Similarly, one can recover the main result of [73] from (6.55), which improves the high temperature condition of (6.42) in the case of ferromagnetic Ising models on graphs with maximal degree $d$ (for the mixing time and spectral gap). For the ferromagnetic Ising model, $A$ is the negative adjacency matrix of the graph which we now (for comparison) do not normalise to have spectrum contained in $[0, 1]$. The condition of (6.42) cannot be improved for general non-ferromagnetic interactions with bounded spectral radius, but for ferromagnetic models the condition established in [73] is $\beta < \text{artanh}(1/(d-1))$, whereas the condition of (6.42) translates in this case to $\beta < 1/(2d)$. The value $\beta_u = \text{artanh}(1/(d-1))$ is the uniqueness threshold for the ferromagnetic Ising model on the infinite $d$-regular tree and also the critical point for ferromagnetic Ising models on random $d$-regular graphs [73]. This application is summarised in the next example.

**Example 6.15.** Let $A$ be the negative adjacency matrix of a finite graph of maximal degree $d$ (not normalised to have spectrum in $[0, 1]$). If $(d-1) \tanh \beta < 1$ then $\gamma_{\beta,h}$ is uniformly bounded below (and diverges polynomially in $\beta \uparrow \beta_u = \text{artanh}(1/(d-1))$ and in the size of the graph if $\beta = \beta_u$).

*Sketch.* From [73, Section 5.4, Eq. (39)], we deduce that for any sites $x, y$ in the graph:

$$ (6.56) \qquad \mathbb{E}_{\mu_{\beta,0}}[\sigma_x \sigma_y] \le \sum_{w \in S(y)} (\tanh \beta)^{\text{dist}(x,w)}, $$

where the bound is obtained by comparing the graph with a $d$-regular tree as in [88] and $S(y)$ stands for the set of sites associated with $y$ in this tree. Summing over the sites $y$ in the graph boils down to sum over all the sites in $\cup_y S(y)$, i.e., in the $d$-regular tree. Thus we get

$$ (6.57) \qquad \chi_{\beta} = \sup_x \sum_y \mathbb{E}_{\mu_{\beta,0}}[\sigma_x \sigma_y] \le 1 + C_d \sum_{\ell \ge 1} ((d-1) \tanh \beta)^{\ell} \le \frac{C_d}{1 - (d-1) \tanh \beta}, $$

i.e., one finds a divergence of the susceptibility as in Example 6.14 as $\beta \uparrow \beta_u = \text{artanh}(1/(d-1))$. Moreover, when $\beta$ approaches $\beta_u$, one can use that for a graph with $N$ sites then trivially $\chi_{\beta} \le N$, so that

$$ (6.58) \qquad \chi_{\beta} \le \frac{C_d}{1 - (d-1) \tanh \beta + N^{-1}}. $$

By Theorem 6.13, we deduce a polynomial lower bound for the log-Sobolev constant in the size of the graph at $\beta = \beta_u = \text{artanh}(1/(d-1))$. $\square$

**6.2.5. Choice of Dirichlet form.** As in the original references [11, 18], the above discussion is formulated in terms of the standard Dirichlet form (6.34). There exist general comparison arguments between Dirichlet forms that ensure that one can transfer the log-Sobolev inequality obtained for a certain dynamics to another one, see, e.g., Chapter 4 in [81]. Namely, if $c_1, c_2$ are families of jump rates reversible with respect to the same measure $\nu$ on $\{-1, 1\}^\Lambda$ and $\gamma_1, \gamma_2$ denote the associated log-Sobolev constants, then, for $K > 0$:

$$ (6.59) \qquad \forall \sigma, \sigma' \in \{-1, 1\}^\Lambda, \quad K^{-1}c_2(\sigma, \sigma') \le c_1(\sigma, \sigma') \le Kc_2(\sigma, \sigma') \quad \Rightarrow \quad K\gamma_2 \le \gamma_1 \le \gamma_2/K. $$

