# Universal cumulants of the current in diffusive systems on a ring

Cécile Appert-Rolland, Bernard Derrida, Vivien Lecomte, Frédéric van Wijland

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Cécile Appert-Rolland, Bernard Derrida, Vivien Lecomte, Frédéric van Wijland. Universal cumulants of the current in diffusive systems on a ring. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics [2001-2015], 2008, 78 (2), pp.021122. 10.1103/PhysRevE.78.021122. hal-00273794

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Universal cumulants of the current
in diffusive systems on a ring

C. Appert-Rolland$^{(1)}$, B. Derrida$^{(2)}$, V. Lecomte$^{(3,4)}$ and F. van Wijland$^{(3)}$

$^{(1)}$Laboratoire de Physique Théorique (CNRS UMR 8627),
Université Paris-Sud, bâtiment 210, 91405 Orsay, France

$^{(2)}$Laboratoire de Physique Statistique (CNRS UMR 8550),
École Normale Supérieure, 24 rue Lhomond, 75231 Paris cedex 05, France

$^{(3)}$Laboratoire Matière et Systèmes Complexes (CNRS UMR 7057),
10 rue Alice Domon et Léonie Duquet, Université Paris VII, 75205 Paris cedex 13, France and

$^{(4)}$Département de Physique de la Matière Condensée,
Université de Genève, 24 quai Ernest Ansermet, 1211 Genève, Switzerland

We calculate exactly the first cumulants of the integrated current and of the activity (which is the total number of changes of configurations) of the symmetric simple exclusion process (SSEP) on a ring with periodic boundary conditions. Our results indicate that for large system sizes the large deviation functions of the current and of the activity take a universal scaling form, with the same scaling function for both quantities. This scaling function can be understood either by an analysis of Bethe ansatz equations or in terms of a theory based on fluctuating hydrodynamics or on the macroscopic fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim.

PACS numbers: 82.70.Dd,64.70.Dv

# I. INTRODUCTION

The symmetric simple exclusion process (SSEP) [1, 2, 3, 4] is one of the simplest lattice gas models studied in the theory of non-equilibrium systems. It consists of hard-core particles hopping with equal rates to either of their nearest neighbor sites, on a regular lattice. At equilibrium, when isolated, the system reaches in the long time limit an equilibrium where all accessible configurations are equally likely. Also, when equilibrium is achieved by contact with one or several reservoirs at a single density $ρ$, all sites are occupied with this density $ρ$ and the occupation numbers of different sites are uncorrelated.

As soon as the system is maintained out of equilibrium, by contact with reservoirs at unequal densities, there is a current of particles and one observes long range correlations in the steady state [5]. In this out of equilibrium case several approaches have been developed to calculate steady state properties, such as the fluctuations or the large deviations of the density or of the current [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18].

A lot of progress has been made over the last decades on the study of the fluctuations and the large deviation functions of the current in equilibrium or non equilibrium systems. The large deviation function of the current can be viewed as the dynamical analog of a free energy, as discussed by Ruelle in the early seventies [19]. The idea back then was to build up a thermodynamic formalism based upon probabilities over time realizations rather than over instantaneous configurations. Generic properties of these large deviation functions were later discovered such as the fluctuation theorem which determines how the large deviation function of the current is changed under time reversal symmetry [20, 21, 22, 23, 24, 25, 26, 27, 28].

In the present work, we obtain exact expressions for the first cumulants of the integrated current and of the activity (which is the number of changes of configurations) during a long time

$t$ for the SSEP consisting of $N$ particles on a ring of $L$ sites. For large system sizes, these cumulants and the associated large deviation functions take universal scaling forms. We show how these scaling forms can be calculated for the SSEP by the Bethe ansatz or for more general diffusive systems on a ring by a theory based on fluctuating hydrodynamics or on the macroscopic fluctuation theory developed by Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim [9, 10, 16, 17, 18]. In the Bethe ansatz approach these scaling forms can be extracted from a detailed analysis of finite size effects similar to what was developed recently for quantum spin chains in the context of string theory [29, 30]. In the fluctuating hydrodynamics approach, it results from the discreteness of the wave vectors of the fluctuating modes on the ring.

Universal distributions of the current characteristic of the universality class of the KPZ (Kardar-Parisi-Zhang) equation [31, 32, 33, 34], have been calculated in the past [35, 36, 37, 38] for the asymmetric exclusion process (ASEP). The distributions obtained in the present paper are different and belong to the Edwards-Wilkinson universality class [39].

We begin by presenting in Sec.II exact expressions of the first cumulants of the current and of the activity for the SSEP on a ring. This is where we see that the cumulants of the integrated current and of the activity take scaling forms when the size of the ring becomes large and where emerges the idea that the large deviation function of the current and of the activity obey the same universal scaling function. This is confirmed in Sec.III by Bethe ansatz calculations. By resorting to fluctuating hydrodynamics in Sec.IV we are able to formulate the particular case of the SSEP within a more general framework using the Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim approach and to show that the same universal distribution of the current fluctuations of the current are present in a larger family of diffusive systems.
## II. EXACT EXPRESSIONS OF THE FIRST CUMULANTS

We consider a system of $N$ particles on a one-dimensional lattice of $L$ sites with periodic boundary conditions. Each site is either empty or occupied by a single particle. A microscopic configuration $\mathcal{C} = \{n_i\}_{i,1,...,L}$ can be specified by $L$ occupation numbers $n_i$ (where $n_i = 1$ if site $i$ is occupied and $n_i = 0$ if site $i$ is empty). In the simple symmetric exclusion process, SSEP, each particle hops to its right neighbor at rate 1 or to its left neighbor at rate 1, provided the target site is empty. In the present paper we try to determine the distribution of the total integrated current $Q(t)$ and of the total number $K(t)$ of changes of configuration (that we will call the activity [40]) during a time interval $(0, t)$. To do so we define the generating functions of the cumulants of $Q$ and $K$ as

$$ \psi_Q(s) = \lim_{t \to \infty} \frac{\ln\langle e^{-sQ} \rangle}{t}, \quad \psi_K(s) = \lim_{t \to \infty} \frac{\ln\langle e^{-sK} \rangle}{t}, \quad (1) $$

where the brackets denote an average over the time evolutions during the time interval $(0, t)$. As the evolution is an irreducible Markov process with a finite number of states, the long time limits in (1) do not depend on the initial configuration and the generating functions defined in (1) can be calculated as the largest eigenvalue of a matrix [20, 36, 41].

Because the calculations are very similar for both observables $K$ and $Q$, we shall first focus on the activity $K$ and explain how to calculate the cumulant generating function $\psi_K(s)$ as a perturbation series in powers of $s$. We will then present only the results for $\psi_Q(s)$.

### A. The cumulants of the activity $K(t)$

In order to determine $\psi_K$, as in [36], one can write a Master equation for the probability $P(\mathcal{C}, K, t)$ to find the system in configurations $\mathcal{C}$ at time $t$, given that the activity at time $t$ is $K$ (i.e. given that the system has changed $K$ times of configurations during the time interval $(0, t)$).

$$ \partial_t P(\mathcal{C}, K, t) = -r(\mathcal{C})P(\mathcal{C}, K, t) + \sum_{C'} W(C' \to C)P(C', K-1, t) \quad (2) $$

where $W(C \to C')$ is the transition rate from configuration $C$ to $C'$, and $r(\mathcal{C}) = \sum_{C'} W(C \to C')$ is the escape rate from configuration $C$.

If one introduces the generating function $\hat{P}(\mathcal{C}, s, t) = \sum_K e^{-sK} P(\mathcal{C}, K, t)$, its evolution satisfies

$$ \partial_t \hat{P}(\mathcal{C}, s, t) = \sum_{C'} \mathbb{W}_K(\mathcal{C}, C') \hat{P}(C', s, t) \quad (3) $$

where

$$ \mathbb{W}_K(\mathcal{C}, C') = e^{-s}W(C' \to C) - r(\mathcal{C})\delta_{C,C'}. \quad (4) $$

In the long time limit, $\hat{P}(\mathcal{C}, s, t)$ grows (or decays) exponentially with time, with a rate given by the eigenvalue with largest real part [36] of the modified matrix $\mathbb{W}_K$. Thus $\psi_K(s)$

can be calculated as this largest eigenvalue of $\mathbb{W}_K$. For $s = 0$, $\mathbb{W}_K$ reduces to the evolution operator of the Master equation $W$ for the symmetric simple exclusion process, and this largest eigenvalue (which is 0) as well as the related eigenvector are known. We now present a way of obtaining the large deviation function $\psi_K$, by a perturbative expansion [41, 42] in powers of $s$.

The idea is to start from the eigenvalue equation for $\psi_K$ and its eigenvector $\tilde{P}$,

$$ \psi_K(s) \tilde{P}(\mathcal{C}, s) = \sum_{C'} \mathbb{W}_K(\mathcal{C}, C') \tilde{P}(C', s) \quad (5) $$

normalized such that $\sum_C \tilde{P}(\mathcal{C}, s) = 1$. One can then define the average $\langle A(\mathcal{C}) \rangle_s$ of an observable $A(\mathcal{C})$ in the corresponding eigenstate, (i.e. $\langle A(\mathcal{C}) \rangle_s = \sum_C A(\mathcal{C}) \tilde{P}(\mathcal{C}, s)$ and this is the same as averaging, in the limit of a long time interval $(0, t)$, over all trajectories weighted by a coefficient $e^{-sK(t)}$). Note that, though the value of $K(t)$ is defined on trajectories running from 0 to $t$, the observable $A(\mathcal{C})$ is evaluated at the final time $t$. From the eigenvalue equation, one gets

$$ \psi_K(s) \langle A(\mathcal{C}) \rangle_s = e^{-s} \left\langle \sum_{C'} W(\mathcal{C} \to C') A(C') \right\rangle_s - \langle A(\mathcal{C}) r(\mathcal{C}) \rangle_s \quad (6) $$

where the escape rate $r(\mathcal{C})$ is twice the number of clusters of adjacent particles in the system

$$ r(\mathcal{C}) = \sum_{C'} W(\mathcal{C} \to C') = 2 \sum_{j=1}^{L} n_j (1 - n_{j+1}). \quad (7) $$

Choosing $A(\mathcal{C}) = 1$ in (6) leads to

$$ \psi_K(s) = (e^{-s} - 1) \langle r(\mathcal{C}) \rangle_s = 2L(e^{-s} - 1)(\rho - C_s(1)) \quad (8) $$

where $C_s(r) = \langle n_i n_{i+r} \rangle_s$ is the correlation function (which by translational invariance does not depend on $i$) computed within the eigenstate $\tilde{P}(\mathcal{C}, s)$, and $\rho = N/L$ is the average density.

