}_{2n,per}$, we can find a constant $c \in \mathbb{R}$ such that

$$ p(x) = c \prod_{j=1}^{2n} \sin \left( \frac{x - x_j}{2} \right). $$

This gives the following identity for $p \in \mathbb{R}_{2n,per}$,

$$ \frac{p'(x)}{p(x)} = \frac{1}{2} \sum_{i=1}^{2n} \cotan \left( \frac{x - x_i}{2} \right). \qquad (4.1) $$

Let us notice that this identity is the counterpart of the decomposition

$$ \frac{P'(X)}{P(X)} = \sum_{i=1}^{n} \frac{1}{X - x_i}, $$

for $P \in \mathbb{C}_n[X]$ such that $P(X) = \prod_{i=1}^n (X - x_i)$.

**4.2. Comparison principles on the particles.** We first state a comparison principle for the roots of real polynomials before stating it for trigonometric polynomials.

**Proposition 4.1 (Discrete comparison principle).** Let $x_1 < x_2 < \dots < x_N$, $y_1 < y_2 < \dots < y_N$ be two families of $N$ real numbers and consider $P_N(X) = \prod_{i=1}^N (X - x_i)$ and $Q_N(X) = \prod_{i=1}^N (X - y_i)$. Let $x_1 < x'_1 < x_2 < x'_2 < \dots < x'_{N-1}$ the $N-1$ real roots of $P'$ and $y_1 < y'_1 < y_2 < y'_2 < \dots < y'_{N-1}$ the $N-1$ real roots of $Q'$. Assume that for all $1 \le i \le N$, $x_i \le y_i$, then for all $1 \le i \le N-1$, $x'_i \le y'_i$

*Proof.* Let $1 \le k \le N-1$. As for the Gauss-Lucas theorem we start from the expression

$$ \frac{P'(X)}{P(X)} = \sum_{i=1}^{N} \frac{1}{X - x_i} $$

in $x'_k$ and the same thing for $Q$. We get the following identities:

$$ \left\{ \begin{array}{l} 0 = \sum_{i=1}^{N} \frac{1}{x'_{k} - x_{i}} \\ 0 = \sum_{i=1}^{N} \frac{1}{y'_{k} - y_{i}} \end{array} \right. \qquad (4.2) $$

Subtracting the two equations of (4.2):

$$ 0 = \sum_{i=1}^{N} \frac{1}{x'_{k} - x_{i}} - \frac{1}{y'_{k} - y_{i}} = \sum_{i=1}^{N} \frac{y'_{k} - x'_{k} + x_{i} - y_{i}}{(x'_{k} - x_{i})(y'_{k} - y_{i})}. \qquad (4.3) $$
We obtain the following identity:

$$y'_{k} - x'_{k} = \frac{\sum_{i=1}^{N} \frac{y_i - x_i}{(x'_{k} - x_i)(y'_{k} - y_i)}}{\sum_{i=1}^{N} \frac{1}{(x'_{k} - x_i)(y'_{k} - y_i)}}$$

Let us remark that for all $1 \le k \le N-1$ and for all $1 \le i \le N$, $(x'_{k} - x_i)(y'_{k} - y_i) \ge 0$ because the two terms of the product have the same sign. Thanks to the hypothesis and the equation (4.3) we get that $y'_{k} \ge x'_{k}$. $\square$

Even if there is no order on the circle we can state a comparison principle for the roots of trigonometric polynomials.

Given $N$ points $(x_k)_{0 \le k \le N-1}$ in an interval of length $2\pi$ we periodize these points by defining $(x_k)_{k \in \mathbb{Z}}$ as $x_k = x_{k[N]} + 2 \left\lfloor \frac{k}{N} \right\rfloor \pi$ where $k[N]$ is the rest of the euclidean division of $k$ by $N$.

**Proposition 4.2** (Periodic discrete comparison principle). Let $a \in \mathbb{R}$ and $a \le x_0 < x_1 < \dots < x_{2N-1} < a + 2\pi$, $a \le y_0 < y_1 < \dots < y_{2N-1} < a + 2\pi$ be two families of $2N$ real numbers and consider the trigonometric polynomials $P_N(X) = \prod_{i=0}^{2N-1} \sin\left(\frac{x-x_i}{2}\right)$ and $Q_N(X) = \prod_{i=0}^{2N-1} \sin\left(\frac{x-y_i}{2}\right)$. We consider the periodic families $(x_i)_{i \in \mathbb{Z}}$ and $(y_i)_{i \in \mathbb{Z}}$ associated to the $(x_i)_{0 \le i \le 2N-1}$ and $(y_i)_{0 \le i \le 2N-1}$. Let $(x'_i)_{i \in \mathbb{Z}}$ and $(y'_i)_{i \in \mathbb{Z}}$ be such that $x'_i$ is the only root of the trigonometric polynomial $P'_N$ in $]x_i, x_{i+1}[$ and $y'_i$ is the only root of the trigonometric polynomial $Q'_N$ in $]y_i, y_{i+1}[$. Assume that for all $i \in \mathbb{Z}$, $x_i \le y_i$, then for all $i \in \mathbb{Z}$, $x'_i \le y'_i$.

*Proof.* Let $k \in \mathbb{Z}$. We follow the proof of the real case. We evaluate

$$\frac{P'(X)}{P(X)} = \frac{1}{2} \sum_{i=0}^{2N-1} \cotan \left( \frac{X - x_i}{2} \right)$$

in $x'_k$ and the same thing for $Q$. We get the following identities:

$$\left\{ \begin{aligned} 0 &= \sum_{i=0}^{2N-1} \cotan \left( \frac{x'_{k} - x_{i}}{2} \right) \\ 0 &= \sum_{i=0}^{2N-1} \cotan \left( \frac{y'_{k} - y_{i}}{2} \right) \end{aligned} \right. \qquad (4.4)$$

Since the $(x_i)_{i \in \mathbb{Z}}$ and $(y_i)_{i \in \mathbb{Z}}$ are periodic we can rewrite the previous equations as:

$$\left\{ \begin{aligned} 0 &= \sum_{i=1}^{2N} \cotan \left( \frac{x'_{k} - x_{k+i}}{2} \right) \\ 0 &= \sum_{i=1}^{2N} \cotan \left( \frac{y'_{k} - y_{k+i}}{2} \right) \end{aligned} \right. \qquad (4.5)$$

Subtracting the two equations of (4.5):

$$0 = \sum_{i=1}^{2N} \left[ \cotan \left( \frac{x'_{k} - x_{k+i}}{2} \right) - \cotan \left( \frac{y'_{k} - y_{k+i}}{2} \right) \right] \qquad (4.6)$$
For all $i \in \mathbb{Z}$, let $a_i = (x'_k - x_{k+i})/2$ and $b_i = (y'_k - y_{k+i})/2$. By definition, for all $i \in \{1, \dots, 2N\}$, $a_i, b_i \in (-\pi, 0)$. Since cotan is decreasing on $(-\pi, 0)$ and

$$0 = \sum_{i=1}^{2N} [\cotan(a_i) - \cotan(b_i)],$$

there exists $i$ such that $b_i \ge a_i$. For this $i$ we have $y'_k - y_{k+i} \ge x'_k - x_{k+i}$ which implies $y'_k - x'_k \ge y_{k+i} - x_{k+i} \ge 0$ by hypothesis. $\square$

### 4.3. An heuristic of the derivation of the viscosity solutions of the PDE.
In this section we tried to obtain the solutions of the PDE as a limit of a system of particles as in the heuristic of Section 2.3. The problem is technical and difficult to obtain. In [27] authors succeed to obtain a notion of convergence of a system of particle towards the smooth solution of the PDE under some assumptions at initial time on the particles and the smoothness of the initial data of the PDE. Quite often the theory of viscosity solutions allows to avoid such assumptions by an extensive use of discrete comparison principles [5]. Even if we were not able to apply this strategy in its entirety, we present the partial proof that we were able to find. Let us also mention that this approach follows the steps of the proof of a similar result obtained in the study of the eigenvalues of large random matrices [6, 8].