One can for instance check that the heat-bath- and canonical jump rates associated with an Ising measure with interaction $\beta A$ and external field $h \in \mathbb{R}^\Lambda$ satisfy such a bound, with a constant $K$ that depends only on $\|h\|_\infty$ and $\beta \max_i \sum_j |A_{i,j}|$. The heat-bath Dirichlet form is [68, (3.4)-(3.6)]:

$$ (6.60) \qquad D_{\mu}^{\text{HB}}(F) = \frac{1}{2} \sum_{\sigma \in \{\pm 1\}^{\Lambda}} \sum_{x \in \Lambda} \Psi(\mu(\sigma), \mu(\sigma^x))(F(\sigma) - F(\sigma^x))^2, \qquad \Psi(a, b) = \frac{ab}{a+b}. $$

There are however situations in which the comparison argument (6.59) is not applicable. This observation was made in [41] in the situation of the SK model treated in [11], which is an Ising
model with random couplings which can take arbitrarily large values. This makes the constant K in (6.59) arbitrarily small. In such cases, if one is interested in a dynamics different from the one induced by the canonical jump rates, it is desirable to directly obtain the log-Sobolev inequality (or modified log-Sobolev inequality) for this dynamics. This was done for the heat-bath dynamics, for the modified log-Sobolev inequality, in [3, 28, 41].

One can however check that the arguments of [11, 18] sketched above are not specific to the standard Dirichlet form. It is in fact straightforward to apply them to other dynamics as sketched below in the heat-bath case, provided:

* the associated single-spin LSI constant (or modified LSI constant) is uniform in the field;

* the associated Dirichlet form is a concave function of the measure (this can be generalised).

Let us show how this works in the heat-bath case (6.60), for which both points are satisfied in view of Proposition 6.5 and the concavity of $\Psi(a, b) = ab/(a+b)$. Instead of using the single-spin log-Sobolev inequality for the standard Dirichlet form of Proposition 6.4, one can instead apply the single-spin modified LSI with the heat-bath Dirichlet form of Proposition 6.5 since it also holds uniformly in the external field. For instance, for $\mathbb{E}_{\nu_{0,\beta}}[\text{Ent}_{\mu_0^\varphi}(F)]$ the bound becomes:

$$ 
\begin{align*}
& \mathbb{E}_{\nu_{0,\beta}}[\text{Ent}_{\mu_0}(F(\sigma))] \\
\le & \frac{1}{2} \sum_{\sigma} \sum_x \mathbb{E}_{\nu_{0,\beta}}[\Psi(\mu_0(\sigma), \mu_0(\sigma^x))](F(\sigma) - F(\sigma^x))(\log F(\sigma) - \log F(\sigma^x)) \\
(6.61) \quad & \le \frac{1}{2} \sum_{\sigma} \sum_x \Psi(\mu(\sigma), \mu(\sigma^x))(F(\sigma) - F(\sigma^x))(\log F(\sigma) - \log F(\sigma^x)) = D_{\mu}^{\text{HB}}(F, \log F),
\end{align*}
$$

where the first inequality is the single-spin modified log-Sobolev inequality from Proposition 6.5, and the second inequality is Jensen's inequality, using $\mu(\sigma) = \mathbb{E}_{\nu_{0,\beta}}[\mu_0(\sigma)]$ and that $\Psi$ is concave. The bound for the other term in (6.33) works analogously and gives $4D_{\mu}^{\text{HB}}(\sqrt{F}) \le D_{\mu}^{\text{HB}}(F, \log F)$ instead of $D_{\mu}(\sqrt{F})$ with the standard Dirichlet form.

The conclusion is that the modified log-Sobolev constant for the heat-bath Dirichlet form satisfies exactly the same bound (up to an overall factor 4 from different normalisations):

$$ (6.62) \qquad \text{Ent}_{\mu}(F) \le \frac{1}{2\gamma_0^{\text{HB}}} D_{\mu}^{\text{HB}}(F, \log F), \qquad \frac{1}{\gamma_0^{\text{HB}}} \le 2 + 4 \int_0^{\beta} e^{-2\lambda_t} dt. $$

This strategy can be further generalised to Dirichlet forms which are not concave in the measure, see [16].