For the leading contribution as $s \to 0$, we can use the fact that at $s = 0$ the eigenvector is known (this is the equilibrium distribution, for which all allowed microscopic configurations are equally likely), so that $\psi_K(s) = -2N \left(1 - \frac{N-1}{L-1}\right) s + O(s^2)$. In order to compute the $O(s^2)$ contribution from (8), we need to evaluate $C_s(1)$ at order $s$, which can be done by choosing $A(\mathcal{C}) = n_i n_j$ in (6). This requires the knowledge of the correlation function $C_s(r) = \langle n_i n_{i+r} \rangle_s$ at order $O(s)$. For $A(\mathcal{C}) = n_i n_j$ in (6) one gets

$$ C_s(1) - C_s(2) = sA_{N,L} + O(s^2) $$

$$ \text{where } A_{N,L} = \frac{N(N-1)(L-N)(L-N-1)}{L(L-1)^2(L-2)} $$

$$ C_s(r+1)+C_s(r-1)-2C_s(r)=s\frac{2A_{N,L}}{L-3}+O(s^2) \quad (9) $$

for $2 \le r \le L-2$,
which have the following solution

$$C_s(r) = \frac{N(N-1)}{L(L-1)} - sA_{N,L} \frac{6r(L-r) - L(L+1)}{6(L-3)} + \mathcal{O}(s^2). \quad (10)$$

We can therefore extract $\psi_K$ up to $\mathcal{O}(s^2)$ and $\langle K^2 \rangle_c/t$ follows.

To obtain higher cumulants, we have repeated the same procedure, with the observables $\mathcal{A}(C) = n_i n_j n_k$ and $\mathcal{A}(\mathcal{C}) = n_i n_j n_k n_l$. The calculations are longer but very similar. We

found that the first cumulants of $K$, $\lim_{t \to \infty} \langle K^n \rangle_c/t = (-1)^n \frac{d^n \psi_K}{ds^n}|_{s=0}$, when expressed in terms of the system size $L$ and of

$$\sigma(\rho) = 2\rho(1 - \rho) = \frac{2N(L - N)}{L^2} \quad (11)$$

are given by (in the $t \to \infty$ limit)

$$\begin{gathered} \frac{\langle K \rangle}{t} = L^2 \frac{\sigma}{L-1}, \qquad \frac{\langle K^2 \rangle_c}{t} = \frac{L^2 \sigma (L^2 \sigma + 4L - 4)}{6(L-1)^2} \\ \frac{\langle K^3 \rangle_c}{t} = \frac{L^2 \sigma [-L^5 \sigma^2 + L^4 \sigma (2 + 3\sigma) - 2L^3 \sigma + 48(L-1)^2]}{60(L-1)^3} \\ \frac{\langle K^4 \rangle_c}{t} = L^2 \sigma (\sigma^3 L^6 (10L^3 - 70L^2 + 175L - 153) - 4\sigma^2 L^4 (L-1)(11L^3 - 69L^2 + 154L - 126) \\ + 16\sigma L^2 (L-1)^2 (3L^3 - 17L^2 + 46L - 63) + 2112(L-1)^3 (L-3)) (2520(L-1)^4 (L-3))^{-1}. \end{gathered} \quad (12)$$

When $L$ becomes large, while $\rho = N/L$ is kept fixed, the asymptotic behavior of the above cumulants reads

$$\begin{gathered} \frac{\langle K \rangle}{t} \approx \sigma L, \qquad \frac{\langle K^2 \rangle_c}{t} \approx \frac{\sigma^2}{6} L^2, \quad (13) \\ \frac{\langle K^3 \rangle_c}{t} \approx -\frac{\sigma^3}{60} L^4, \qquad \frac{\langle K^4 \rangle_c}{t} \approx \frac{\sigma^4}{252} L^6 \end{gathered}$$

One might have expected the derivatives at $s=0$ of the eigenvalue $\psi_K$ to become extensive for a large system size $L$ (after all, as we shall see in section III, it is always possible to view $\psi_K$ as the ground state energy of a short range Hamiltonian). Yet this is not the case since the second and higher cumulants grow faster than linearly with $L$ at fixed density $\rho$. This suggests that, in the large $L$ limit, $\psi_K/L$ becomes a singular function of $s$ at $s=0$.

Also one can guess from (13) that for $n \ge 2$

$$\frac{\langle K^n \rangle_c}{t} \sim \sigma^n L^{2n-2}$$

and that for $L \to \infty$ and $s \to 0$, the eigenvalue $\psi_K$ takes a scaling form

$$\lim_{L \to \infty} L^2 \left[ \psi_K(s) + s \frac{\langle K \rangle}{t} \right] = \mathcal{F}_K \left( \frac{\sigma}{2} L^2 s \right) \quad (14)$$

where the scaling function $\mathcal{F}_K$ is given by

$$\mathcal{F}_K(u) = \frac{1}{3}u^2 + \frac{1}{45}u^3 + \frac{1}{378}u^4 + \mathcal{O}(u^5). \quad (15)$$

We shall see in sections III and IV that this scaling function can be fully determined and written as

$$\mathcal{F}_K(u) = -4 \sum_{n \ge 1} \left[ n\pi \sqrt{n^2\pi^2 - 2u} - n^2\pi^2 + u \right] \quad (16)$$

or equivalently (see appendix A) as

$$\mathcal{F}_K(u) = \sum_{k \ge 2} \frac{B_{2k-2}}{(k-1)! k!} (-2u)^k \quad (17)$$

where the Bernoulli numbers $B_n$ are known to be simply the coefficients of the expansion $x(e^x - 1)^{-1} = \sum_n B_n x^n/n!$. As a consequence, the generalization of (13) will be for $n \ge 2$

$$\frac{\langle K^n \rangle_c}{t} \approx \frac{B_{2n-2}}{(n-1)!} \sigma^n L^{2n-2}. \quad (18)$$

## B. The cumulants of the current

The same procedure can be followed for the total integrated current $Q$ (which can be defined by $Q = \sum_{j=1}^N x_j(t)$ where $x_j(t)$ is the total displacement of the $j$th particle during the time interval $(0, t)$). Its cumulant generating function $\psi_Q$ defined in (1) is the eigenvalue (with largest real part) of the matrix

$$\mathbb{W}_Q(C, C') = W(C' \to C)e^{-sj(C', C)} - r(C)\delta_{C, C'} \quad (19)$$

where $j(C', C)$ is +1 or -1 depending on whether a particle has moved to the right or to the left when the system jumps from configuration $C'$ to configuration $C$. Using an expansion in powers of $s$ as in II A we have obtained (in the limit $t \to \infty$)
$$ \frac{\langle Q^2 \rangle}{t} = \frac{L^2 \sigma}{L-1}, \qquad \frac{\langle Q^4 \rangle_c}{t} = \frac{1}{2} \frac{L^4 \sigma^2}{(L-1)^2} \qquad (20) $$

$$ \frac{\langle Q^6 \rangle_c}{t} = - \frac{L^6 \sigma^2 ((L^2 - L + 2)\sigma - 2(L-1))}{4(L-1)^3(L-2)} $$ $$ \frac{\langle Q^8 \rangle_c}{t} = \frac{L^8 \sigma^2 ((10L^4 - 2L^3 + 27L^2 - 15L + 18)\sigma^2 - 4(L-1)(11L^2 - L + 12)\sigma + 48(L-1)^2)}{24(L-1)^4(L-2)(L-3)} \qquad (21) $$

with the corresponding large L behaviors (for $\rho = N/L$ fixed)

$$ \frac{\langle Q^2 \rangle_c}{t} \approx \sigma L, \quad \frac{\langle Q^4 \rangle_c}{t} \approx \frac{\sigma^2}{2} L^2, $$ $$ \frac{\langle Q^6 \rangle_c}{t} \approx -\frac{\sigma^3}{4} L^4, \quad \frac{\langle Q^8 \rangle_c}{t} \approx \frac{5\sigma^4}{12} L^6. \qquad (22) $$

As for K, these results indicate that for $n \ge 2$

$$ \frac{\langle Q^{2n} \rangle_c}{t} \sim \sigma^n L^{2n-2} $$

and that $\psi_Q$ takes a scaling form, in the limit $L \to \infty$ and $s \to 0$

$$ \lim_{L \to \infty} L^2 \left[ \psi_Q(s) - \frac{s^2}{2} \frac{\langle Q^2 \rangle_c}{t} \right] = \mathcal{F}_Q \left( -\frac{\sigma}{4} L^2 s^2 \right) \qquad (23) $$

where, according to (22), the expansion of $\mathcal{F}_Q(u)$ in powers of $u$ coincides with the expansion (15) of $\mathcal{F}_K(u)$, at least up to the 4th order in $u$.

We will see, in section IV, that these two scaling functions (which appear in (14) and in (23)) are in fact the same. Therefore the formula which generalizes (22) will be for $n \ge 2$

$$ \frac{\langle Q^{2n} \rangle_c}{t} \approx \frac{(2n)! B_{2n-2}}{2^n (n-1)! n!} \sigma^n L^{2n-2}. \qquad (24) $$

### III. BETHE ANSATZ

It is well known that the Bethe ansatz allows one to calculate the eigenvalues of matrices such as $W_K(C, C')$ and $W_Q(C, C')$ defined in (4,19) for exclusion processes [35, 36, 37, 38, 43, 44, 45, 46, 47, 48, 49, 50]. In this section we show how to obtain the scaling forms (14,23) from the Bethe ansatz equations.

#### A. Relation to spin chains

It is possible to write the matrices $W_K(C, C')$ and $W_Q(C, C')$ as quantum spin-chain Hamiltonians [51]. We use the correspondence in which the $z$ component of a two state spin operator is up when a particle is present at site $i$, and is

down otherwise. In this basis one finds that

$$ \hat{H}_K = \frac{L}{2} - \frac{1}{2} \sum_{i=1}^{L} \left[ e^{-s} (\sigma_i^x \sigma_{i+1}^x + \sigma_i^y \sigma_{i+1}^y) + \sigma_i^z \sigma_{i+1}^z \right] = -W_K $$ $$ \hat{H}_Q = \frac{L}{2} - \frac{1}{2} \sum_{i=1}^{L} \left[ \cosh s (\sigma_i^x \sigma_{i+1}^x + \sigma_i^y \sigma_{i+1}^y) + \sigma_i^z \sigma_{i+1}^z \right] \qquad (25) $$ $$ -i \sinh s (\sigma_i^x \sigma_{i+1}^y - \sigma_i^y \sigma_{i+1}^x) \right] = -W_Q $$

where we have resorted to the Pauli matrices $\sigma_i^{x,y,z}$. In this language, the quantities $\psi_K$ and $\psi_Q$ are the ground state energies of these operators. It also suggests that the methods of one-dimensional exactly solvable models apply in our case, such as the Bethe ansatz, as was exploited for similar systems in the past [36, 52, 53].