For $(\lambda_0, \dots, \lambda_{2N-1}) \in \mathbb{T}^{2N}$ we introduce the empirical measure $\mu_{2N} \in \mathcal{P}(\mathbb{T})$ associated to $(\lambda_0, \dots, \lambda_{2N-1})$:

$$\mu_{2N} = \frac{1}{2N} \sum_{i=0}^{2N-1} \delta_{\lambda_i} \in \mathcal{P}(\mathbb{T}).$$

This definition shall also be used if we consider a $2N$ periodic system of particles $(\lambda_i)_{i \in \mathbb{Z}} \in \mathbb{T}^\mathbb{Z}$ (i.e. for all $i \in \mathbb{Z}$, $\lambda_i = \lambda_{i[2N]}$ modulo $2\pi$ with $i[2N]$ the rest of the euclidean division of $i$ by $2N$) by defining the empirical measure on $\mathcal{P}(\mathbb{T})$ associated to these particles as

$$\mu_{2N} = \frac{1}{2N} \sum_{i \in \mathbb{Z}/2N\mathbb{Z}} \delta_{\lambda_i} \in \mathcal{P}(\mathbb{T}).$$

For $a \in \mathbb{R}$, let $a \le \lambda_0^0 < \lambda_0^1 < \dots < \lambda_0^{2N-1} < a + 2\pi$ be $2N$ initial particles. We associate to this family the trigonometric polynomial

$$p_{2n}(x) = \prod_{j=0}^{2N-1} \sin \left( \frac{x - \lambda_0^j}{2} \right).$$

We define for $i \in \mathbb{Z}$ the $i^{th}$ initial particle $\lambda_0^i = \lambda_0^{i[2N]} + 2 \lfloor \frac{i}{2N} \rfloor \pi$.

We construct the following particles $(\lambda_t^i)_{i \in \mathbb{Z}, t \ge 0}$ by : for $t \in [0, \frac{1}{2N})$, $\lambda_t^i = \lambda_0^i$ and for $k \in \mathbb{N}_{\ge 1}$ and $t \in [\frac{k}{2N}, \frac{k+1}{2N})$, $\lambda_t^i$ is the unique root of $p_{2N}^{(k)}$ in $(\lambda_{\frac{k-1}{2N}}^i, \lambda_{\frac{k-1}{2N}}^{i+1})$. Formally, we start from the family $(\lambda_0^i)_{i \in \mathbb{Z}}$ which are the roots of $p_{2N}$, we wait a time $\Delta t_N = \frac{1}{2N}$, then at the time $t = \frac{1}{2N}$, the particles jump to the roots of the derivative of $p_{2N}$ and we repeat the procedure.

For all $t \ge 0$, for all $\theta \in \mathbb{R}$, we define

$$F_{2N}(t, \theta) := F_{\mu_{2N}(t)}(\theta),$$

where $\mu_{2N}(t)$ is the empirical measure associated to $(\lambda_t^i)_{i \in \mathbb{Z}}$ and where we recall that $F_\mu$ is the cumulative distribution function of $\mu \in \mathcal{P}(\mathbb{T})$ introduced in Section 3.2. We also define the upper semi continuous function:

$$F^*(t, \theta) := \limsup_{N \to \infty, t_N \to t, \theta_N \to \theta} F_{2N}(t_N, \theta_N).$$
The goal of this section is to give the heuristic of the following statement.

Assume that the empirical measure $\mu_N^0$ of initial conditions defined by: $\mu_N^0 = \frac{1}{2N} \sum_{i=0}^{2N-1} \delta_{\lambda_0^i}$ converges toward a measure $\mu_0 \in \mathcal{P}(\mathbb{T})$ and that there exists $m > 0$ such that $F^*$ satisfies $(H_m)$. Then, $F^*$ is the unique viscosity solution of (3.5) which satisfies $F(0, \theta) = F_{\mu_0}(\theta)$ almost everywhere.

We cut the heuristic in different steps.

### 4.3.1. Step 1: Notation and setting.
We shall use the following notations: $u_N \lesssim v_N$, $u_N = O(v_N)$, $u_N = o(v_N)$ and $u_N \sim v_N$ if there exists a constant $C$ independent of $N$ such that $u_N \le Cv_N$, $(u_N/v_N)_N$ is bounded, $(u_N/v_N)_N$ goes to 0 when $N$ goes to $+\infty$ and $u_N = O(v_N)$ and $v_N = O(u_N)$. These notations will always refer to the number of particles $2N$ that shall go to $+\infty$.

We want to heuristically explain why $F^*$ is a subsolution of (3.5). The supersolution statement should follow by the same arguments.

First, $F^*$ is upper semi continuous and is in $\mathcal{F}_{t,2\pi}$.

Take a small $\delta > 0$ to be specified later on, $\phi \in C_{t,2\pi}^{1,1}$ and $(t_0, \theta_0) \in (0, +\infty) \times \mathbb{R}$ such that $(F^* - \phi)(t_0, \theta_0) = 0$ and $(F^* - \phi)(t, \theta) < 0$ for any $(t, \theta) \in B((t_0, \theta_0), \delta) - (t_0, \theta_0)$ and $\partial_\theta(\phi(t_0, \theta_0)) > m$ as explained in Remark 3.11.

We want to prove that:

$$ \partial_t \phi(t_0, \theta_0) + \frac{1}{\pi} \left( \arctan \left( \frac{I_{1,\delta}[\phi(t_0, \cdot)](\theta_0) + I_{2,\delta}[F^*(t_0, \cdot)](\theta_0)}{\partial_\theta \phi(t_0, \theta_0)} \right) + \frac{\pi}{2} \right) \le 0. $$

In spite of $\phi$ we can consider a function $\phi_1$ such that $\phi_1$ is equal to $\phi$ in $B((t_0, \theta_0), \delta)$, is in $C_{t,2\pi}^{1,1}$, is non-decreasing and satisfies that $\phi(t, \theta + 2\pi) = \phi(t, \theta) + 1$ for all $t > 0$ and $\theta \in \mathbb{R}$. If we prove the last inequality with $\phi_1$ instead of $\phi$ it will implies the result for $\phi$ because $\phi$ and $\phi_1$ are equal around $(t_0, \theta_0)$. We shall now use $\phi_1$ in spite of $\phi$ but we still call it $\phi$. We consider $I_\delta = (\theta_0 - \delta, \theta_0 + \delta)$. We can choose $\delta$ small enough such that $\phi(t, \cdot)$ is strictly increasing on $I_\delta$ for $t$ near of $t_0$ since $\partial_\theta \phi(t_0, \theta_0) > 0$. In what follows we choose such $\delta$. We write $J_\delta(t) = \phi(t, I_\delta)$ which is an open interval of positive length that contains $\phi(t_0, \theta_0)$ again since $\partial_\theta \phi(t_0, \theta_0) > 0$ for $t$ near $t_0$.