**6.3. Applications to conservative dynamics.** The criterion of Theorem 3.6 in principle also applies to dynamics with a conservation law. This is for instance the case for spin models with constrained magnetisation:

$$ (6.63) \qquad \mu_{N,m}(d\varphi) \propto e^{-V_0(\varphi)} d\varphi|_{X_{N,m}}, $$

with $X_{N,m}$ the hyperplane of spins with magnetisation $m \in \mathbb{R}$:

$$ (6.64) \qquad X_{N,m} := \left\{ \varphi \in \mathbb{R}^N : \sum_i \varphi_i = Nm \right\}. $$

For instance, an infinite temperature example is given by $V_0 = \sum_{i=1}^N V(\varphi_i)$ and $V : \mathbb{R} \to \mathbb{R}$ a $C^2$ potential, assumed to be strictly convex outside of a segment for definiteness. The associated conservative dynamics reads:

$$ (6.65) \qquad d\varphi_t = -\nabla V_0(\varphi_t) dt + \sqrt{2} dB_t, $$

where $(B_t)$ is a standard Brownian motion on $X_{N,0}$.

See the forthcoming works [15] and [16] for results in continuous and discrete spin settings, and [13] for such results for the continuum sine-Gordon model.
# A Classical renormalised potential and Hamilton–Jacobi equation

**A.1. Hamilton–Jacobi equation.** In classical field theory, fields are typically minimisers of an action functional S:

$$ (A.1) \qquad \varphi_0 \in \argmin_{\varphi} S(\varphi), \qquad S(\varphi) = \frac{1}{2} |\varphi|^2 + V(\varphi). $$

These can be related to the Hamilton–Jacobi equation

$$ (A.2) \qquad \frac{\partial V_t}{\partial t} = -\frac{1}{2}(\nabla V_t)^2, \qquad V_0(\varphi) = V(\varphi). $$

Indeed, its unique viscosity solution is given by the Hopf–Lax formula [42, Section 3.3.2]:

$$ (A.3) \qquad V_t(\varphi) = \min_{\zeta} \left( V_0(\zeta) + \frac{t}{2} \left| \frac{\varphi - \zeta}{t} \right|^2 \right). $$

In particular, the minimum of the action S is given by $V_1(0)$. Note the analogy with the renormalised potential from (3.11). The constructions of Sections 4 and 5 have classical analogues in which the role of the Polchinski equation is replaced by that of the Hamilton–Jacobi equation, as we will see in this appendix.

Unlike solutions of the Polchinski equation, which are smooth at least for $\dot{C}_t$ nondegenerate and a finite number of variables (as in our discussion), the Hamilton–Jacobi equation can develop shocks and the appropriate weak solutions are not necessarily smooth. However, we can assume that V is locally Lipschitz continuous and that (A.2) holds almost everywhere. We refer to [42, Chapters 3 and 10] for an introduction. In statistical physics, Hamilton–Jacobi equations are well-known to arise in mean-field limits of statistical mechanical models, see [2] and in particular [74] and references for recent work in the context of disordered models, as well as Section A.2 below. Shocks of the Hamilton–Jacobi equations are related to phase transitions. Note that the Polchinski equation in a finite number of variables describes finite systems (rather than limits) and thus has smooth solutions. It therefore provides a complete description of the models (no information is lost) while the Hamilton–Jacobi equations describing mean-field systems are effective equations describing macroscopic information. In the thermodynamic limit, where the number of variables tends to infinity, shocks can also form in the Polchinski equation and then likewise correspond to phase transitions in the statistical mechanical models. We illustrate the above in the simple example of the mean-field Ising model in Section A.2 below.

We now discuss the ‘classical’ analogues of the constructions of Sections 4–5 for the Hamilton–Jacobi equation. Our goal is to emphasise the analogy and to provide a different intuition also for the stochastic constructions, and we will therefore impose convenient regularity assumptions in all statements.