As the number of particles on the ring is fixed, we need to find the ground state with a fixed particle density $\rho$, that is, at fixed transverse magnetization $\sum_i \sigma_i^z$. The quantum operators appearing in (25) have of course been extensively studied [54], including within the framework of stochastic dynamics [55]. For instance, following the notations of Baxter [54] the operator $e^{s\hat{H}_K}$ is the ferromagnetic XXZ chain with anisotropy parameter $\Delta = e^s$. Similarly, $\hat{H}_Q$ corresponds to an XXZ chain with additional Dzyaloshinskii-Moriya interactions. A study of an operator closely related to $\hat{H}_Q$ was carried out by Kim [52] in 1995. His results will be recalled at the end of the present section.

The Bethe ansatz consists in looking for the ground state of $\hat{H}_{K \text{ or } Q}$ in the form of a linear combination of $N$-particle plane waves (see [43, 52]). We denote by $\{x_j\}_{j=1,...,N}$ the positions of the $N$ particles and we postulate that the right eigenvector of $W_K$ can be cast in the form

$$ P(\{x_j\}, s) = \sum_{\mathcal{P}} \mathcal{A}(\mathcal{P}) \prod_{i=j}^{N} [\zeta_{p(j)}]^{x_j} \qquad (26) $$

where $\mathcal{P} = (p(1), \dots, p(N))$ is a permutation over the first $N$ integers, and the $\zeta_j$'s are a priori complex numbers. This is an exact eigenstate provided these parameters satisfy the so-called Bethe equations. These take different forms for $K$ and $Q$. We now discuss how to implement the Bethe ansatz to calculate $\psi_K(s)$ and $\psi_Q(s)$ defined in (1). Technical details have been gathered in the appendices.
## B. Bethe ansatz for K

For the expression (26) to be an eigenvector of $\hat{H}_K$ or $W_Q$ the $\zeta_j$'s have to satisfy a number of constraints [56], the so-called Bethe (see for example [49]) equations

$$ \zeta_i^L = \prod_{\substack{j=1 \\ j \neq i}}^{N} \left[ -\frac{1 - 2e^s \zeta_i + \zeta_i \zeta_j}{1 - 2e^s \zeta_j + \zeta_i \zeta_j} \right], \quad (27) $$

The expression of $\psi_K(s)$ is given by

$$ \psi_K(s) = e^{-s} \sum_{j=1}^{N} \left( \zeta_j + \frac{1}{\zeta_j} \right) - 2N \quad (28) $$

Our goal is to obtain (14) from (27) and (28) in the double limit $s \to 0$ and $L \to \infty$ keeping $sL^2$ and $N/L = \rho$ fixed. Because of the particle-hole symmetry the discussion below is limited to the case $\rho \le \frac{1}{2}$.

In the large $L$ limit, the $\zeta_j$'s accumulate on a curve which depends on $s$ and as $s \to 0^-$ becomes a finite arc of the unit circle (see [54, 56] and references therein). Note however that the $s > 0$ case can be approached by similar methods.

If one writes

$$ e^s = \cos \delta \quad (29) $$

and

$$ \zeta_j = e^{ik_j \delta} \quad (30) $$

(27) becomes

$$ k_i = \frac{1}{L} \sum_{\substack{j=1 \\ j \neq i}}^{N} U(k_i, k_j) \quad (31) $$

where

$$ U(k_i, k_j) = \frac{1}{i\delta} \ln \left[ \frac{1 - 2e^{ik_i\delta} \cos\delta + e^{i(k_j+k_i)\delta}}{1 - 2e^{ik_j\delta} \cos\delta + e^{i(k_j+k_i)\delta}} \right] \quad (32) $$

In the limit $\delta \to 0$, one can check that when $k_i - k_j = \mathcal{O}(1)$

$$ U(k_i, k_j) = 2 \frac{1 - k_i k_j}{k_i - k_j} + \mathcal{O}(\delta^2) . \quad (33) $$

In the large $L$ limit, however, the distance between consecutive $k_i$ becomes of order $1/L \sim \delta$ and for $i-j$ of order 1 one should use instead

$$ U(k_i, k_j) = \frac{1}{i\delta} \ln \left[ \frac{k_i - k_j + i\delta(1 - k_i^2) + i\delta k_i(k_i - k_j) - \delta^2 k_i(1 - k_i^2)}{k_i - k_j - i\delta(1 - k_i^2) - i\delta k_i(k_i - k_j) + \delta^2 k_i(1 - k_i^2)} \right] \quad (34) $$

Therefore one can rewrite (31) as

$$ Lk_i \approx \sum_{\substack{j \le n_0 \\ j \ne i}} \frac{1}{i\delta} \ln \left[ \frac{k_i - k_j + i\delta(1 - k_i^2) + i\delta k_i(k_i - k_j) - \delta^2 k_i(1 - k_i^2)}{k_i - k_j - i\delta(1 - k_i^2) - i\delta k_i(k_i - k_j) + \delta^2 k_i(1 - k_i^2)} \right] + \sum_{j \notin [i-n_0, i+n_0]} 2 \frac{1 - k_i k_j}{k_i - k_j} \quad (35) $$

where $n_0$ is a fixed large number $1 \ll n_0 \ll L$, so that one can use expression (33) for $|j-i| > n_0$ and (34) for $|j-i| \le n_0$. As shown in appendix B, the two sums (89,100) in (35) depend on the cut-off $n_0$ but this dependence disappears when the two terms in the right hand side of (35) are added.

In the large $L$ limit, the $k_i$ become dense on an interval $(-\theta, \theta)$ of the real axis, with some density $g(k)$. In what follows we will assume that the $k_i$ are regularly spaced according to this density, meaning that

$$ L \int_{k_i}^{k_j} g(k) dk = j - i \quad \text{and} \quad L \int_{-\theta}^{\theta} g(k) dk = N. \quad (36) $$

Replacing the two sums in (35) by their expressions (89,100) obtained in Appendix B, one gets that for $k = k_i$ the density

$g(k)$ should satisfy

$$ k = 2P \int_{-\theta}^{\theta} dk' g(k') \frac{1 - k'^2}{k - k'} \frac{1}{L} \left[ \left( \frac{g'(k)(1 - k^2)}{g(k)} - 2k \right) \times \pi(1 - k^2) g(k) L \delta \coth[\pi(1 - k^2) g(k) L \delta] \right] \quad (37) $$

If we make the change of variable $k' = \theta y$, $k = \theta x$, and

$$ g(k)(1 - k^2) = \phi(x) \quad (38) $$

equation (37) becomes

$$ P \int_{-1}^{1} dy \frac{\phi(y)}{y - x} = f(x) \quad (39) $$
where

$$f(x) = -\frac{\theta x}{2} + \frac{\pi(1 - \theta^2 x^2)\phi'(x)}{2\theta}\delta \coth[L\delta\pi\phi(x)] + \dots \quad (40)$$

As explained in (101,102) of Appendix C one can invert (39) and express $\phi(x)$ in terms of $f(x)$

$$\phi(x) = \frac{C}{\sqrt{1-x^2}} - \frac{1}{\pi^2\sqrt{1-x^2}} \mathcal{P} \int_{-1}^{1} \frac{\sqrt{1-y^2}}{y-x} f(y) dy \quad (41)$$

where the constant C is so far an arbitrary constant.

For small $\delta$, one can write (28), using (30,38,41), as

$$\begin{aligned} \psi_K(s) &\approx \sum_{j=1}^{N} \delta^2(1-k_i^2) \approx L\delta^2 \int_{-\theta}^{\theta} g(k)(1-k^2)dk \\ &= L\delta^2\theta \left[ \int_{-1}^{1} \frac{dx}{\sqrt{1-x^2}} \frac{C}{\pi^2\sqrt{1-x^2}} - \frac{1}{\pi^2\sqrt{1-x^2}} \mathcal{P} \int_{-1}^{1} \frac{\sqrt{1-y^2}}{y-x} f(y)dy \right] \end{aligned}$$

which gives using (113,115)

$$\psi_K(s) \approx L\delta^2\theta C\pi \quad (42)$$

Also, as (36)

$$\int_{-\theta}^{\theta} g(k)dk = \rho$$

one has (38,41)

$$\rho = \theta \int_{-1}^{1} dx \left[ \frac{C}{(1 - \theta^2 x^2)\sqrt{1 - x^2}} - \frac{1}{\pi^2(1 - \theta^2 x^2)\sqrt{1 - x^2}} \mathcal{P} \int_{-1}^{1} \frac{\sqrt{1 - y^2}}{y - x} f(y) dy \right]$$

which can be simplified using (116,120)

$$\rho = \frac{C\theta\pi}{\sqrt{1-\theta^2}} + \frac{\theta^3}{\pi\sqrt{1-\theta^2}} \int_{-1}^{1} \frac{f(y)y\sqrt{1-y^2}}{1-\theta^2 y^2} dy. \quad (43)$$

### C. The leading order in the large $L$ limit

For large $L$ (at fixed $L\delta$), (40) reduces to $f(x) = -\theta x/2$, so that (41) becomes to leading order using (110)

$$\phi(x) = \frac{4\pi C - \theta}{4\pi\sqrt{1-x^2}} + \frac{\theta}{2\pi}\sqrt{1-x^2} + \mathcal{O}\left(\frac{1}{L}\right) \quad (44)$$

whereas (43) becomes using (120)

$$\rho = \frac{C\theta\pi}{\sqrt{1-\theta^2}} + \frac{1}{2} + \frac{\theta^2 - 2}{4\sqrt{1-\theta^2}} \quad (45)$$

Therefore for a fixed density $\rho$ of particles, the constant $C$ in (41,44) and the eigenvalue (42) are given, to leading order in $1/L$, by

$$C = \frac{1}{\pi\theta} \left[ \left( \rho - \frac{1}{2} \right) \sqrt{1 - \theta^2} + \frac{2 - \theta^2}{4} \right] \quad (46)$$

and

$$\psi_K(s) = L\delta^2 \left[ \left( \rho - \frac{1}{2} \right) \sqrt{1 - \theta^2} + \frac{2 - \theta^2}{4} \right] \quad (47)$$

So far, the constant $C$ remains undetermined.

The leading order corresponds to using expression (33) in (31) even when $i$ and $j$ differ by a few units. For the continuum description to be valid, we are now going to argue that $\phi(x)$ should remain finite as $x \to \pm 1$, or, in terms of the original density $g$, that $g(k)$ remains finite as $k \to \pm \theta$. This will impose (see (44)) that

$$C = \frac{\theta}{4\pi}.$$

Indeed if we order the $N$ solutions $k_i$ and focus on the ones closest to $\theta$, ..., $k_{N-1} < k_N \le \theta$, then we may estimate using (36) the difference between $k_N$ and $\theta$, or between $k_{N-1}$ and $k_N$. If $C \ne \frac{\theta}{4\pi}$, then $g(k) \sim (\theta - k)^{-1/2}$ as $k \to \theta$ implies that $k_N - k_{N-1} \sim L^{-2}$. This is not compatible with $k_N > \frac{2(1-\theta^2)}{L(k_N - k_{N-1})}$ (which follows from (31,33)), where the right hand side of this inequality would be $O(L)$ in contradiction with the fact that $k_N \le \theta$. Hence we must have $4\pi C = \theta$, in which case $k_N - k_{N-1} \sim L^{-2/3}$ and there is no contradiction.