We also consider an interval $I$ of length $2\pi$ such that $I_\delta \subset \mathring{I}$.

We consider $s_N := \frac{k(N)}{2N}$ that shall go to 0 and that will be specified later on.

We change the numbering of our $\lambda_{t_0-s_N}$ with respect to the interval $I$ which means that if $I$ is $[a, a + 2\pi)$ for an $a \in \mathbb{R}$, we write:

$$ a \le \lambda_{t_0-s_N}^0 < \lambda_{t_0-s_N}^1 < \dots < \lambda_{t_0-s_N}^{2N-1} < a + 2\pi. $$

For all $i \bmod 2N$, $\lambda_{t_0-s_N}^i \bmod 2\pi$ is associated to a class $\frac{i}{2N} \bmod 1$ defined by $F_N(t_0 - s_N, \lambda_{t_0-s_N}^i) = \frac{i}{2N}[1]$.

For the $i \bmod 2N$ such that $\frac{i}{2N} \bmod 1$ is in $J_\delta(t_0 - s_N) \bmod 1$, we define $\gamma_{t_0-s_N}^i \bmod 2\pi$ which is equal to $(\phi(t_0 - s_N))^{-1}(\frac{i}{2N}) \bmod 2\pi$ (this definition makes sense since $\phi(t_0 - s_N, \theta + 2\pi) = \phi(t_0 - s_N, \theta) + 1$ for all $\theta$ and we recall that $\phi(t_0 - s_N, \cdot)$ is strictly increasing on $I_\delta$ for $N$ large enough). By abuse of notation we write $\gamma_{t_0-s_N}^i$ the unique real in $I$ which represents the previous class modulo $2\pi$. More exactly $\gamma_{t_0-s_N}^i$ is in $I_\delta$ by definition of $J_\delta$.

We define the new system of particles $(\mu^i)_{i \in \mathbb{Z}}$ by the initial data given by:

$$ \left\{ \begin{array}{l} \mu_{t_0-s_N}^i = \lambda_{t_0-s_N}^i \text{ if } \frac{i}{2N} \bmod 1 \text{ is not in } J_\delta(t_0-s_N) \bmod 1 \\ \\ \mu_{t_0-s_N}^i = \min(\gamma_{t_0-s_N}^i, \lambda_{t_0-s_N}^i) \text{ if } \frac{i}{2N} \bmod 1 \text{ is in } J_\delta(t_0-s_N) \bmod 1 \end{array} \right. \qquad (4.7) $$
Then $(\mu_t^i)_{i \in \mathbb{Z}, t \ge t_0-s_N}$ is defined with the same dynamic as the $(\lambda_t^i)_{i \in \mathbb{Z}, t \ge 0}$. More exactly let $q_{2N}$ be the trigonometric polynomial whose roots are $(\mu_{t_0-s_N}^i)_{i \in \mathbb{Z}}$. During a time $\Delta t_N = \frac{1}{2N}$, the particles do not move and at time $t_0 - s_N + \frac{1}{2N}$ they are replaced by the roots of $q'_{2N}$ and we repeat the process.

### 4.3.2. Step 2: Consequence of the discrete comparison principle.
Since $I_\delta \subset \mathring{I}$ and $F^* \le \phi$ in $I_\delta$ we have that for all $i \in \mathbb{Z}$, $\mu_{t_0-s_N}^i \le \lambda_{t_0-s_N}^i$. Hence, by the discrete comparison principle of Proposition 4.2 we deduce that $\mu_{t_0}^i \le \lambda_{t_0}^i$. By definition of $F^*$, for all $N$ we consider an index $i_0(N)$ which satisfies:

$$ \left\{ \begin{array}{l} \limsup_N \lambda_{t_0-s_N}^{i_0(N)} \le \theta_0 \\ \\ \frac{i_0(N)}{2N} \to F^*(t_0, \theta_0). \end{array} \right. \qquad (4.8) $$

Hence for $N$ large enough we have:

$$ \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) - \phi(t_0, \mu_{t_0}^{i_0(N)}) \ge \frac{i_0(N)}{2N} - \phi(t_0, \lambda_{t_0}^{i_0(N)}), $$

by definition of $\mu_{t_0-s_N}^i$ and since $\phi$ is non-decreasing. By construction we get

$$ \liminf_N \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) - \phi(t_0, \mu_{t_0}^{i_0(N)}) \ge F^*(t_0, \theta_0) - \phi(t_0, \theta_0) = F^*(t_0, \theta_0) - F^*(t_0, \theta_0) = 0. $$

Choosing $s_N$ of the form $s_N = \frac{k(N)}{2N}$ that goes to 0 sufficiently slowly we obtain

$$ \liminf_N \frac{\phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) - \phi(t_0, \mu_{t_0}^{i_0(N)})}{s_N} \ge 0. \qquad (4.9) $$

### 4.3.3. Step 3: Dynamic of the particles.
We compute the evolution of $\phi$ along the flow of the particles:

$$ \frac{\phi(t_0, \mu_{t_0}^{i_0(N)}) - \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)})}{s_N} = \frac{\phi(t_0, \mu_{t_0}^{i_0(N)}) - \phi(t_0 - s_N, \mu_{t_0}^{i_0(N)})}{s_N} + \frac{\phi(t_0 - s_N, \mu_{t_0}^{i_0(N)}) - \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)})}{s_N} $$

Taking the superior limit in the previous equality, (4.9) yields:

$$ \partial_t \phi(t_0, \theta_0) + \limsup_N \frac{\phi(t_0 - s_N, \mu_{t_0}^{i_0(N)}) - \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)})}{s_N} \le 0. \qquad (4.10) $$

Doing a Taylor expansion with respect to the space variable:

$$ \frac{\phi(t_0 - s_N, \mu_{t_0}^{i_0(N)}) - \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)})}{s_N} = \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N} \partial_\theta \phi(t_0-s_N, \mu_{t_0-s_N}^{i_0(N)}) + O \left( \frac{(\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)})^2}{s_N} \right). \qquad (4.11) $$

We see that the quantity that we have to understand is:

$$ \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N} = \frac{1}{k(N)} \sum_{j=0}^{k(N)-1} 2N \left[ \mu_{t_0-s_N+\frac{j+1}{2N}}^{i_0(N)} - \mu_{t_0-s_N+\frac{j}{2N}}^{i_0(N)} \right]. \qquad (4.12) $$