We begin with a classical analogue of Corollary 4.2. Define the classical renormalised action by

$$ (A.4) \qquad S_t(\varphi) = \frac{|\varphi|^2}{2(1-t)} + V_t(\varphi). $$

Minimisers of $S_t$ will take the role of the renormalised measure introduced in (3.13).

**Proposition A.1.** Assume that $(\varphi_t)_{t \in [0,1]}$ is differentiable in $t \in (0, 1)$, that $\varphi_t \to 0$ as $t \to 1$, that V is smooth along $(\varphi_t)_{t \in (0,1)}$, and that $\varphi_t$ is an isolated local minimum of $S_t$ for each $t \in [0, 1)$. Then for $t \in [0, 1)$,

$$ (A.5) \qquad \varphi_t = - \int_t^1 \nabla V_u(\varphi_u) du. $$
Different from the situation in Corollary 4.2, the choice of $\varphi$ is not necessarily unique because $S$ may have multiple minimisers and $V$ need not be globally smooth. Indeed, the equation (A.5) for $\varphi$ is nothing but the equation for the characteristics of Hamilton–Jacobi equation (A.2), see [42, Section 3.3]. These are the curves $(\varphi_t)$ such that $\frac{1}{2}U_t(\varphi_t)^2$ is constant in $t$, where $U_t = \nabla V_t$. Since by the Hamilton–Jacobi equation (A.2),

$$ (A.6) \qquad \frac{\partial U_t}{\partial t} = -(\nabla U_t, U_t), $$

provided $U$ is smooth at $(t, \varphi)$, then

$$ (A.7) \qquad \frac{\partial}{\partial t} U_t(\varphi_t) = -(\nabla U_t(\varphi_t), U_t(\varphi_t)) + (\nabla U_t(\varphi_t), \dot{\varphi}_t) = -(U_t(\varphi_t) - \dot{\varphi}_t, \nabla U_t(\varphi_t)) = 0. $$

*Sketch of Proposition A.1.* Since $\varphi_t \to 0$ as $t \to 1$, the claim is equivalent to proving that

$$ (A.8) \qquad \dot{\varphi}_t = \nabla V_t(\varphi_t). $$

Since $\varphi_t$ minimises $S_t$, it satisfies the Euler–Lagrange equation

$$ (A.9) \qquad \frac{\varphi_t}{1-t} + \nabla V_t(\varphi_t) = \nabla S_t(\varphi_t) = 0. $$

Differentiating this equation in $t$,

$$ (A.10) \qquad \frac{\dot{\varphi}_t}{1-t} + \frac{\varphi_t}{(1-t)^2} + (\partial_t \nabla V_t)(\varphi_t) + \text{Hess } V_t(\varphi_t) \dot{\varphi}_t = 0. $$

Using the Euler–Lagrange equation $\varphi_t/(1-t) = -\nabla V_t(\varphi_t)$ again and $\partial_t \nabla V_t = -\text{Hess } V_t \nabla V_t$ (which follows from (A.2)), we obtain

$$ (A.11) \qquad \frac{\dot{\varphi}_t - \nabla V_t(\varphi_t)}{1-t} - \text{Hess } V_t(\varphi_t) \nabla V_t(\varphi_t) + \text{Hess } V_t(\varphi_t) \dot{\varphi}_t = 0. $$

Therefore

$$ (A.12) \qquad \text{Hess } S_t(\varphi_t)[-\nabla V_t(\varphi_t) + \dot{\varphi}_t] = \left(\frac{1}{1-t} + \text{Hess } V_t(\varphi_t)\right)[-\nabla V_t(\varphi_t) + \dot{\varphi}_t] = 0. $$

Since $\varphi_t$ is an isolated local minimum of $S_t$, the Hessian on the left-hand side is strictly positive definite so that necessarily $\nabla V_t(\varphi_t) - \dot{\varphi}_t = 0$. $\square$