It then follows that

$$\theta = 2\sqrt{\rho(1 - \rho)} \quad (48)$$

and therefore $\psi_K(s) = L\delta^2\rho(1 - \rho)$ and (44)

$$\phi(x) = \frac{\theta\sqrt{1-x^2}}{2\pi} + \mathcal{O}\left(\frac{1}{L}\right) \quad (49)$$

### D. The next order

Once $\phi$ is known to leading order (49), one can update the expression (40)

$$f(x) = -\frac{\theta x}{2} - \frac{(1 - \theta^2 x^2)x}{4\sqrt{1 - x^2}} \delta \coth \left[ \frac{L \delta \theta \sqrt{1 - x^2}}{2} \right] + \dots \quad (50)$$

and one gets from (43)

$$\rho = \frac{C\theta\pi}{\sqrt{1-\theta^2}} + \frac{1}{2} - \frac{\theta^2 - 2}{4\sqrt{1-\theta^2}} - \frac{\theta^3\delta}{4\pi\sqrt{1-\theta^2}} \int_{-1}^{1} y^2 \coth \left[ \frac{L\delta\theta\sqrt{1-y^2}}{2} \right] dy \quad (51)$$

Then using the fact that (see (80) in appendix A)

$$\int_{-1}^{1} y^2 \coth(u\sqrt{1-y^2}) dy = \frac{\pi}{2u} + \frac{\pi}{2u^3} \mathcal{F}\left(-\frac{u^2}{2}\right) \quad (52)$$

we get

$$\rho = \frac{C\theta\pi}{\sqrt{1-\theta^2}} + \frac{1}{2} + \frac{\theta^2 - 2}{4\sqrt{1-\theta^2}} - \frac{\theta^2}{4L\sqrt{1-\theta^2}} - \frac{1}{L^3\delta^2\sqrt{1-\theta^2}} \mathcal{F}\left(-\frac{L^2\delta^2\theta^2}{8}\right) \quad (53)$$
and this gives (42)

$$ \psi_K(s) = L\delta^2 C\pi\theta = L\delta^2 \left[ \left(\rho - \frac{1}{2}\right) \sqrt{1 - \theta^2} + \frac{2 - \theta^2}{4} + \frac{\theta^2}{4L} + \frac{1}{L^3\delta^2} \mathcal{F} \left(-\frac{L^2\delta^2\theta^2}{8}\right) \right] \quad (54) $$

The leading order (the first two terms of (54)) has a minimum for $\theta$ given by (48). Therefore to obtain $\psi_K(s)$ at first order in $1/L$ one can simply replace $\theta$ by (48) in (54) and one gets

$$ \psi_K(s) = \frac{L\delta^2\theta^2}{4} \left(1 + \frac{1}{L}\right) + \frac{1}{L^2} \mathcal{F} \left(-\frac{L^2\delta^2\theta^2}{8}\right) \quad (55) $$

which is equivalent (see (29,48)) to (14).

It is shown in (85) of appendix A that for large negative $u$

$$ \mathcal{F}_K(u) \approx \frac{2^{7/2}}{3\pi} (-u)^{3/2}, \quad u \to -\infty \quad (56) $$

This implies that (14) becomes for small negative $s$ (but large negative $L^2s$)

$$ \psi_K(s) \approx L \left[ -2s\rho(1-\rho) + \frac{2^{7/2}}{3\pi} (-s\rho(1-\rho))^{3/2} + \dots \right] \quad (57) $$

So for $s$ small, but $L^2s$ large, the extensivity of $\psi_K(s)$ is recovered and (57) gives the beginning of the small $s$ expansion in the large $L$ limit.

One can also notice that the function $\mathcal{F}(u)$ (16) becomes singular as $u \to \frac{\pi^2}{2}$. This indicates the occurrence of a phase transition discussed at the end of section IV: for $u > \frac{\pi^2}{2}$ the optimal profile to reduce $K$ is no longer flat and the system adopts a deformed profile as in [16]. In fact in the limit $s \to +\infty$ the configurations which dominate are those formed of a single cluster of particles and the activity is limited to the two boundaries of this cluster.

The result (55) or equivalently (14) with $\mathcal{F}$ given by (16)

$$ \mathcal{F}_K(u) = -4 \sum_{n \ge 1} \left[ n\pi \sqrt{n^2\pi^2 - 2u} - n^2\pi^2 + u \right] \quad (58) $$

gives the leading finite-size correction to $\psi_K(s)$. These finite corrections have been calculated recently, starting from the Bethe ansatz equations, for several spin chains in the context of string theory and expressions very similar to our $\mathcal{F}$ have been obtained [29]. Note also that a more systematic approach has been developed to calculate the next finite size correction [30].

### E. Bethe ansatz for $Q$

The eigenvector corresponding to the largest eigenvalue of $W_Q$ can be written as in (26), with the Bethe equations (27)

replaced by

$$ \zeta_i^L = \prod_{\substack{j=1 \\ j \neq i}}^{N} \left[ - \frac{e^s - 2\zeta_i + e^{-s}\zeta_i\zeta_j}{e^s - 2\zeta_j + e^{-s}\zeta_i\zeta_j} \right], \quad (59) $$

Given the solutions $\zeta_j$ to (59), the expression of $\psi_Q$ reads

$$ \psi_Q(s) = -2N + e^{-s}[\zeta_1 + \dots + \zeta_N] + e^s \left[ \frac{1}{\zeta_1} + \dots + \frac{1}{\zeta_N} \right] \quad (60) $$

By a method following closely the steps of the Bethe ansatz for $K$, the basic ingredients of which are provided in appendix E, we arrive at the following result for $\psi_Q$,

$$ \psi_Q(s) = \frac{1}{2}\sigma(\rho)s^2(L+1) + L^{-2}\mathcal{F}\left(-\frac{L^2s^2\sigma(\rho)}{4}\right) \quad (61) $$

which leads to the asymptotic behavior as $L \to \infty$,

$$ \frac{\psi_Q(s)}{L} \approx \frac{1}{2}\sigma(\rho)s^2 + \frac{2^{1/2}}{3\pi}\sigma^{3/2}|s|^3 \quad (62) $$

The Bethe equations (59) are very close to that considered by Kim [52] who worked out the asymmetric exclusion process case. As outlined in appendix E, it seems that Kim's results cannot be extended to the SSEP. We think that this is at the origin of the discrepancy between our expression (62) and what was found earlier (expression (A.12) of [20]) for the same quantity $\psi_Q(s)$.

Before concluding this section devoted to the Bethe ansatz, let us mention that, both for the current or the activity, one can obtain $\psi_Q(s)$ or $\psi_K(s)$ in the $s \to \infty$ limit by directly solving (59) or (27). We do not give these expressions here because they are out of the universal regime.

## IV. FLUCTUATING HYDRODYNAMICS AND THE MACROSCOPIC FLUCTUATION THEORY

In this section we are going to show that the expressions (14,23) can be recovered by a macroscopic theory based on hydrodynamical large deviations [1, 2, 4].

### A. Calculation of $\psi_Q$ for a general diffusive system and derivation of (23)

The macroscopic fluctuation theory developed by Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim [6, 7, 8, 9, 10] is based on the fact that, for a large system of size $L$, the density and the current of a diffusive system take scaling forms. If one defines $\hat{\rho}_i(t)$, the density averaged in the neighborhood of site $i$ at time $t$, and $\hat{Q}_i(t)$, the total flux between site $i$ and $i+1$ during time $t$, these quantities take scaling forms [17, 18]

$$ \hat{\rho}_i(t) = \rho \left( \frac{i}{L}, \frac{t}{L^2} \right) \quad (63) $$
$$ \hat{Q}_i(t) = LQ\left(\frac{i}{L}, \frac{t}{L^2}\right) \qquad (64) $$

This allows one to define a rescaled current $j(x, \tau)$ as

$$ j(x, \tau) = \frac{\partial Q(x, \tau)}{\partial \tau} = L \frac{d}{dt} \hat{Q}_{Lx} (L^2 \tau) . $$

The average microscopic current between site $i$ and $i + 1$ is related to the rescaled current $j$ by

$$ \frac{d\hat{Q}_i(t)}{dt} = \frac{1}{L} j\left(\frac{i}{L}, \frac{t}{L^2}\right) $$

From the macroscopic fluctuation theory [6, 7, 8, 17, 18], the probability of observing a rescaled current $j(x, \tau)$ and a density profile $\rho(x, \tau)$ over a time $t = TL^2$ is given by

$$ \text{Pro}(\{\rho(x, \tau), j(x, \tau)\}) \sim \exp \left[ -L \int_0^T d\tau \int_0^1 dx \frac{[j(x, \tau) + D(\rho(x, \tau))\rho'(x, \tau)]^2}{2\sigma(\rho(x, \tau))} \right] \quad (65) $$

where the current $j(x, \tau)$ and the density profile $\rho(x, \tau)$ satisfy the conservation law

$$ \frac{d\rho}{d\tau} = -\frac{dj}{dx}. \qquad (66) $$

and the diffusive system under study is characterized by the two functions $D(\rho)$ and $\sigma(\rho)$. For the SSEP, these functions are known: $D(\rho) = 1$ and $\sigma(\rho) = 2\rho(1 - \rho)$ (see [2]).

Note that (65) can be seen as the fact that the macroscopic density $\rho(x, \tau)$ and the macroscopic current $j(x, \tau)$ satisfy in addition to the conservation law (66) a Langevin equation of the form [5].

$$ j(x, \tau) = -\partial_x \rho(x, t) + \xi(x, \tau) \qquad (67) $$

where $\xi(x, \tau)$ is a Gaussian white noise

$$ \langle \xi(x, \tau) \xi(x', \tau') \rangle = L^{-1} \sigma(\rho(x, \tau)) \delta(x-x') \delta(\tau-\tau'). \quad (68) $$

The contribution of a small time dependent perturbation to a constant profile $\rho_0$ and a constant rescaled current $j_0$,

$$ \rho(x, \tau) = \rho_0 + \delta\rho(x, \tau) $$

$$ j(x, \tau) = j_0 + \delta j(x, \tau) $$

to the quadratic form in (65) is

$$ \frac{[j(x, t) + D(\rho(x, t))\rho'(x, t)]^2}{2\sigma(\rho(x, t))} = \frac{j_0^2}{2\sigma} + \frac{j_0}{\sigma}\delta j - \frac{j_0^2\sigma'}{2\sigma^2}\delta\rho + \frac{j_0D}{\sigma}\delta\rho' + \frac{\delta j^2 + 2D\delta j\delta\rho' + D^2\delta\rho'^2 + 2j_0D'\delta\rho\delta\rho'}{2\sigma} - \frac{j_0\sigma'(\delta j\delta\rho + D\delta\rho\delta\rho')}{\sigma^2} + j_0^2\left(\frac{\sigma'^2}{2\sigma^3} - \frac{\sigma''}{4\sigma^2}\right)\delta\rho^2 \quad (69) $$

where the functions $D, \sigma, \sigma, \sigma''$ are evaluated at the density $\rho_0$.