We expect that

$$ \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N} \lesssim 1. \qquad (4.13) $$
We prove this estimate in Appendix in Lemma (A.9) for the case $s_N = \frac{1}{2N}$ (so the case $k(N) = 1$) which corresponds to when there is just one derivation.
Assuming (4.13), this implies that:

$$ \limsup_N \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N} < +\infty \quad \text{and} \quad \frac{(\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)})^2}{s_N} = o(1). $$

Using (4.10) and (4.11), we should get:

$$ \partial_t \phi(t_0, \theta_0) + \limsup_N \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N} \partial_\theta \phi(t_0, \theta_0) \le 0. \quad (4.14) $$

We have to understand how the particles interact to study $\limsup_N \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N}$ and more precisely to understand rigorously the flow of polynomial roots under differentiation. Set

$$ L(t_0, \theta_0) := \limsup_N \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N}. $$

Up to considering a subsequence, from now on we will consider that:

$$ L(t_0, \theta_0) = \lim_{N \to \infty} \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N}. $$

### 4.3.4. Step 4: Interaction between particles.
The main difficult part is to show the following inequality for the behaviour of the flow of particles. We present a proof in the case $s_N = \frac{1}{2N}$ that corresponds to one derivation.

**Proposition 4.3.** Assume that $s_N = \frac{1}{2N}$, we have the following inequality:

$$ \begin{aligned} & 0 \ge \partial_{\theta} \phi(t_0, \theta_0) \cotan(\pi L(t_0, \theta_0) \partial_{\theta} \phi(t_0, \theta_0)) + \\ & I_{1,\delta}[\phi(t_0, \cdot)](\theta_0) - \cotan\left(\frac{\delta}{2}\right)(\phi(t_0, \theta_0 + \delta) + \phi(t_0, \theta_0 - \delta)) + \\ & I_{2,\delta}[F^*(t_0, \cdot)](\theta_0) + \cotan\left(\frac{\delta}{2}\right)(F^*(t_0, \theta_0 + \delta) + F^*(t_0, \theta_0 - \delta)) \\ & =: \partial_{\theta} \phi(t_0, \theta_0) \cotan(\pi L(t_0, \theta_0) \partial_{\theta} \phi(t_0, \theta_0)) + \mathcal{I}(\delta, t_0, \theta_0). \end{aligned} \quad (4.15) $$

This implies that:

$$ L(t_0, \theta_0) \ge \frac{1}{\pi \partial_{\theta} \phi(t_0, \theta_0)} \arctan \left( - \frac{\mathcal{I}(\delta, t_0, \theta_0)}{\partial_{\theta} \phi(t_0, \theta_0)} \right), \quad (4.16) $$

where $\arctan$ is the reciprocal function of $\cotan: ]0, \pi[ \to \mathbb{R}$.

The proof of this result in the case $s_N = \frac{1}{2N}$ is referred in Appendix A.5. Using that $\arctan(-\theta) = \arctan(\theta) + \pi/2$ for all $\theta \in \mathbb{R}$, Proposition 4.3 and (4.14), we get the following result.

**Proposition 4.4.** Assume that $s_N = \frac{1}{2N}$, we have

$$ \partial_t \phi(t_0, \theta_0) + \frac{1}{\pi} \left( \arctan \left( \frac{\mathcal{I}(\delta, t_0, \theta_0)}{\partial_{\theta} \phi(t_0, \theta_0)} \right) + \frac{\pi}{2} \right) \le 0 \quad (4.17) $$
## APPENDIX A. PROOF OF PROPOSITION 4.3 IN THE PARTICULAR CASE $s_N = \frac{1}{2N}$

Before starting, let us mention that the computations of this part are similar to the estimates obtained in [27]. We try to keep some of their notation to make it easier for the reader.

### A.1. Notation.
We recall that $I$ is the interval of length $2\pi$ that contains $I_\delta$ introduced in the first step of the heuristic. For all $N$, we cut the interval $I$ in three parts: a very near part: $\mathcal{N}_N := \{\mu \in \mathbb{R}, |\mu - \mu_{t_0}^{i_0(N)}| \le 1/\sqrt{N}\}$, a near part: $I_\delta - \mathcal{N}_N$ and a far away part: $I - I_\delta$. We define the sets of indexes associated to these sets: $J_{1,N} = \{j \in \mathbb{Z} | \mu_{t_0-s_N}^j \in \mathcal{N}_N\}$, $J_{2,N} = \{j \in \mathbb{Z} | \mu_{t_0-s_N}^j \in I_\delta - \mathcal{N}_N\}$ and $J_{3,N} = \{j \in \mathbb{Z} | \mu_{t_0-s_N}^j \in I - I_\delta\}$. We write explicitly $J_{1,N} = \{j-, \dots, j_+ - 1\}$ for after since we shall have to control the size of $J_{1,N}$.
As in the heuristic of the proof, the starting point is the relation (4.1):

$$ 
\begin{aligned}
0 &= \frac{1}{2N} \sum_{j=1}^{2N} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j}{2} \right) \\
&= \frac{1}{2N} \sum_{j \in J_{1,N}} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j}{2} \right) + \frac{1}{2N} \sum_{j \in J_{2,N}} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j}{2} \right) \\
&\qquad + \frac{1}{2N} \sum_{j \in J_{3,N}} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j}{2} \right) \quad (A.1) \\
&=: S_{1,N} + S_{2,N} + S_{3,N},
\end{aligned}
$$

where $S_{1,N}$ corresponds to the interaction between $\mu_{t_0}^{i_0(N)}$ and the particles that are very near it at time $t_0 - s_N$, $S_{2,N}$ corresponds to the interaction between $\mu_{t_0}^{i_0(N)}$ and the particles that are near it at time $t_0 - s_N$ and $S_{3,N}$ corresponds to the interaction between $\mu_{t_0}^{i_0(N)}$ and the particles that are far away it at time $t_0 - s_N$. We first deal with the near and the far away part.