The classical version of Föllmer's problem is the following control problem. We continue to use the convention that the ODE is backwards in time so that the Hamilton–Jacobi equation has initial rather than terminal condition. Suppose that $U = (U_u(\cdot))_{u \in [0,1]}$ are given smooth functions and that $\varphi^U$ solves the classical analogue of (5.2): $\varphi_t^U \to 0$ as $t \to 1$ and

$$ (A.13) \qquad \varphi_t^U = - \int_t^1 U_u(\varphi_u^U) du, $$

with $\varphi_0^U$ an absolute minimum of $S$ (which we recall is the classical analogue of demanding that $\varphi_0^U$ is a random sample from a desired target measure $\nu_0 \propto e^{-\beta S}$). The classical analogues of Theorem 5.1 and Proposition 5.3 are as follows.

**Proposition A.2.** The optimal drift is given by $U = \nabla V$ in the following sense. For any smooth $U$ and associated trajectory $\varphi^U$ that satisfies (A.13),

$$ (A.14) \qquad \frac{1}{2} |\varphi_0^U|^2 \le \frac{1}{2} \int_0^1 |U_u(\varphi_u^U)|^2 du, $$

with equality if $U = \nabla V$.
Comparing with Theorem 5.1, the classical analogue of $\mathbb{H}(\nu_0|\gamma_0)$ is simply $\frac{1}{2}|\varphi_0|^2$ and the classical analogue of the path space entropy $\mathbb{H}(\mathbf{Q}|\mathbf{P})$ is the cost $\frac{1}{2}\int_0^1 |U_u(\varphi_u^U)|^2 du = \frac{1}{2}\int_0^1 |\dot{\varphi}_u^U|^2 du$.

*Sketch.* The proof is analogous to that of Theorem 5.1. Indeed, by the Hopf–Lax formula,

$$ (A.15) \qquad V_1(0) - V_0(\varphi_0^U) \le V_0(\varphi_0^U) + \frac{1}{2}|\varphi_0^U|^2 - V_0(\varphi_0^U) = \frac{1}{2}|\varphi_0^U|^2, $$

with equality if and only if $\varphi_0^U$ is an absolute minimum of $S$. On the other hand, assuming $V$ is locally Lipschitz continuous (so that the fundamental theorem of calculus hold), one has

$$ (A.16) \qquad V_1(0) - V_0(\varphi_0^U) = \int_0^1 dt \frac{\partial}{\partial t} V_t(\varphi_t^U), $$

with $\varphi^U$ evolving according to (A.13), and therefore (whenever the derivative exists classically)

$$ \begin{align} \frac{\partial}{\partial t} V_t(\varphi_t^U) &= \left(\frac{\partial}{\partial t} V_t\right)(\varphi_t^U) + \left(\nabla V_t(\varphi_t^U), U_t(\varphi_t^U)\right) \nonumber \\ &= -\frac{1}{2}(\nabla V_t(\varphi_t^U))^2 + \left(\nabla V_t(\varphi_t^U), U_t(\varphi_t^U)\right) \nonumber \\ (A.17) \qquad &= -\frac{1}{2}(\nabla V_t(\varphi_t^U) - U_t(\varphi_t^U))^2 + \frac{1}{2}(U_t(\varphi_t^U))^2, \end{align} $$

where we used the Hamilton–Jacobi equation on the second line. In particular, if $\varphi_0^U$ is an absolute minimum of $S$,

$$ (A.18) \qquad \frac{1}{2} \int_0^1 (\nabla U_t(\varphi_t^U))^2 dt = \frac{1}{2} |\varphi_0^U|^2 + \frac{1}{2} \int_0^1 (\nabla V_t(\varphi_t^U) - U_t(\varphi_t^U))^2 dt \ge \frac{1}{2} |\varphi_0^U|^2, $$

and the gradient of the renormalised potential $V_t$ provides the optimal drift. $\square$

As in (4.3), instead of (A.13), one can also consider the equations for the characteristics in a reduced time interval $[0, t]$ with $t \le 1$ and $\varphi_t^U = \varphi$,

$$ (A.19) \qquad \varphi_s^U = \varphi - \int_s^t U_u(\varphi_u^U) du, \quad (s \le t). $$