If one considers a fluctuation of the form

$$ \delta\rho = k[a_{k,\omega}e^{i\omega\tau+ikx} + a_{k,\omega}^*e^{-i\omega t-ikx}] \quad (70) $$

one has

$$ \delta\rho' = ik^2[a_{k,\omega}e^{i\omega\tau+ikx} - a_{k,\omega}^*e^{-i\omega t-ikx}] $$

and due to (66)

$$ \delta j = -\omega[a_{k,\omega}e^{i\omega\tau+ikx} + a_{k,\omega}^*e^{-i\omega t-ikx}] . $$

The ring geometry ($x \equiv x+1$) imposes that the wave numbers $k$ are discrete

$$ k = 2\pi n \quad \text{with} \quad n \ge 1 $$

Also because one considers a finite time interval $T$, the frequencies $\omega$ are also discrete and

$$ \omega = \frac{2\pi m}{T} \quad \text{with} \quad m \in \mathbb{Z} $$

Integrating over the time interval $0 < \tau < T$ and over space, one gets one has

$$ \langle \delta\rho^2 \rangle = 2k^2 |a_{k,\omega}|^2 T $$

$$ \langle \delta\rho'^2 \rangle = 2k^4 |a_{k,\omega}|^2 T $$

$$ \langle \delta j^2 \rangle = 2\omega^2 |a_{k,\omega}|^2 T $$

$$ \langle \delta j \delta \rho \rangle = -2k\omega |a_{k,\omega}|^2 T $$

$$ \langle \delta\rho\delta\rho' \rangle = \langle \delta j\delta\rho' \rangle = 0 $$

Therefore the superposition of all the fluctuations (70) leads to

$$ \text{Pro}(j_0, \{a_{k,\omega}\}) \sim \exp \left[ -\frac{j_0^2}{2\sigma L} \frac{t}{L} - \frac{t}{L} \sum_{\omega,k} |a_{k,\omega}|^2 \left( \frac{(\sigma\omega + j_0\sigma'k)^2}{\sigma^3} + \frac{D^2k^4}{\sigma} - \frac{j_0^2\sigma''k^2}{2\sigma^2} \right) \right] $$

where some terms independent of $j_0$ have been forgotten (they will be fixed later by normalization). After integrating over the Gaussian fluctuations and if one replaces the sum over $\omega$ by an integral one gets

$$ \text{Pro}(j_0) \sim \exp \left[ -\frac{j_0^2}{2\sigma L} \frac{t}{L} - \frac{t}{2\pi L^2} \sum_{1 \le k \le k_{\text{max}}} \int_{-\omega_{\text{max}}}^{\omega_{\text{max}}} d\omega \ln \left( \frac{(\omega\sigma + j_0\sigma'k)^2}{\sigma^3} + \frac{D^2k^4}{\sigma} - \frac{j_0^2\sigma''k^2}{2\sigma^2} \right) \right] \quad (71) $$
where we have introduced cut-offs $k_{max}$ and $\omega_{max}$. The reason for these cut-offs is that the macroscopic fluctuation theory (65) is valid only on hydrodynamic space and time scales. For $x = O(L^{-1})$ or $\tau = O(L^{-2})$ it has no validity at all, meaning that the cut-offs should satisfy $k_{max} < L$ and $\omega_{max} < L^2$.

For large $L$, i.e. for large $k_{max}$ and $\omega_{max}$, one can see by integrating over $\omega$ that only the constant term and the term proportional to $j_0^2$ depend on the cut-offs so that

$$ \frac{1}{2\pi} \sum_{1 \le k \le k_{\text{max}}} \int_{-\omega_{\text{max}}}^{\omega_{\text{max}}} d\omega \ln \left( \frac{(\omega\sigma + j_0\sigma'k)^2}{\sigma^3} + \frac{D^2k^4}{\sigma} - \frac{j_0^2\sigma''k^2}{2\sigma^2} \right) \\ \approx A(k_{\text{max}}, \omega_{\text{max}}) + B(k_{\text{max}}, \omega_{\text{max}})j_0^2 \\ + \sum_{n=1}^{\infty} \left\{ \sqrt{D^2(2\pi n)^4 - \frac{j_0^2\sigma''}{2\sigma}(2\pi n)^2 - 4\pi^2 n^2 D + \frac{j_0^2\sigma''}{4D\sigma}} \right\} \\ = A(k_{\text{max}}, \omega_{\text{max}}) + B(k_{\text{max}}, \omega_{\text{max}})j_0^2 - \mathcal{D}\mathcal{F} \left( \frac{j_0^2\sigma''}{16D^2\sigma} \right) \quad (72) $$

where we have used the definition (16) of $\mathcal{F}$.

If the averaged rescaled current is $j_0$ over a macroscopic time $T$, the sum of the microscopic flux over all the bonds is $Q = TL^2j_0 = tj_0$. Thus as $\lim_{t\to\infty} \frac{\langle Q^2 \rangle}{t} = \frac{L^2}{L-1}\sigma$ (see (21)) one can determine the cut-off dependent constants and get

$$ \text{Pro}(j_0) \sim \exp \left[ - \frac{j_0^2(L-1)}{2\sigma L^2} \frac{t}{L} + \frac{t}{L^2} \mathcal{D}\mathcal{F} \left( \frac{j_0^2\sigma''}{16D^2\sigma} \right) \right] \quad (73) $$

where $\mathcal{F}$ is defined in (16). This becomes, at order $1/L^2$, using the fact that $\psi_Q(s) = \max_{j_0}[-j0s + t^{-1}\ln \text{Pro}(j_0)]$

$$ \psi_Q(s) - \frac{s^2 \langle Q^2 \rangle}{2t} = \frac{1}{L^2} \mathcal{D}\mathcal{F} \left( \frac{\sigma\sigma''}{16D^2} L^2 s^2 \right) . \quad (74) $$

This formula is in principle valid for arbitrary diffusive systems, i.e. for arbitrary functions $\sigma(\rho)$ and $D(\rho)$. As $\sigma = 2\rho(1-\rho)$, $D = 1$, $\sigma'' = -4$ for the SSEP this leads to the announced result (23,16).

For a general diffusive system the expressions of the cumulants (22) would therefore become

$$ \lim_{t \to \infty} \frac{\langle Q^{2n} \rangle_c}{t} = B_{2n-2} \frac{(2n)!}{n! (n-1)!} D \left( \frac{-\sigma\sigma''}{8D^2} \right)^n L^{2n-2} \quad (75) $$

where $\sigma(\rho)$ and $D(\rho)$ are the two functions which appear in (65) and the $B_n$'s are the Bernoulli numbers.

## B. Calculation of $\psi_K$ for the SSEP and derivation of (14)

To obtain (14), one can first write the activity $K$ as

$$ K = 2L^3 \int_0^T d\tau \int_0^1 dx \rho(x, \tau)(1 - \rho(x, \tau)). $$

Then one has

$$ K - \langle K \rangle \approx 2L^3 \int_0^T d\tau \int_0^1 dx [\langle \delta\rho^2 \rangle - \delta\rho(x, \tau)^2] $$

Then one can proceed as above (69-74) and get, up to terms constant or proportional to $s$, in the exponential

$$ \langle e^{-s(K-\langle K \rangle)} \rangle \sim \int dj_0 \int da_{k,\omega} \exp \left[ -\frac{j_0^2}{2\sigma} \frac{t}{L} - \frac{t}{L} \sum_{\omega,k} |a_{k,\omega}|^2 \left( \frac{(\sigma\omega + j_0\sigma'k)^2}{\sigma^3} + \frac{D^2k^4}{\sigma} - \frac{j_0^2\sigma''k^2}{2\sigma^2} + 4k^2sL^2 \right) \right] $$

The rest of the calculation is the same as (72-74), with a maximum over $j_0$ achieved at $j_0 = 0$, and one finally gets

$$ \psi_K(s) = -s \frac{\langle K \rangle}{t} + L^{-2} \mathcal{F}_K \left( \frac{\sigma}{2} L^2 s \right) \quad (76) $$

which is exactly (14).

## C. Calculation of $\psi_Q$ in the case of a weak asymmetry

One can also repeat the above calculation in the case of weakly driven systems, i.e. for systems where there is an additional driving force of strength $1/L$. This would in particular be the case for the weakly asymmetric exclusion process (WASEP) [16] for which the hopping rates to the right and to the left are respectively $\exp(\frac{\nu}{L})$ and $\exp(-\frac{\nu}{L})$.

For such systems, (65) becomes

$$ \text{Pro}(\{\rho(x, \tau), j(x, \tau)\}) \sim \exp \left[ -L \int_0^T d\tau \int_0^1 dx \frac{[j(x, \tau) + D(\rho(x, \tau))\rho'(x, \tau) - \nu\sigma(\rho(x, \tau))]^2}{2\sigma(\rho(x, \tau))} \right] \quad (77) $$

Following exactly the same steps as before, one gets an additional term $\frac{\nu^2\sigma''}{4}\delta\rho^2$ in (69), everything else remaining the same. Then (73) becomes in this case:

$$ \text{Pro}(j_0) \sim \exp \left[ - \frac{(j_0 - \nu\sigma)^2(L-1)}{2\sigma L^2} \frac{t}{L} + \frac{t}{L^2} \mathcal{D}\mathcal{F} \left( \frac{j_0^2 - \nu^2\sigma^2}{16D^2\sigma} \right) \right] \quad (78) $$

where we have adjusted as in (73) the terms linear and quadratic in $j_0$ which are cut-off dependent.

## D. Phase transitions

The function $\mathcal{F}(u)$ becomes singular as $u \to \frac{\pi^2}{2}$ (see (16)). For systems for which $\sigma'' < 0$, this implies the occurrence of a phase transition in the expression (76) of $\psi_K(s)$ in or in the large deviation function (78) of the current in the case of a weak asymmetry. These phase transitions are exactly the same
as the one discussed in [9, 10, 16, 17]: beyond the transition the system does not fluctuate anymore about a flat density profile, but the profile becomes deformed on a macroscopic scale.

For systems such as the Kipnis Marchioro Presutti model [57, 58] which have $\sigma'' > 0$, a similar phase transition occurs in $\psi_Q$ even in absence of a weak asymmetry.