### A.2. Near part.

**Proposition A.1.** We have the following inequality:

$$ \liminf_N S_{2,N} \ge 2\pi \left[ I_{1,\delta}[\phi(t_0, \cdot)](\theta_0) - \cotan \left(\frac{\delta}{2}\right) \left(\phi(t_0, \theta_0 + \delta) + \phi(t_0, \theta_0 - \delta)\right) \right]. \quad (A.2) $$

*Proof.* We start from the definition of the $\mu_{t_0-s_N}^j$:

$$ S_{2,N} = \frac{1}{2N} \sum_{j \in J_{2,N}} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j}{2} \right) = \frac{1}{2N} \sum_{j \in J_{2,N}} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - (\phi(t_0 - s_N))^{-1}(\frac{j}{2N})}{2} \right). $$

By monotonicity of $\phi(t_0 - s_N, \cdot)$, we get:

$$ \sum_{j \in J_{2,N}} \int_{\frac{j-1}{2N}}^{\frac{j}{2N}} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - (\phi(t_0 - s_N))^{-1}(y)}{2} \right) dy \le S_{2,N} $$

The change of variable $y = \phi(t_0 - s_N, x)$ yields:

$$ \sum_{j \in J_{2,N}} \int_{\mu_{t_0-s_N}^{\frac{j}{2N}}(\frac{j-1}{2N})}^{\mu_{t_0-s_N}^{\frac{j}{2N}}(\frac{j}{2N})} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - x}{2} \right) \partial_x \phi(t_0 - s_N, x) dx \le S_{2,N} \quad (A.3) $$
Since

$$
\begin{aligned}
\left| P.V. \int_{\mathcal{N}_N} \partial_x \phi(t_0 - s_N, x) \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - x}{2} \right) dx \right| &= \\
&\left| \int_{\mathcal{N}_N} \left[ \partial_x \phi(t_0 - s_N, x) - \partial_x \phi(t_0 - s_N, \mu_{t_0}^{i_0(N)}) \right] \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - x}{2} \right) dx \right| \\
&\lesssim \frac{\sup_{t \in [(t_0-1)_+, t_0+1]} \|\partial_{xx}^2 \phi(t, \cdot)\|_{\infty}}{\sqrt{N}}
\end{aligned}
$$

goes to 0 when N goes to $+\infty$, taking the limit in (A.3) gives:

$$ \liminf_N S_{2,N} \ge P.V. \int_{I_\delta} \partial_x \phi(t_0, x) \cotan \left( \frac{\theta_0 - x}{2} \right) dx = 2\pi H_{1,\delta}[\partial_x \phi(t_0, \cdot)](\theta_0). $$

Integrating by part yields the result.

<figure><img src="image_2.png" /></figure>

A.3. **Far away part.** By similar computations as in Section A.2 we get the following result for the far away term.

**Proposition A.2.** We have the following inequality:

$$ \liminf_N S_{3,N} \ge 2\pi \left[ I_{2,\delta}[F^*(t_0, \cdot)](\theta_0) + \cotan\left(\frac{\delta}{2}\right) (F^*(t_0, \theta_0 + \delta) + F^*(t_0, \theta_0 - \delta)) \right]. \quad (A.4) $$

A.4. **Very near part.**

**Proposition A.3.** We have the following convergence:

$$ \lim_{N} S_{1,N} = 2\pi \partial_{\theta} \phi(t_0, \theta_0) \cotan(\pi L(t_0, \theta_0) \partial_{\theta} \phi(t_0, \theta_0)) \quad (A.5) $$

The proof is quite technical and we give it in Appendix A.5.

A.5. **Proof of Proposition A.3.**

A.5.1. *General Lemmas.* This part must be considered as independent of the others (nonethe-
less, the notations have been chosen consistently with the proof). We fix an integer $N > 0$,
$\delta > 0$ and $\theta_0 \in \mathbb{R}$ and we define $I_\delta = ]\theta_0 - \delta, \theta_0 + \delta[$. Let $\phi : \mathbb{R} \to \mathbb{R}$ be a smooth function
and let $\psi = \phi'$. Let $K_\delta = \inf_{x \in I_\delta} \psi(x)$. In all this part we suppose that $K_\delta > 0$.

We define $J_\delta = \phi(I_\delta)$ and $E_{\delta,N} = \{i \in \mathbb{Z} \mid \frac{i}{2N} \in J_\delta\}$. We suppose that $E_{\delta,N}$ is not empty (this hypothesis is not really important since for $N$ large enough it is the case because of the fact that $\text{Leb}(J_\delta) > 0$ since $K_\delta > 0$).

Since $\phi$ is strictly increasing on $I_\delta$ we can define $\mu^i = \phi^{-1}(\frac{i}{2N})$ for $i \in E_{\delta,N}$. Finally, we fix $i_0 \in E_{\delta,N}$.

**Lemma A.4.** For all $j \in E_{\delta,N}$, we have the following bounds:

$$ |\mu^j - \mu^{i_0}| \ge \frac{1}{||\phi'|_{L^\infty(I_\delta)} \frac{|i_0 - j|}{2N}} \quad (A.6) $$

$$ |\mu^j - \mu^{i_0}| \le \frac{1}{K_\delta} \frac{|i_0 - j|}{2N} \quad (A.7) $$

*Proof.* By definition, we have the following equality:

$$ \left| \int_{\mu^j}^{\mu^{i_0}} \phi'(x) dx \right| = \frac{|i_0 - j|}{2N}. $$
Since

$$K_{\delta}|\mu^j - \mu^{i_0}| \leq \left| \int_{\mu^j}^{\mu^{i_0}} \phi'(x)dx \right| \leq ||\phi'||_{L^{\infty}(I_{\delta})} |\mu^j - \mu^{i_0}|,$$

we get the result.

<figure><img src="image_3.png" /></figure>

For all $j \in E_{\delta,N}$, we define $\tilde{\mu}^j = \mu^{i_0} + \frac{j - i_0}{2N\psi(\mu^{i_0})}$.

**Lemma A.5.** For all $j \in E_{\delta,N}$, we have the following bounds:

$$|\mu^j - \tilde{\mu}^j| \leq \frac{||\phi''||_{L^{\infty}(I_{\delta})} |i_0 - j|^2}{K_{\delta}^3} \quad (A.8)$$

$$|V_j| := \left| \mu^{j+1} - \mu^j - \frac{1}{2N\psi(\mu^j)} \right| \leq \frac{||\phi''||_{L^{\infty}(I_{\delta})}}{K_{\delta}^3} \frac{1}{8N^2} \quad (A.9)$$

$$|\mu^j - \tilde{\mu}^j| \leq \frac{||\phi''||_{L^{\infty}(I_{\delta})} |j - i_0|}{8N^2} + \frac{||\phi''||_{L^{\infty}(I_{\delta})}}{8N^2 K_{\delta}^3} |i_0 - j|^2 \quad (A.10)$$

*Proof.* We directly compute:

$$\begin{aligned} |\tilde{\mu}^j - \mu^j| &= |\mu^{i_0} + \frac{j - i_0}{2N\psi(\mu^{i_0})} - \mu_j| \\ &= \left| \mu^{i_0} - \mu^j + \frac{\int_{\mu^{i_0}}^{\mu^j} \psi(x)dx}{\psi(\mu^{i_0})} \right| \\ &\leq \frac{1}{K_{\delta}} \left| (\mu^{i_0} - \mu^j)\psi(\mu^{i_0}) + \int_{\mu^{i_0}}^{\mu^j} \psi(x)dx \right| \\ &= \frac{1}{K_{\delta}} \left| \int_{\mu^{i_0}}^{\mu^j} \left[ \psi(x) - \psi(\mu^{i_0}) \right] dx \right| \\ &\leq \frac{||\psi''||_{L^{\infty}(I_{\delta})}}{K_{\delta}} \left| \int_{\mu^{i_0}}^{\mu^j} |x - \mu^{i_0}| dx \right| \\ &\leq \frac{||\psi''||_{L^{\infty}(I_{\delta})}}{2K_{\delta}} (\mu^j - \mu^{i_0})^2. \end{aligned}$$

Using (A.5), yields (A.8).