**Proposition A.3.** Under certain regularity conditions on $V_0$, and for $t \le 1$ then

$$ (A.20) \qquad V_t(\varphi) = \inf_U \left[ V_0\left(\varphi - \int_0^t U_s(\varphi_s^U) ds\right) + \frac{1}{2} \int_0^t |U_s(\varphi_s^U)|^2 ds \right]. $$

This is discussed in [46, Section I.9 and I.10]. Namely, with $L(t, x, v) = \frac{1}{2}v^2$ and $\psi = V_0$ (and opposite time direction), the statement follows from [46, Theorem 10.1]. This minimiser does not have to be unique if the Hamilton–Jacobi equation has a shock, see [46, Theorem 9.1 and 10.2].

*Sketch.* The Hopf–Lax formula implies that

$$ (A.21) \qquad V_t(\varphi) = \min_{\zeta} \left( \frac{t}{2} |\varphi - \zeta|^2 + V_0(\zeta) \right) \le \frac{1}{2t} |\varphi - \varphi_0^U|^2 + V_0(\varphi_0^U) $$

where given any drift $U_s$, we let $\varphi_0^U$ be the final condition of (A.19). The first term on the right-hand side of (A.21) is bounded as in (A.14):

$$ (A.22) \qquad \frac{1}{2t} |\varphi - \varphi_0^U|^2 \le \frac{1}{2} \int_0^t |U_s(\varphi_s^U)|^2 ds. $$
This shows

$$
(A.23) \qquad V_t(\varphi) \le \frac{1}{2} \int_0^t |U_s(\varphi_s^U)|^2 ds + V_0\left(\varphi - \int_0^t U_s(\varphi_s^U) ds\right).
$$

On the other hand, equality if $U = \nabla V$ follows from the Hamilton–Jacobi equation as in
Proposition 4.3: if $\varphi$ is a solution to (A.5),

$$
(A.24) \quad \frac{\partial}{\partial s} \left[ V_s(\varphi_s) + \frac{1}{2} \int_s^t (\nabla V_u(\varphi_u))^2 ds \right] = \frac{\partial V_s}{\partial s}(\varphi_s) + (\nabla V_s(\varphi_s))^2 - \frac{1}{2} (\nabla V_s(\varphi_s))^2 = 0,
$$

i.e.,

$$
(A.25) \qquad V_t(\varphi) - V_0(\varphi_0) = \frac{1}{2} \int_0^t (\nabla V_u(\varphi_u))^2 du.
$$

This solution need not be unique.

<figure><img src="image_10.png" /></figure>

**A.2. Example: Mean-field Ising model.** We conclude this section with the example of the mean-field Ising model, which can be described both in terms of a Polchinski equation and, in the limit, by a Hamilton–Jacobi equation. The mean-field Ising model is given by the measure

$$
(A.26) \quad \mathbb{E}_{\nu}[G] = \frac{1}{Z_N(\beta, \mathbf{h})} \sum_{\sigma \in \{\pm 1\}^N} e^{-\frac{\beta}{4N} \sum_{i,j} (\sigma_i - \sigma_j)^2 + (\sigma, \mathbf{h})} G(\sigma),
$$

where the vector $\mathbf{h} \in \mathbb{R}^N$ is a possibly site-dependent external field. It is convenient to rewrite it
as

$$
(A.27) \qquad \mathbb{E}_{\nu}[G] = \frac{1}{Z_N(\beta, \mathbf{h})} \sum_{\sigma \in \{\pm 1\}^N} e^{-\frac{\beta}{2}(\sigma, P\sigma) + (\sigma, \mathbf{h})} G(\sigma),
$$