## V. CONCLUSION

In the present paper we have obtained exact expressions (12,21) of the first cumulants of the activity $K$ and of the integrated current $Q$ for the SSEP. In the large $L$ limit, these cumulants take scaling forms (13,22).

We have shown in section III that these scaling forms can be understood starting from the Bethe ansatz equations (27,59), by calculating the leading finite size corrections. These finite size corrections are similar to the ones calculated recently for spin chains in the context of quantum strings [29, 30].

We have also shown in section IV that they can also be understood starting from the macroscopic fluctuation theory (65) of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim. This enabled us to extend (74,73,75) our results for the SSEP to arbitrary diffusive systems and to see that the occurrence of phase transitions can be predicted from the scaling form of the cumulants of the current. In order to better understand these phase transitions it might be interesting to characterize the eigenstate of the s-dependent evolution operator by, e.g., determining correlation functions in those states.

We have discussed here systems governed by diffusive dynamics with a single conserved field. How the universal scaling forms would be modified for systems with several conserved fields is an interesting open question.

We thank N. Gromov, H.J. Hilhorst, V. Kazakov, K. Mallick, S. Prohlac, H. Spohn, P. Vieira, R.K.P. Zia, for several useful discussions. This work was supported by the French Ministry of Education through an ANR-05-JCJC-44482 grant and LHMSHE.

## APPENDIX A: SEVERAL REPRESENTATIONS OF THE FUNCTION $\mathcal{F}$

In this appendix we show the equivalence between several representations (16, 17,52) of the function $\mathcal{F}$ defined in (16)

$$ \mathcal{F}(u) = -4 \sum_{n \ge 1} \left[ n\pi \sqrt{n^2\pi^2 - 2u} - n^2\pi^2 + u \right] \quad (79) $$

To do so consider the integral $I$

$$ I = \frac{2u^3}{\pi} \int_{-1}^{1} y^2 dy \coth(u\sqrt{1-y^2}) $$

Then by using the fact that

$$ \coth z = \frac{1}{z} + \sum_{n=1}^{\infty} \frac{2z}{z^2 + n^2 \pi^2} $$

and by integrating over $y$, one gets

$$ \begin{aligned} I &= \frac{2u^3}{\pi} \int_{-1}^{1} y^2 dy \coth(u\sqrt{1-y^2}) \\ &= u^2 + \sum_{n \ge 1} \left[ 2u^2 + 4n^2\pi^2 - 4n\pi \sqrt{n^2\pi^2 + u^2} \right] \quad (80) \\ &= u^2 + \mathcal{F}\left(-\frac{u^2}{2}\right) \end{aligned} $$

This establishes (52). Now as

$$ \frac{x}{e^x - 1} = \sum_{n \ge 0} \frac{B_n}{n!} x^n = 1 - \frac{x}{2} + \frac{x^2}{12} - \frac{x^4}{720} + \frac{x^6}{30240} + \dots \quad (81) $$

which is simply the definition of the Bernoulli numbers $B_n$ (so that $B_2 = \frac{1}{6}$, $B_4 = -\frac{1}{30}$, $B_6 = \frac{1}{42}$, ...), one can show that

$$ \coth x = \frac{1}{x} + \sum_{k \ge 2} 2^{2k-2} x^{2k-3} \frac{B_{2k-2}}{(2k-2)!} \quad (82) $$

Therefore

$$ \begin{aligned} I &= \frac{2u^3}{\pi} \int_{-1}^{1} y^2 dy \coth(u\sqrt{1-y^2}) \\ &= \frac{2u^2}{\pi} \int_{-1}^{1} \frac{y^2}{\sqrt{1-y^2}} dy \\ &\quad + \sum_{k \ge 2} \frac{2^{2k-1}}{\pi} \frac{B_{2k-2}}{(2k-2)!} u^{2k} \int_{-1}^{1} y^2 (1-y^2)^{\frac{2k-3}{2}} dy \end{aligned} $$

i.e.

$$ I = u^2 + \sum_{k \ge 2} \frac{B_{2k-2}}{\Gamma(k)\Gamma(k+1)} u^{2k} \quad (83) $$

Comparing (80) and (83), one gets

$$ \mathcal{F}(u) = \sum_{k \ge 2} \frac{B_{2k-2}}{\Gamma(k)\Gamma(k+1)} (-2u)^k = \frac{u^2}{3} + \frac{u^3}{45} + \frac{u^4}{378} + \frac{u^5}{2700} + \dots \quad (84) $$

so that (17) and (15) are consistent with (16).

For large negative $u$, one gets, by replacing in (79) the sum over $n$ by an integral,

$$ \mathcal{F}(u) \approx \frac{2^{7/2}(-u)^{3/2}}{3\pi} \quad (85) $$

## APPENDIX B: CALCULATION OF THE TWO SUMS APPEARING IN (35)

In this appendix we calculate the two sums which appear in (35) when $\delta \to 0$ and $L \to \infty$ keeping $L\delta$ fixed.

### The first sum in (35):

If the $k_i$ are distributed according to a density $g(k)$ on the real axis one can write that

$$ L \int_{k_i}^{k_{i+1}} g(k') dk' = n \quad (86) $$
Therefore for $n$ fixed and large $L$, one has

$$L(k_{i+n} - k_i)g(k_i) + L(k_{i+n} - k_i)^2 \frac{g'(k_i)}{2} + \dots = n$$

so that

$$k_{i+n} - k_i = \frac{n}{g(k_i)L} - \frac{n^2 g'(k_i)}{2g(k_i)^3 L^2} + \dots \quad (87)$$

Replacing $k_j$ by expression (87) into the first sum in (35) one gets

$$\sum_{n=1}^{n_0} \left( 4k_i - \frac{2g'(k_i)(1-k_i^2)}{g(k_i)} \right) \frac{n^2}{n^2 + (1-k_i^2)^2 g(k_i)^2 L^2 \delta^2}$$

Using the fact that for $n_0 \gg 1$ (and $b < \mathcal{O}(1)$)

$$\sum_{n=1}^{n_0} \frac{1}{n^2 + b^2} = -\frac{1}{2b^2} + \frac{\pi}{2b} \coth \pi b \quad (88)$$

the first sum in (35) can be replaced by

$$\sum_{j=i-n_0}^{i-1} + \sum_{j=i+1}^{i+n_0} U(k_i, k_j) \approx \left(4k_i - \frac{2g'(k_i)(1-k_i^2)}{g(k_i)}\right) n_0 \\ - \left(2k_i - \frac{g'(k_i)(1-k_i^2)}{g(k_i)}\right) \left[-1 + \pi(1-k_i^2)g(k_i)L\delta \coth[\pi(1-k_i^2)g(k_i)L\delta]\right] \quad (89)$$

**The second sum in (35):**

Let us consider the following integral.

$$I = \mathcal{P} \int_{-\theta}^{\theta} g(k') dk' \frac{1 - k_i k'}{k_i - k'} \quad (90)$$

We are now going to compare this integral with the sum

$$S = \sum_{j \notin [i-n_0, i+n_0]} \frac{1 - k_i k_j}{k_i - k_j}$$

We assume (86) that the $k_j$ are given by

$$L \int_{-\theta}^{k_j} g(q) dq = j - \alpha \quad (91)$$

and for the moment $\alpha$ is arbitrary. Therefore

$$k_{j+1} - k_j \approx \frac{1}{g(k_i)L} \quad (92)$$

One can decompose the integral $I$ as

$$I = \mathcal{P} \int_{k_{i-n_0}}^{k_{i+n_0}} g(q) dq \frac{1 - k_i q}{k_i - q} + \sum_{j=1}^{i-n_0-1} \int_{k_j}^{k_{j+1}} g(q) dq \frac{1 - k_i q}{k_i - q} + \sum_{j=i+n_0}^{N-1} \int_{k_j}^{k_{j+1}} g(q) dq \frac{1 - k_i q}{k_i - q} \\ + \int_{-\theta}^{k_1} g(q) dq \frac{1 - k_i q}{k_i - q} + \int_{k_N}^{\theta} g(q) dq \frac{1 - k_i q}{k_i - q} \quad (93)$$

As $k_{j+1} - k_j$ is small and of order $1/L$ and because of (91,92)

$$\int_{k_j}^{k_{j+1}} g(q) dq \frac{1 - k_i q}{k_i - q} \approx \frac{1}{L} \frac{1 - k_i k_j}{k_i - k_j} + \frac{g(k_j)(k_{j+1} - k_j)^2}{2} \frac{d}{dk_j} \left( \frac{1 - k_i k_j}{k_i - k_j} \right) \\ \approx \frac{1}{L} \frac{1 - k_i k_j}{k_i - k_j} + \frac{1}{2L^2 g(k_j)} \frac{d}{dk_j} \left( \frac{1 - k_i k_j}{k_i - k_j} \right) \\ \approx \frac{1}{L} \frac{1 - k_i k_{j+1}}{k_i - k_{j+1}} - \frac{1}{2L^2 g(k_{j+1})} \frac{d}{dk_{j+1}} \left( \frac{1 - k_i k_{j+1}}{k_i - k_{j+1}} \right) \quad (94)$$

Therefore using (94) in the sum $1 \le j \le i - n_0 - 1$ and (94) in the sum $i + n_0 \le j \le N - 1$, one can rewrite (93) as

$$I \approx \mathcal{P} \int_{k_{i-n_0}}^{k_{i+n_0}} g(q) dq \frac{1 - k_i q}{k_i - q} + \frac{1}{L} \sum_{j=1}^{i-n_0-1} \frac{1 - k_i k_j}{k_i - k_j} + \frac{1}{L} \sum_{j=i+n_0+1}^{N} \frac{1 - k_i k_j}{k_i - k_j} + \frac{1}{2L^2} \sum_{j=1}^{i-n_0-1} \frac{1}{g(k_j)} \frac{d}{dk_j} \left( \frac{1 - k_i k_j}{k_i - k_j} \right) \\ - \frac{1}{2L^2} \sum_{j=i+n_0+1}^{N} \frac{1}{g(k_j)} \frac{d}{dk_j} \left( \frac{1 - k_i k_j}{k_i - k_j} \right) + \int_{-\theta}^{k_1} g(q) dq \frac{1 - k_i q}{k_i - q} + \int_{k_N}^{\theta} g(q) dq \frac{1 - k_i q}{k_i - q}$$