To prove (A.9) we do the same computation as for (A.8), we get that for all $j \in E_{\delta,N}$,

$$|V_j| \leq \frac{||\phi''||_{L^{\infty}(I_{\delta})}}{2K_{\delta}} (\mu^{j+1} - \mu^j)^2 \leq \frac{||\phi''||_{L^{\infty}(I_{\delta})}}{K_{\delta}^3} \frac{1}{8N^2},$$

using Lemma A.4 Without loss of generality, assume that $j > i_0$, we have:

$$\begin{aligned} \tilde{\mu}^j - \mu^j &= \mu^{i_0} + \frac{j - i_0}{2N\psi(\mu^{i_0})} - \mu_j \\ &= -\sum_{k=i_0}^{j-1} \left( V_k + \frac{1}{2N\psi(\mu^k)} \right) + \frac{j - i_0}{2N\psi(\mu^{i_0})} \\ &= -\sum_{k=i_0}^{j-1} V_k + \sum_{k=i_0}^{j-1} \left[ \frac{1}{2N\psi(\mu^{i_0})} - \frac{1}{2N\psi(\mu^k)} \right] \end{aligned}$$
Hence using the (A.7):

$$
\begin{align*}
|\tilde{\mu}^j - \mu^j| &\le \frac{\|\phi''\|_{L^\infty(I_\delta)}}{K_\delta^3} \frac{|j-i_0|}{8N^2} + \frac{\|\phi''\|_{L^\infty(I_\delta)}}{2NK_\delta^2} \sum_{k=i_0}^{j-1} |\mu^{i_0} - \mu_k| \\
&\le \frac{\|\phi''\|_{L^\infty(I_\delta)}}{K_\delta^3} \frac{|j-i_0|}{8N^2} + \frac{\|\phi''\|_{L^\infty(I_\delta)}}{4N^2K_\delta^3} \sum_{k=i_0}^{j-1} |i_0 - k| \\
&\le \frac{\|\phi''\|_{L^\infty(I_\delta)}}{K_\delta^3} \frac{|j-i_0|}{8N^2} + \frac{\|\phi''\|_{L^\infty(I_\delta)}}{8N^2K_\delta^3} |i_0 - j|^2
\end{align*}
$$

□

**Remark A.6.** The terms $V_j$ introduced in (A.9) can be interpreted as error terms that measure the error when we approximate the function $\phi$ by the $\mu^i$. This error was also introduced in [27].

A.5.2. *Consequences of Section A.5.1.* As a consequence of Section A.5.1 and regularity of $\phi$ in space and time, we get the following results.

**Lemma A.7.** There exists a constant $K > 0$ (independent on $N$) such that:

(1) for all $N$, for all $j \in J_{1,N} \cup J_{2,N}$

$$
\frac{1}{K} \frac{|i_0(N) - j|}{N} \le |\mu_{t_0-s_N}^j - \mu_{t_0-s_N}^{i_0(N)}| \le K \frac{|i_0(N) - j|}{N} \quad (A.11)
$$

(2) for all $N$, for all $j \in J_{1,n} \cup J_{2,N} - \{i_0(N), i_0(N) + 1\}$

$$
\frac{1}{K} \frac{|i_0(N) - j|}{N} \le |\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{j}| \le K \frac{|i_0(N) - j|}{N} \quad (A.12)
$$

*Proof.* The first point is a direct application of Lemma A.4. The second one is a consequence of (A.11) and the fact that the roots of a trigonometric polynomial and its derivative interlace.
□

The bound (A.12) will also be true for $j = i_0(N)$ and $j = i_0(N)+1$ but the estimate are more difficult to obtain. Nonetheless the upper bound is immediate again by interlacing but the lower bound is less immediate.

**Lemma A.8.** We have the following bound:

$$
\begin{align*}
|J_{1,N}| &\sim \sqrt{N} \\
j_+ - i_0(N) &\sim \sqrt{N} \\
2i_0 - j_+ - j_- &= O(1)
\end{align*}
$$

*Proof.* The two first points are immediate since the length of $\mathcal{N}_N$ is of order $1/\sqrt{N}$ and the space between two $\mu_{t_0-s_N}^i$ is of order $1/N$ by Lemma A.7.

For the second estimate, let us notice that by definition of $j_+$ and $j_-$ and by Lemma A.7, we have

$$
\mu_{t_0-s_N}^{j_+} - \mu_{t_0-s_N}^{i_0(N)} = N^{-1/2} + O(N^{-1}), \quad \mu_{t_0-s_N}^{i_0(N)} - \mu_{t_0-s_N}^{j_-} = N^{-1/2} + O(N^{-1}).
$$

But we also have that

$$
\mu_{t_0-s_N}^{j_+} - \mu_{t_0-s_N}^{i_0(N)} = \sum_{k=i_0(N)}^{j_+-1} (\mu_{t_0-s_N}^{k+1} - \mu_{t_0-s_N}^{k}) \quad (A.13)
$$

$$
\mu_{t_0-s_N}^{i_0(N)} - \mu_{t_0-s_N}^{j_-} = \sum_{k=j_-}^{i_0(N)-1} (\mu_{t_0-s_N}^{k+1} - \mu_{t_0-s_N}^{k}). \quad (A.14)
$$
Using (A.9) we get

$$ \mu_{t_0-s_N}^{k+1} - \mu_{t_0-s_N}^{k} = \frac{1}{2N \partial_{\theta} \phi(t_0 - s_N, \mu_{t_0-s_N}^{k})} + O(N^{-2}). $$

Subtracting the two equalities of (A.13) yields:

$$ O(N^{-1}) = \frac{2i_0(N) - j_+ - j_-}{2N \partial_{\theta} \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)})} + O(N^{1/2}N^{-2} + N^{-1}). $$

This gives the result.

□

**Lemma A.9.** There exists a constant $K > 0$ such that:

$$ \frac{1}{KN} \le \min\{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}, \mu_{t_0-s_N}^{i_0(N)+1} - \mu_{t_0}^{i_0(N)}\} \le \max\{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}, \mu_{t_0-s_N}^{i_0(N)+1} - \mu_{t_0}^{i_0(N)}\} \le \frac{K}{N}. $$

*Proof.* The upper bound is immediate using the interlacing property of the roots.