where $P = \text{id} - Q$ and $Q$ is the orthogonal projection onto constants: $Qf = (\frac{1}{N} \sum_i f_i)\mathbf{1}$ with
$\mathbf{1} = (1, \dots, 1) \in \mathbb{R}^N$. For $\mathbf{h} = h\mathbf{1}$ with $h \in \mathbb{R}$, the free energy $F(\beta, h)$ is the limit $N \to \infty$ of
$F_N(\beta, h)$ where

$$
(A.28) \qquad F_N(\beta, h) = -\frac{1}{N} \log Z_N(\beta, h1).
$$

(Physically more correctly, the right-hand side should have been divided by $\beta$, but it is here more
convenient to omit this.) It is well-known and easy to check that

$$
(A.29) \qquad \frac{\partial F_N}{\partial \beta} = \frac{1}{2N} \frac{\partial^2 F_N}{\partial h^2} - \frac{1}{2} \left(\frac{\partial F_N}{\partial h}\right)^2, \qquad F_N(0, h) = -\log \cosh(h),
$$

and thus that the limiting free energy $F$ is the viscosity solution of the Hamilton–Jacobi equation

$$
(A.30) \qquad \frac{\partial F}{\partial \beta} = -\frac{1}{2} \left(\frac{\partial F}{\partial h}\right)^2, \qquad F(0, h) = -\log \cosh(h),
$$

see in particular [74]. Equivalently, $F$ is given by the Hopf–Lax formula which coincides with the
well-known variational formula for the free energy alternatively obtained from Laplace’s Principle
(see for example [17, Chapter 1] or [85]):

$$
(A.31) \quad F(\beta, h) = \min_{g \in \mathbb{R}} \left[ \frac{1}{2\beta} (g-h)^2 - \log \cosh(g) \right] = \min_{\varphi \in \mathbb{R}} \left[ \frac{\beta}{2} \varphi^2 - \log \cosh(\beta\varphi + h) \right].
$$
This Hamilton–Jacobi equation for the free energy can be related, as follows, to the Polchinski equation studied earlier in Section 6.2.1. Recall that $id = P+Q$ with $Q$ the orthogonal projection onto constant vectors in $\mathbb{R}^N$ and $PQ = 0$. For $\alpha > \beta$, let

$$ (A.32) \qquad C_t = (tP + (\alpha - t))^{-1}, \qquad \dot{C}_t = (\alpha - t)^{-2}Q, $$

where we used $PQ = 0$ to simplify $\dot{C}_t$. For $\varphi \in \mathbb{R}^N$, the renormalised potential (6.23) then is

$$ (A.33) \qquad V_t(\varphi) = -\log \sum_{\sigma \in \{\pm 1\}} e^{-\frac{1}{2}(\sigma - \varphi, (tP + (\alpha - t))(\sigma - \varphi))} + (\text{constant}) $$

and satisfies the Polchinski equation (for appropriate otherwise irrelevant choice of the constants):

$$ (A.34) \qquad \frac{\partial V_t}{\partial t} = \frac{1}{2} \Delta \dot{C}_t V_t - \frac{1}{2} (\nabla V_t)^2 \dot{C}_t = (\alpha - t)^{-2} \left[ \frac{1}{2} \Delta_Q V_t - \frac{1}{2} (\nabla V_t)^2 \right]. $$

Note that the right-hand side only depends on derivatives of $V_t$ in constant directions (i.e., in the image of $Q$). By (A.32), the covariance $C_\beta - C_t$ of the renormalised measure $\nu_t = \nu_{t,\beta}$ defined in (6.26) is also proportional to $Q$ and therefore supported on constant fields. Thus one can restrict the renormalised potential $V_t$ to constant fields and the restriction satisfies a closed equation. Explicitly, for $\tilde{\varphi} \in \mathbb{R}$, define $\tilde{V}(\tilde{\varphi}) = \frac{1}{N} V(\tilde{\varphi} \mathbf{1})$. In other words, $V_t(\varphi) = N \tilde{V}_t(Q\varphi) = N \tilde{V}_t(\frac{1}{N} \sum_i \varphi_i)$ holds for constant fields $\varphi = Q\varphi$ and