This becomes

$$I \approx \mathcal{P} \int_{k_{i-n_0}}^{k_{i+n_0}} g(q) dq \frac{1 - k_i q}{k_i - q} + \frac{1}{L} \sum_{j=1}^{i-n_0-1} \frac{1 - k_i k_j}{k_i - k_j} + \frac{1}{L} \sum_{j=i+n_0+1}^{N} \frac{1 - k_i k_j}{k_i - k_j} + \frac{1}{2L} \left[ \frac{1 - k_i k_{i-n_0-1}}{k_i - k_{i-n_0-1}} + \frac{1 - k_i k_{i+n_0+1}}{k_i - k_{i+n_0+1}} - \frac{1 - k_i k_1}{k_i - k_1} - \frac{1 - k_i k_N}{k_i - k_N} \right] \\ + \int_{-\theta}^{k_1} g(q) dq \frac{1 - k_i q}{k_i - q} + \int_{k_N}^{\theta} g(q) dq \frac{1 - k_i q}{k_i - q}$$
which can be rewritten as

$$
\begin{align}
I \approx \mathcal{P} \int_{k_{i-n_0}}^{k_{i+n_0}} g(q) dq \frac{1-k_i q}{k_i - q} &+ \frac{1}{L} \sum_{j=1}^{i-n_0-1} \frac{1-k_i k_j}{k_i - k_j} \nonumber \\
&+ \frac{1}{2L} \left[ \frac{1-k_i k_{i+n_0+1}}{k_i - k_{i+n_0+1}} + \frac{1-k_i k_{i-n_0-1}}{k_i - k_{i-n_0-1}} \right] \nonumber \\
&+ \frac{1-k_i k_1}{k_i - k_1} \left[ -\frac{1}{2L} + \int_{-\theta}^{k_1} g(q) dq \right] \tag{95} \\
&+ \frac{1-k_i k_N}{k_i - k_N} \left[ -\frac{1}{2L} + \int_{k_N}^{\theta} g(q) dq \right] \nonumber
\end{align}
$$

From (87) one can show that

$$
\mathcal{P} \int_{k_{i-n_0}}^{k_{i+n_0}} g(q) dq \frac{1-k_i q}{k_i - q} \approx \frac{2k_i n_0}{L} - \frac{k_i n_0 (1-k_i^2) g'(k_i)}{L g(k_i)} \quad (96)
$$

and that

$$
\frac{1 - k_i k_{i+n_0}}{k_i - k_{i+n_0}} + \frac{1 - k_i k_{i-n_0-1}}{k_i - k_{i-n_0-1}} \approx 2k_i - (1 - k_i^2) \frac{g'(k_i)}{g(k_i)} + \mathcal{O}\left(\frac{1}{L}\right) \quad (97)
$$

Lastly because one expects the symmetry $k_j = -k_{N+1-j}$ and
because $L \int_{-\theta}^{\theta} g(q) dq = N$, one gets that $\alpha = 1/2$ in (91) and
therefore the last two terms of (95) vanish.
Then using (96,97) into (95), one gets that

$$
\begin{align}
& \frac{1}{L} \sum_{j=1}^{i-n_0-1} \frac{1-k_i k_j}{k_i - k_j} + \frac{1}{L} \sum_{j=i+n_0+1}^{N} \frac{1-k_i k_j}{k_i - k_j} \tag{98} \\
& \approx I - \frac{1}{L} \left( 2k_i - (1-k_i^2) \frac{g'(k_i)}{g(k_i)} \right) \left( n_0 + \frac{1}{2} \right) \nonumber
\end{align}
$$

where the integral *I* is defined in (90). Lastly using the fact
that *g*(*k*) = *g*(*−*(*k*), one can rewrite the integral *I* in (90) as

$$
I = \mathcal{P} \int_{-\theta}^{\theta} g(k') dk' \frac{1-k'^2}{k_i - k'} \quad (99)
$$

so that (98) becomes

$$
\begin{align}
& \frac{1}{L} \sum_{j=1}^{i-n_0-1} \frac{1-k_i k_j}{k_i - k_j} + \frac{1}{L} \sum_{j=i+n_0+1}^{N} \frac{1-k_i k_j}{k_i - k_j} \nonumber \\
& \approx \mathcal{P} \int_{-a}^{a} g(k') dk' \frac{1-k'^2}{k_i - k'} \tag{100} \\
& - \frac{1}{L} \left( 2k_i - (1-k_i^2) \frac{g'(k_i)}{g(k_i)} \right) \left( n_0 + \frac{1}{2} \right) \nonumber
\end{align}
$$

Note that (91) is not accurate for *i* close to 1 or *N*, i.e. near
the singularities of *g*(*k*). A more detailed analysis of these
two neighborhoods would only contribute to higher orders in
the 1/*L* expansion [30].

APPENDIX C: SOLUTION OF THE AIRFOIL EQUATION
(39)

In this appendix we show, in the spirit of [59], that the solution
$\phi(x)$ of

$$
f(x) = \mathcal{P} \int_{-1}^{1} dy \frac{\phi(y)}{y-x} \quad (101)
$$

is

$$
\phi(x) = \frac{C}{\sqrt{1-x^2}} - \frac{1}{\pi^2} \mathcal{P} \int_{-1}^{1} dy \sqrt{\frac{1-y^2}{1-x^2}} \frac{f(y)}{y-x} \quad (102)
$$

This solution is used to obtain (41) as the solution of (39).
Let us choose

$$
\phi(x) = \frac{\sqrt{1-x^2}}{x-\alpha} \quad (103)
$$

Then for $x \notin [-1, 1]$ and $\alpha \notin [-1, 1]$ one can see using (111)

$$
\int_{-1}^{1} dy \frac{\phi(y)}{y-x} = \pi \left[ \frac{\sqrt{\alpha^2 - 1}}{\alpha - x} - \frac{\sqrt{x^2 - 1}}{\alpha - x} - 1 \right] \quad (104)
$$

and therefore

$$
f(x) = \mathcal{P} \int_{-1}^{1} dy \frac{\phi(y)}{y-x} = \pi \left[ \frac{\sqrt{\alpha^2 - 1}}{\alpha - x} - 1 \right] \quad (105)
$$

Now the following integral of this function $f(x)$ can be computed (using (109,111) for $x \notin [-1, 1]$)

$$
-\frac{1}{\pi^2} \int_{-1}^{1} dy \frac{\sqrt{1-y^2}}{y-x} f(y) = \sqrt{\alpha^2 - 1} \left( \frac{\sqrt{\alpha^2 - 1}}{\alpha - x} \right. \\
\left. - \frac{\sqrt{x^2 - 1}}{\alpha - x} - 1 \right) + \sqrt{x^2 - 1} - x \quad (106)
$$

so that

$$
\begin{align}
-\frac{1}{\pi^2} \mathcal{P} \int_{-1}^{1} dy \frac{\sqrt{1-y^2}}{y-x} f(y) &= \frac{\alpha^2 - 1}{\alpha - x} - \sqrt{\alpha^2 - 1} - x \nonumber \\
&= \alpha - \sqrt{\alpha^2 - 1} - \frac{1 - x^2}{\alpha - x} \tag{107}
\end{align}
$$

Comparing with (103) we see that

$$
\begin{align}
-\frac{1}{\pi^2} \mathcal{P} \int_{-1}^{1} dy \sqrt{\frac{1-y^2}{1-x^2}} \frac{f(y)}{y-x} &= \frac{\alpha - \sqrt{\alpha^2 - 1}}{\sqrt{1-x^2}} + \frac{\sqrt{1-x^2}}{x-\alpha} \nonumber \\
&= \frac{\alpha - \sqrt{\alpha^2 - 1}}{\sqrt{1-x^2}} + \phi(x) \tag{108}
\end{align}
$$

Therefore (102) is the solution of (101) with a constant *C*
which depends through *α* on *φ(x)* when one chooses (103)
for *φ(x)*.
As the inversion formula (102) is valid for arbitrary $\alpha$, it would also be valid when $f(x)$ is any polynomial in $x$, and as the polynomials are dense in the set of continuous functions on $(-1, 1)$, one can consider that (101,102) are valid for "arbitrary functions" $f(x)$.

## APPENDIX D: USEFUL INTEGRALS

In this appendix we list a few integrals which are used in various places of the paper.

First for $x \notin [-1, 1]$ one has

$$ \frac{1}{\pi} \int_{-1}^{1} \frac{\sqrt{1-y^2}}{y-x} dy = \sqrt{x^2-1-x} \quad (109) $$

so that

$$ \frac{1}{\pi} \mathcal{P} \int_{-1}^{1} \frac{\sqrt{1-y^2}}{y-x} dy = -x \quad (110) $$

As a consequence of (111) one has for $x \notin [-1, 1]$ and $\alpha \notin [-1, 1]$

$$ \frac{1}{\pi} \int_{-1}^{1} \frac{\sqrt{1-y^2}}{y-x} \frac{dy}{y-\alpha} = \frac{\sqrt{\alpha^2-1}}{\alpha-x} - \frac{\sqrt{x^2-1}}{\alpha-x} - 1 \quad (111) $$

and thus for $x \in [-1, 1]$ and $\alpha \notin [-1, 1]$

$$ \frac{1}{\pi} \mathcal{P} \int_{-1}^{1} \frac{\sqrt{1-y^2}}{y-x} \frac{dy}{y-\alpha} = \frac{\sqrt{\alpha^2-1}}{\alpha-x} - 1 \quad (112) $$

One can also show that

$$ \int_{-1}^{1} \frac{dx}{\sqrt{1-x^2}} = \pi \quad (113) $$

and that for $y \notin [-1, 1]$

$$ \int_{-1}^{1} \frac{dx}{\sqrt{1-x^2}} \frac{1}{y-x} = \frac{\pi}{\sqrt{y^2-1}} \quad (114) $$

As a consequence of (112,114), one has

$$ \int_{-1}^{1} \frac{dx}{\sqrt{1-x^2}} \mathcal{P} \int_{-1}^{1} \frac{dy}{y-x} F(y) = 0 \quad (115) $$

for an arbitrary function $F(y)$ as it is valid for any polynomial

For $\theta < 1$ one can show using (114) that

$$ \int_{-1}^{1} \frac{dx}{(1 - \theta^2 x^2) \sqrt{1 - x^2}} = \frac{\pi}{\sqrt{1 - \theta^2}} \quad (116) $$

one can also show

$$ \int_{-1}^{1} \frac{\sqrt{1-x^2}}{(1-\theta^2 x^2)} dx = \pi \frac{1-\sqrt{1-\theta^2}}{\theta^2} \quad (117) $$

and that

$$ \int_{-1}^{1} \frac{y^2 \sqrt{1-y^2}}{1-\theta^2 y^2} dy = \pi \left( \frac{1}{\theta^4} - \frac{1}{2\theta^2} - \frac{\sqrt{1-\theta^2}}{\theta^4} \right) \quad (118) $$

and for $y \notin [-1, 1]$

$$ \int_{-1}^{1} \frac{1}{(1 - \theta^2 x^2) \sqrt{1 - x^2}} \frac{dy}{y-x} = \frac{\pi}{(1 - \theta^2 y^2) \sqrt{y^2 - 1}} - \frac{\pi \theta^2 y}{(1 - \theta^2 y^2) \sqrt{1 - \theta^2}} \quad (119) $$

and therefore for any function $F(y)$

$$ \int_{-1}^{1} \frac{dx}{(1 - \theta^2 x^2) \sqrt{1 - x^2}} \mathcal{P} \int_{-1}^{1} \frac{dy}{y-x} F(y) = - \frac{\pi \theta^2}{\sqrt{1 - \theta^2}} \int_{-1}^{1} \frac{yF(y)}{1 - \theta^2 y^2} dy \quad (120) $$