We now focus on the lower bound. We suppose that $\mu_{t_0}^{i_0(N)}$ is closer to $\mu_{t_0-s_N}^{i_0(N)}$ (the other can be done by similar computations). We first obtain thanks to this hypothesis and Lemma A.7 that

$$ |\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)+1}| \ge \frac{|\mu_{t_0-s_N}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)+1}|}{2} \ge \frac{1}{N}. \quad (A.15) $$

It remains to show that

$$ \mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)} \ge \frac{1}{N}. $$

Starting from (A.1), using Lemma A.8 and Lemma A.7 yields

$$ \begin{aligned} 0 = S_{1,N} + S_{2,N} + S_{3,N} &= \frac{1}{2N} \sum_{j \in J_{1,N}} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j}{2} \right) + S_{2,N} + S_{3,N} \\ &= \frac{1}{2N} \sum_{j \in J_{1,N}} \left( \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j} + O(\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j) \right) + S_{2,N} + S_{3,N} \\ &= \frac{1}{2N} \sum_{j \in J_{1,N}} \left( \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j} + O(N^{-1/2}) \right) + S_{2,N} + S_{3,N} \\ &= \frac{1}{2N} \sum_{j \in J_{1,N}} \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j} + O(N^{-1}) + S_{2,N} + S_{3,N} \end{aligned} \quad (A.16) $$

Since now we suppose that $j_+ - i_0(N) \ge i_0(N) - j_-$ (the other case can be done by symmetry doing the same computations). Isolating the index $i_0(N)$ in $S_{1,N}$ gives:

$$ \begin{aligned} \frac{1}{N} \frac{1}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}} &= -\frac{1}{2N} \sum_{l=j_{-}}^{2i_0(N)-j_{+}} \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{l}} \\ &\quad -\frac{1}{2N} \sum_{l=1}^{j_{+}-i_0(N)-1} \left[ \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)+l}} + \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)-l}} \right] + O(N^{-1}) - S_{2,N} - S_{3,N} \end{aligned} \quad (A.17) $$

First we estimate the first sum. By definition of $j_+$ and $j_-$ and by Lemma A.8 we have:

$$ \left| \frac{1}{2N} \sum_{l=j_{-}}^{2i_0(N)-j_{+}} \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{l}} \right| \le \frac{1}{N}(2i_0(N) - j_{+} - j_{-} + 1)O(N^{1/2}) = O(N^{-1/2}) \quad (A.18) $$
For the second one, compute

$$ \frac{1}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)+l}} + \frac{1}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)-l}} = \frac{2\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)+l} - \mu_{t_0-s_N}^{i_0(N)-l}}{(\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)+l})(\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)-l})} $$

We first estimate the numerator. As in Section A.5.1, for all $j \in J_{1,N} \cup J_{2,N}$, let

$$ V_{j,N} = \mu_{t_0-s_N}^{j+1} - \mu_{t_0-s_N}^{j} - \frac{1}{2N \partial_x \phi(t_0 - s_N, \mu_{t_0-s_N}^{j})} $$

By doing similar computations as in Lemma A.4 and Lemma A.5 show that there exist $K > 0$ such that for all $j \in J_{1,N} \cup J_{2,N}$, for all $N$

$$ V_{j,N} \le \frac{K}{N^2}. $$

Using the regularity of $\phi$ and the estimate for $V_{j,N}$, we get:

$$ \begin{aligned} \mu_{t_0-s_N}^{i_0(N)+l} + \mu_{t_0-s_N}^{i_0(N)-l} &= 2\mu_{t_0-s_N}^{i_0(N)} + \sum_{k=1}^{l} \left[ \frac{1}{2N \partial_x \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)+k-1})} + V_{i_0(N)+k-1,N} - \frac{1}{2N \partial_x \phi(t_0 - s_N, \mu_{t_0}^{i_0(N)-k})} + V_{i_0(N)+k,N} \right] \\ &= 2\mu_{t_0-s_N}^{i_0(N)} + O(lN^{-2}) + O\left(\frac{1}{N} \sum_{k=1}^{l} \left| \mu_{t_0-s_N}^{i_0(N)+k-1} - \mu_{t_0-s_N}^{i_0(N)-k} \right|\right) \\ &= 2\mu_{t_0-s_N}^{i_0(N)} + O(lN^{-2}) + O\left(\frac{1}{N} \sum_{k=1}^{l} \frac{2k}{N}\right) \\ &= 2\mu_{t_0-s_N}^{i_0(N)} + O(lN^{-2}) + O(l^2N^{-2}) \end{aligned} $$

Using (A.15) and (A.7) and Lemma A.8, it yields

$$ \begin{aligned} \frac{1}{2N} \sum_{l=1}^{j_+-i_0(N)-1} \left| \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)+l}} + \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)-l}} \right| &\le \frac{1}{N} \sum_{l=1}^{\sim n^{1/2}} O \left( \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)} + l^2 N^{-2} + lN^{-2}}{l^2 N^{-2}} \right) \\ &= O(1 + N^{-1/2} + \log(N)N^{-1}) = O(1) \end{aligned} \quad (A.19) $$

So, (A.17), (A.18), (A.19) give

$$ \frac{1}{N} \frac{1}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}} = -S_{2,N} - S_{3,N} + O(1) \quad (A.20) $$

Using similar estimates as for Proposition A.3 and for Proposition A.2 we can obtain

$$ \left\{ \begin{array}{l} |S_{2,N}| \le \pi H_{1,\delta}[\partial_x \phi(t_0, \cdot)](\mu_{t_0-s_N}^{i_0(N)}) + O(1) \\ |S_{3,N}| \le 4\pi I_{2,\delta}[F^*(t_0 - s_N, \cdot)](\mu_{t_0-s_N}^{i_0(N)}) + \cotan\left(\frac{\delta}{2}\right) (F^*(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)} + \delta) + F^*(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)} - \delta)) \end{array} \right. \quad (A.21) $$

Since $\phi$ is smooth and $F$ is non-decreasing and satisfies for every $t \ge 0$ $F(t, \cdot + 2\pi) = F(t, \cdot) + 1$ we get that the previous quantities are bounded

$$ \left\{ \begin{array}{l} \left| H_{1,\delta}[\partial_x \phi(t_0 - s_N, \cdot)](\mu_{t_0-s_N}^{i_0(N)}) \right| \lesssim \sup_{t \in [(t_0-1),t]} ||\partial_{xx}\phi(t, \cdot)||_{\infty} = O(1) \\ \\ \left| I_{2,\delta}[F^*(t_0 - s_N, \cdot)](\mu_{t_0-s_N}^{i_0(N)}) \right| \le O(1) \int_{\mathbb{T} \cap |\theta| > \delta} \frac{1}{\sin^2(\theta/2)} d\theta = O(1) \end{array} \right. \quad (A.22) $$
Hence (A.20) gives

$$ \frac{1}{N} \frac{1}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}} = O(1), \qquad (A.23) $$

which concludes the proof.

<figure><img src="image_4.png" /></figure>

As in Section A.5.1 and in Section 2.3 we introduce the points

$$ \tilde{\mu}_{t_0-s_N}^j := \mu_{t_0-s_N}^{i_0(N)} + \frac{j-i_0}{2N \partial_{\theta} \phi(t_0-s_N, \mu_{t_0-s_N}^{i_0(N)})} $$

As an immediate consequence of Lemma A.7, Lemma A.9 Lemma A.5 and interlacing of the roots we have the estimate obtained for the bound between $\mu_{t_0}^{i_0(N)}$ and $\mu_{t_0-s_N}^j$ are also valid for $\mu_{t_0}^{i_0(N)}$ and $\tilde{\mu}_{t_0-s_N}^j$.

