We show how to apply the macroscopic fluctuation theory (MFT) of Bertini, De Sole, Gabrielli, Jona-Lasinio, and Landim to study the current fluctuations of diffusive systems with a step initial condition. We argue that one has to distinguish between two ways of averaging (the annealed and the quenched cases) depending on whether we let the initial condition fluctuate or not. Although the initial condition is not a steady state, the distribution of the current satisfies a symmetry very reminiscent of the fluctuation theorem. We show how the equations of the MFT can be solved in the case of non-interacting particles. The symmetry of these equations can be used to deduce the distribution of the current for several other models, from its knowledge [29] for the symmetric simple exclusion process. In the range where the integrated current Qt ∼ √ t, we show that the non-Gaussian decay exp[−Q 3 t /t] of the distribution of Qt is generic.keywords: current fluctuations, step initial condition, fluctuation theorem PACS numbers: 02.50.-r, 05.40.-a, 05.70 Ln, 82.20-w This work is dedicated to our master and friend Joel Lebowitz on the occasion of the 100th Statistical Mechanics Meeting held at Rutgers University in December 2008. * We acknowledge the support of the ANR LHMSHE.

For the symmetric simple exclusion process on an infinite line, we calculate exactly the fluctuations of the integrated current Qt during time t through the origin when, in the initial condition, the sites are occupied with density ρa on the negative axis and with density ρ b on the positive axis. All the cumulants of Qt grow like √ t. In the range where Qt ∼ √ t, the decay exp[−Q 3 t /t] of the distribution of Qt is non-Gaussian. Our results are obtained using the Bethe ansatz and several identities recently derived by Tracy and Widom for exclusion processes on the infinite line.

Consider a collection of random variables attached to the vertices of a graph. The reconstruction problem requires to estimate one of them given 'far away' observations. Several theoretical results (and simple algorithms) are available when their joint probability distribution is Markov with respect to a tree. In this paper we consider the case of sequences of random graphs that converge locally to trees. In particular, we develop a sufficient condition for the tree and graph reconstruction problem to coincide. We apply such condition to colorings of random graphs.Further, we characterize the behavior of Ising models on such graphs, both with attractive and random interactions (respectively, 'ferromagnetic' and 'spin glass').

The ABC model is a simple diffusive one-dimensional non-equilibrium system which exhibits a phase transition. Here we show that the cumulants of the currents of particles through the system become singular near the phase transition. At the transition, they exhibit an anomalous dependence on the system size (an anomalous Fourier's law). An effective theory for the dynamics of the single mode which becomes unstable at the transition allows one to predict this anomalous scaling. 05.70 Ln, A lot of work has been devoted recently to the study of the fluctuations of the current of heat or of particles through nonequilibrium one dimensional systems [1][2][3][4][5][6][7][8][9][10][11][12]. In such studies the basic quantity one considers is the total flux Q(t) of energy or of particles through a section of the system during time t. In the steady state this flux Q(t) fluctuates due to the randomness of the initial condition for purely deterministic models and due to the noisy dynamics in stochastic models (here we only discuss classical systems: see [13][14][15][16] for the quantum case). If one assumes that the total energy or the total number of particles in the system remains bounded, the average current lim t→∞ Q(t) t as well as the higher cumulants lim t→∞ Q(t) n c t of the flux Q(t) do not depend on the section of the system where this flux is measured.For a one dimensional system of length L, a central question is the size dependence of these cumulants [17]. In particular one would like to know whether a given system satisfies Fourier's law, meaning that, for large L, the average current scales like 1/L:where the prefactor A 1 depends on the temperatures T 1 and T 2 of the two heat baths or on the chemical potentials µ 1 and µ 2 of the two reservoirs of particles at the ends of the system. At equilibrium (T 1 = T 2 or µ 1 = µ 2 ) the prefactor A 1 in (1) vanishes but the question of the validity of Fourier's law remains. One then wants to know whether the second cumulant of Q(t) scales like 1/L.One can show that (2) holds for diffusive systems such as the SSEP (symmetric simple exclusion process) [1,4,18,19] or the KMP (Kipnis-Marchioro-Presutti) model [20]. The macroscopic fluctuation theory developed by Bertini et al. [2,21,22] allows one also to determine[1] all the cumulants of the flux Q(t), with the result that they all scale with system size as 1/L.Even corrections of order 1/L 2 have been computed in some cases [6,7].For mechanical systems with deterministic dynamics, in particular systems which conserve momentum, the average current scales as a non-integer power of the system size:The exponent α takes the value 1/2 for some exactly soluble special models [3]. Values ranging from 0.25 to 0.4 have also been reported in simulations depending on the model considered [23][24][25][26][27]. Theoretical predictions based on a mode coupling approach [28,29] or on renormalization group calculations [30] confirm this anomalous Fourier's law. Less is known on the size dependence of the higher cumulants, which are nu...

Momentum-conserving one-dimensional models are known to exhibit anomalous Fourier's law, with a thermal conductivity varying as a power law of the system size. Here we measure, by numerical simulations, several cumulants of the heat flux of a one-dimensional hard particle gas. We find that the cumulants, like the conductivity, vary as power laws of the system size. Our results also indicate that cumulants higher than the second follow different power laws when one compares the ring geometry at equilibrium and the linear case in contact with two heat baths (at equal or unequal temperatures).