## APPENDIX E: BETHE ANSATZ CALCULATION FOR THE CURRENT LARGE DEVIATION FUNCTION $\psi_Q(s)$

This appendix describes how a Bethe ansatz calculation of $\psi_Q(s)$ similar to the one conducted for $\psi_K$ can be implemented. The operator $\mathcal{W}_Q$ whose largest eigenvalue is $\psi_Q$ reads, in the spin language already used in (25),

$$ \mathcal{W}_Q(s) = \sum_{i=1}^{L} \left[ \frac{\sigma_i^z \sigma_{i+1}^z - 1}{2} + e^{-s} \sigma_i^+ \sigma_{i+1}^- + e^s \sigma_i^- \sigma_{i+1}^+ \right] \quad (121) $$

The Bethe ansatz equation analogous to (27) take the form (59)

$$ \zeta_i^L = \prod_{\substack{j=1 \\ j \neq i}}^{N} \left[ - \frac{1 - 2e^{-s}\zeta_i + e^{-2s}\zeta_i\zeta_j}{1 - 2e^{-s}\zeta_j + e^{-2s}\zeta_i\zeta_j} \right] \quad (122) $$

In terms of the $\zeta_j$'s, we have that

$$ \psi_Q(s) = -2N + e^{-s} [\zeta_1 + \dots + \zeta_N] + e^s \left[ \frac{1}{\zeta_1} + \dots + \frac{1}{\zeta_N} \right] \quad (123) $$

Kim [52] has studied the spectrum of $\mathcal{H} = -\mathcal{W}_Q/(\cosh s/2)$ by means of a Bethe ansatz calculation: in the notations of his equation (1), the parameters $\tilde{\Delta}$ and $S$ are given by

$$ \tilde{\Delta} = \frac{1}{\cosh s}, \quad S = \tanh s \quad (124) $$

but unfortunately his results do not apply to our particular case, which turns out to correspond to a critical point of the related six-vertex model. The defining parameters of the latter, denoted by $\Delta$, $H$ and $\nu$, are related to Kim's by $\tilde{\Delta} = \Delta/\cosh(2H)$, $S = \tanh(2H)$, $\Delta = \cosh \nu$. Thus, in terms of our original parameters, we get that

$$ \Delta = 1, \quad 2H = s, \quad \nu = 0 \quad (125) $$

a limiting case explicitly excluded by Kim which lies at the critical point of the six-vertex model.
We choose to write that $\zeta_j = e^{-is(k_j+2i\rho)}$. The two main differences with the calculation of $\psi_K$ is that the $\zeta_j$'s dependence in $s$ is different. We have also shifted them by $2i\rho$ for convenience. Just as was the case previously, the $k_j$'s will be densely distributed on a connected curve $C$ of the complex plane that is invariant upon complex conjugation. Given that the equations for the $\zeta_j$'s are invariant under complex conjugation, we expect the contour $C$ to be symmetric with respect to the vertical axis in the complex $k$ plane. We shall denote the end points of $C$ by $-\theta^*$ and $\theta$.

Given that (122) becomes

$$-i(k_i + i2\rho) = \frac{1}{L} \sum_{j=1, j \neq i}^{N} U(k_i, k_j), \quad \text{where} \\ U(k_i, k_j) = \frac{1}{s} \ln \left[ \frac{1 - 2e^{-s}\zeta_i + e^{-2s}\zeta_i\zeta_j}{1 - 2e^{-s}\zeta_j + e^{-2s}\zeta_i\zeta_j} \right] \quad (126)$$

for $|i-j| \gg 1$ we expect that

$$U(k_i, k_j) = \frac{2i(k_i + i\alpha)(k_j + i\alpha)}{k_i - k_j}, \quad \alpha = 2\rho - 1 \quad (127)$$

while for $i-j$ of order 1, $s$ will be over order $1/L$ and $k_i-k_j$ as well. We define $g(k)$ as the root density along contour $C$, so that

$$L \int_{k_i}^{k_j} g(k)dk = j - i$$

(note that $g(k)$ is in general complex but along the contour $g(k)dk$ is real). If $k_j$ and $k_i$ are $n$ roots apart, we have that

$k_j - k_i = \frac{n}{g(k_i)L} - \frac{n^2g'(k_i)}{2g(k_i)^3L^2} + \dots$. Expanding $U$ at fixed $sL$ in powers of $L^{-1}$ leads to

$$U(k_i, k_j) = \frac{1}{s} \ln \frac{n - ig(k_i)(k_i + i\alpha)^2 sL}{n + ig(k_i)(k_i + i\alpha)^2 sL} \\ -i(k_i + i\alpha) \left( 2 + \frac{g'(k_i)(k_i + i\alpha)}{g(k_i)} \right) \frac{n^2}{n^2 + [g(k_i)(k_i + i\alpha)^2(sL)]^2} \quad (128)$$

Equations (127) and (128) play a role analogous to (33) and (32) in the study of $K$. After using the methods of appendices B and C we arrive at the following equation for $g$ which we express in terms of $\phi(x) = (\theta x + i\alpha)^2 g(\theta x)$ and $r = \theta^*/\theta$:

$$\theta \left( x + i \frac{\alpha + 1}{\theta} \right) = 2\mathcal{P} \int_{-r}^{1} dy \frac{\phi(y) - (y-x)(y + i \frac{\alpha}{\theta})^{-1} \phi(y)}{y-x} \\ - \frac{\theta}{L} \left( x + i \frac{\alpha}{\theta} \right)^2 \frac{\phi'(x)}{\phi(x)} [\pi \phi(x)(sL)] \coth[\pi \phi(x)(sL)] \quad (129)$$

Let us denote $\phi_0(x)$ the solution of the above equation, in the $L \to \infty$ limit

$$\theta x/2 + h = \mathcal{P} \int_{-r}^{1} dy \frac{\phi_0(y)}{y-x} \quad (130)$$

where $h = i(\alpha + 1)/2 + \int dy \phi_0(y)(y + i\alpha/\theta)^{-1}$ is a density-dependent constant to be determined. The general solution of (130) can be written (see (101,102)) as

$$\phi_0(x) = - \frac{C}{\sqrt{(1-x)(r+x)}} - \frac{\theta(r+1)^2}{16\pi\sqrt{(1-x)(r+x)}} \\ + \frac{\theta x^2}{2\pi\sqrt{(1-x)(r+x)}} + \frac{2h(r-1+x)}{2\pi\sqrt{(1-x)(r+x)}} \\ + \frac{\theta x(r-1)}{4\pi\sqrt{(1-x)(r+x)}} \quad (131)$$

The four unknowns $C$, $\theta$, $r$ and $h$ are determined by requiring that $\phi_0$ remains finite as $x \to 1$ and as $x \to -r$, and by noting that by definition

$$\int_{-r}^{+1} dx \frac{\phi_0(x)}{(x + i \frac{\alpha}{\theta})^2} = \theta \rho \quad (132)$$

while $\phi_0$ must verify the self-consistency equation $h = i(\alpha + 1)/2 + \int dy \phi_0(y)(y + i\alpha/\theta)^{-1}$. After explicitly evaluating the latter integral and that appearing in (132), we arrive at $r=1$, $h=0$ and $4\pi C = \theta = 2\sqrt{\rho(1-\rho)}$, which leads to $\phi_0(x) = -\theta \frac{\sqrt{1-x^2}}{2\pi}$. Up to a sign, this is exactly the same function as that found in the study of $K$, and this is the same end point $\theta = 2\sqrt{\rho(1-\rho)}$ for the contour on which the $k_j$'s lie.

We may now simplify (129) into

$$\theta \left( x + i \frac{\alpha + 1}{\theta} \right) = 2\mathcal{P} \int dy \frac{\phi(y) - (y-x)(y + i\alpha/\theta)^{-1} \phi(y)}{y-x} \\ + \frac{1}{\theta L} \frac{x(x + i\alpha/\theta)^2}{1 - x^2} \left( \left[ \theta \sqrt{1 - x^2} (sL)/2 \right] \coth \left[ \theta \sqrt{1 - x^2} (sL)/2 \right] \right) \quad (133)$$

whose solution reads $\phi(x) = \phi_0(x) + \delta\phi(x)$,

$$\delta\phi(x) = - \frac{\delta C}{\sqrt{1-x^2}} + \frac{2\delta h x}{2\pi\sqrt{1-x^2}} \\ - \frac{1}{\pi^2} \frac{1}{\sqrt{1-x^2}} \mathcal{P} \int dy \frac{\sqrt{1-y^2}}{y-x} \delta F(y) \quad (134)$$

We have denoted by $\delta F(x)$ the function

$$\delta F(x) = - \frac{\theta}{2L} \frac{x(x + i\alpha/\theta)^2}{1 - x^2} \\ \times \left( \left[ \theta \sqrt{1 - x^2} (sL)/2 \right] \coth\left[ \theta \sqrt{1 - x^2} (sL)/2 \right] \right) \\ = - \frac{\theta}{2L} \frac{x(x + i\alpha/\theta)^2}{1 - x^2} \left[ \sum_{p \ge 2} \frac{B_p}{p!} (\theta sL)^p (1 - x^2)^{p/2} + 1 \right] \quad (135)$$

The new constants $\delta C$ and $\delta h$ are determined by $\int \frac{\delta \phi}{(x+i\alpha/\theta)^2} = 0$ and $\delta h = \int \frac{\delta \phi}{(x+i\alpha/\theta)}$. After performing
explicit integrations along the lines of appendix D, we obtain
the final result through the following equality

$$
\begin{equation}
\begin{split}
\psi_Q(s)/L = {}& -s^2 \theta \int dx \phi(x) = \frac{\theta^2}{4} s^2 + s^2 \delta C \theta \pi \\
& +s^2 \theta \frac{1}{\pi^2} \int dx \frac{1}{\sqrt{1-x^2}} P \int dy \frac{\sqrt{1-y^2}}{y-x} \delta F(y)
\end{split}
\tag{136}
\end{equation}
$$

where

$$
\begin{align}
\delta C \theta \pi &= \frac{\theta^2}{2\pi L} \int_{-1}^{1} dx \, x^2 \left[ \sum_{p \ge 2} \frac{B_p}{p!} (\theta s L)^p (1-x^2)^{\frac{p-1}{2}} \right. \nonumber \\
&\qquad \left. + \frac{1}{\sqrt{1-x^2}} \right] \nonumber \\
&= \frac{1}{L^3 s^2} \mathcal{F}(-L^2 s^2 \theta^2 / 8) + \frac{\theta^2}{4L} \tag{137}
\end{align}
$$

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