**Lemma A.10.** There exists $K$ independent of $N$ such that for all $j \neq i_0(N)$

$$ \begin{aligned} |\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^j| &\ge \frac{|i_0(N) - j|}{KN} \\ |\mu_{t_0-s_N}^j - \tilde{\mu}_{t_0-s_N}^j| &\le K \frac{|j - i_0(N)|}{N^2} + K \frac{|i_0(N) - j|^2}{N^2}. \end{aligned} $$

We can now prove the main proposition of this section.

**Proposition A.11.** We have the following result:

$$ \lim_{N} S_{1,N} = 2\pi \partial_{\theta} \phi(t_0, \theta_0) \cotan(\pi L(t_0, \theta_0) \partial_{\theta} \phi(t_0, \theta_0)) \quad (A.24) $$

*Proof.* Let us remark that by definition of $S_{1,N}$ and Lemma A.8:

$$ \begin{aligned} S_{1,N} &= \frac{1}{2N} \sum_{j \in J_{1,N}} \cotan \left( \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j}{2} \right) \\ &= \frac{1}{2N} \sum_{j \in J_{1,N}} \left( \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j} + O(\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j) \right) \quad (A.25) \\ &= \frac{1}{2N} \sum_{j \in J_{1,N}} \left( \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j} + O(N^{-1/2}) \right) \\ &= \frac{1}{2N} \sum_{j \in J_{1,N}} \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j} + O(N^{-1}) \end{aligned} $$

Hence to compute the limit of $S_{1,N}$, we have to deal with the limit of

$$ \tilde{S}_{1,N} := \frac{1}{2N} \sum_{j \in J_{1,N}} \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j} $$

As explained in Section 2.3 we shall compare this sum with cotan thanks to the Euler identity using the points $\tilde{\mu}_{t_0-s_N}^j$. We shall prove that

$$ \tilde{S}_{1,N} - 2\pi \partial_{\theta} \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) \cotan \left( \pi \partial_{\theta} \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N} \right) = o(1) \quad (A.26) $$
Taking the limit in (A.26) gives (A.24). Let us now focus on (A.26). Using the Euler identity:

$$
\begin{aligned}
\tilde{S}_{1,N} - 2\pi \partial_{\theta} \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) \cotan \left( \pi \partial_{\theta} \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N} \right) &= \\
&\frac{1}{2N} \sum_{j \in J_{1,N}} \left[ \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{j}} - \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{j}} \right] - \frac{1}{2N} \sum_{j \notin J_{1,N}} \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{j}} \\
&=: \tilde{S}_{1,1,N} + \tilde{S}_{1,2,N}
\end{aligned}
$$

First we estimate $\tilde{S}_{1,2,N}$:

$$
\begin{aligned}
\tilde{S}_{1,2,N} &= \frac{1}{2N} \sum_{j \notin J_{1,N}} \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{j}} \\
&= \frac{1}{2N} \sum_{j \ge j_+ - i_0(N)} \left[ \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{j}} + \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{i_0(N)-j}} \right] + \frac{1}{2N} \sum_{j=2i_0(N)-j_++1}^{j-1} \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{j}}
\end{aligned}
$$

By Lemma A.8, Lemma A.7 and Lemma A.9 and interlacing of the root we deduce that the second term of the sum is of order $O(N^{-1})O(1)O(N^{-1/2}N) = O(N^{-1/2})$. So we get:

$$
\tilde{S}_{1,2,N} = \frac{1}{2N} \sum_{j \ge j_+ - i_0(N)} \left[ \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{j}} + \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{i_0(N)-j}} \right] + O(N^{-1/2}).
$$

Again using Lemma A.7 and Lemma A.9 we have that for all $j \in \mathbb{N}$:

$$
\begin{aligned}
& \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{i_0(N)+j}} + \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^{i_0(N)-j}} = \\
& \qquad \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)} - \frac{j}{2N \partial_{\theta} \phi(t_0-s_N, \mu_{t_0-s_N}^{i_0(N)})}} + \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)} + \frac{j}{2N \partial_{\theta} \phi(t_0-s_N, \mu_{t_0-s_N}^{i_0(N)})}} \\
& \lesssim \frac{|\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}|}{j^2 N^{-2}} \\
& \lesssim \frac{N}{j^2}.
\end{aligned}
\tag{A.27}
$$

It yields that:

$$
|\tilde{S}_{1,2,N}| \lesssim \frac{1}{2N} \sum_{j \ge \sqrt{N}} \frac{N}{j^2} + O(N^{-1/2}) \lesssim O(N^{-1/2}).
$$
Finally let us deal with the term $\tilde{S}_{1,1,N}$. We compute using Lemma A.10, Lemma A.7 Lemma A.9 and Lemma A.8

$$
\begin{aligned}
|\tilde{S}_{1,1,N}| &= \frac{1}{2N} \left| \sum_{j \in J_{1,N} - \{i_0(N)\}} \left[ \frac{2}{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j} - \frac{2}{\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^j} \right] \right| \\
&= \frac{1}{N} \left| \sum_{j \in J_{1,N} - \{i_0(N)\}} \frac{\tilde{\mu}_{t_0-s_N}^j - \mu_{t_0-s_N}^j}{(\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^j)(\mu_{t_0}^{i_0(N)} - \tilde{\mu}_{t_0-s_N}^j)} \right| \\
&\lesssim \frac{1}{N} \sum_{j \in J_{1,N} - \{i_0(N)\}} \frac{|j - i_0(N)|N^{-2}}{(j - i_0(N))^2 N^{-2}} + \frac{|j - i_0(N)|^2 N^{-2}}{(j - i_0(N))^2 N^{-2}} \\
&\lesssim \frac{1}{N} \sum_{j \in J_{1,N} - \{i_0(N)\}} \left[ \frac{1}{(j - i_0(N)) + 1} \right] \\
&\lesssim \frac{1}{N} \sum_{k \le \sqrt{N}} \frac{1}{k} + \frac{|J_{1,N}|}{N} \\
&\lesssim \log(N)N^{-1} + N^{-1/2} = O(N^{-1/2})
\end{aligned}
$$

To summarize we proved that:

$$S_{1,N} - 2\pi \partial_{\theta} \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) \cotan \left( \pi \partial_{\theta} \phi(t_0 - s_N, \mu_{t_0-s_N}^{i_0(N)}) \frac{\mu_{t_0}^{i_0(N)} - \mu_{t_0-s_N}^{i_0(N)}}{s_N} \right) = O(N^{-1/2}).$$

This gives the result.

<figure><img src="image_5.png" /></figure>

**Acknowledgments.** The first author acknowledges a partial support from the Chair FDD (ILB) and the Lagrange Mathematics and Computing Research Center. This work was partially funded by the ERC project PaDiESeM.

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(CB) CEREMADE, CNRS, UNIVERSITÉ PARIS DAUPHINE-PSL, UMR 7534, 75016 PARIS, FRANCE.
*Email address: mailto:bertucci@ceremade.dauphine.fr*

(VP) CMAP, ECOLE POLYTECHNIQUE, UMR 7641, 91120 PALAISEAU, FRANCE.
*Email address: mailto:valentin.pesce@polytechnique.edu*