$$ \begin{align} \frac{\partial \tilde{V}_t}{\partial t}(\tilde{\varphi}) &= \frac{1}{N} \frac{\partial V_t}{\partial t}(\tilde{\varphi} \mathbf{1}) = (\alpha - t)^{-2} \frac{1}{N} \left[ \frac{1}{2} \Delta_Q V_t - \frac{1}{2} (\nabla_Q V_t)^2 \right] (\tilde{\varphi} \mathbf{1}) \nonumber \\ &= (\alpha - t)^{-2} \left[ \frac{1}{2} \frac{\partial V_t}{\partial \varphi_1^2} - \frac{1}{2} (\frac{\partial V_t}{\partial \varphi_1})^2 \right] (\tilde{\varphi} \mathbf{1}) \nonumber \\ (A.35) \qquad &= (\alpha - t)^{-2} \left[ \frac{1}{2N} \tilde{V}_t'' - \frac{1}{2} (\tilde{V}_t')^2 \right] (\tilde{\varphi}). \end{align} $$

Thus the reduced (one-variable) Polchinski equation that $\tilde{V}_t$ satisfies has a prefactor $1/N$ in front of the Laplacian term, and its limit $\overline{V}_t$ as $N \to \infty$ is the unique viscosity solution of the following (one-variable) Hamilton–Jacobi equation (now dropping tilde from the variable $\varphi$):

$$ (A.36) \qquad \frac{\partial \overline{V}_t}{\partial t}(\varphi) = -\frac{1}{2}(\alpha - t)^{-2}(\overline{V}_t'(\varphi))^2, \qquad \overline{V}_0(\varphi) = \frac{\alpha}{2}\varphi^2 - \log \cosh(\alpha\varphi), \qquad (\varphi \in \mathbb{R}). $$

To conclude this section, we relate the Hamilton–Jacobi equation (A.36) for the effective potential to the one satisfied by the free energy (A.30). We follow the argument in Example 3.14 and first note that for a constant field $\varphi \mathbf{1}$ with $\varphi \in \mathbb{R}$, the renormalised potential can be written as (see (3.68))

$$ (A.37) \qquad V_t(\varphi \mathbf{1}) = N \left[ \frac{\alpha - t}{2} \varphi^2 + F_N(t, (\alpha - t)\varphi) \right]. $$

Thus $F(t, h) = -\frac{1}{2}(\alpha - t)^{-1}h^2 + \overline{V}_t((\alpha - t)^{-1}h)$ and the Hamilton–Jacobi equation (A.30) for $F$ follows from the one of $\overline{V}$ exactly as in Example 3.14. Indeed, in the setting of that example with the relation (3.68) between $V_t$ and $F_t$ one has in general that

$$ (A.38) \qquad \frac{\partial}{\partial t} V_t = -\frac{1}{2} (\nabla V_t)^2 \dot{C}_t \quad \Leftrightarrow \quad \frac{\partial}{\partial t} F_t = -\frac{1}{2} (\nabla F_t)^2 \dot{\Sigma}_t, $$

and in the present example the choice of $C_t$ corresponds to $\dot{\Sigma}_t = Q$.
# Acknowledgements

This work was supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 851682 SPINRG).

We thank D. Chafai, R. Eldan, N. Gozlan, J. Lehec, Y. Shenfeld, and H.-T. Yau for various discussions related to the material of this introduction, encouragement, and for pointing out several additional references.

We thank the organisers of the following summer schools at which some of the material was presented: the One World Probability Summer School on “PDE and Randomness” in Bath/Zoom organised by Hendrik Weber and Andris Gerasimovics; the “Summer School on SPDE and Related Fields” in Beijing/Zoom organised by Hao Shen, Scott Smith, Rongchan Zhu, and Xiangchan Zhu; and the Summer School on “PDE and Randomness” at the Max Planck Institute for Mathematics in the Sciences organised by Rishabh Gvalani, Francesco Mattesini, Felix Otto, and Markus Tempelmayr. In particular, we also thank Jiwoon Park for leading the exercise classes at the last summer school.

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