DISORDERED SYSTEMS, RANDOM SPATIAL PROCESSES AND SOME APPLICATIONS

We consider a harmonic chain perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4. We also investigate an interpolation microscopic model which makes the bridge between the heat equation and the skew-fractional heat equation.

Stationary non equilibrium states are characterized by the presence of steady currents flowing through the system and a basic question in statistical mechanics is to understand their structure. Many papers have been devoted to the subject in the context of stochastic interacting particle systems. Usually current density is produced by fixing two different densities at the boundary. We want instead to implement the mass transport by introducing current reservoirs that produce a given current by sending in particles from the left at some rate and taking out particles from the rightmost occupied site at same rate. The removal mechanism is therefore of topological rather than metric nature, since the determination of the rightmost occupied site requires a knowledge of the entire configuration. This prevents from using correlation functions techniques. I will discuss recent results obtained in the study of these topics, whose final purpose is to provide a particle version of a free boundary-type problem.

The cover time is the time it takes for the simple random walk to visit all points of the torus of size NxN. Naively we can hope it compares to the maximum of N^2* independent* i.d. r.v.'s (with the law of hitting time individual points). The two agree for some asymptotics but not for all, and dimension 2 is critical because of long range correlations. Random interlacements yields the description of the covering process at its very end.

We will relate recent work at the interface between interacting particle systems and quantum integrable systems. In particular, we describe an exactly solvable interacting particle system related to higher spin U_q(sl2) vertex models and explain how all of the known exactly solvable models in the KPZ universality class arise from degenerations of this system.

Fluctuations of the current of one dimensional non equilibrium diffusive systems are well understood. After a short review of the one dimensional results, the talk will try to show that the statistics of these fluctuations are exactly the same in higher dimension, for arbitrary geometries. This result can be established by a mapping between the variational problem in finite dimension and in one dimension.
In the particular case of the symmetric exclusion process, the statistical properties of the current are the same as those of disordered quantum wires in the metallic regime, where the distribution of the transmission coefficients of the different channels have been predicted from the theory of random matrices.

We prove that phase transition occurs in the dilute ferromagnetic nearest-neighbour q-state clock model in Z^d, for every q≥2 and d≥2. This follows from the fact that the Edwards-Sokal random-cluster representation of the clock model stochastically dominates a supercritical Bernoulli bond percolation probability, a technique that has been applied to show phase transition for the low-temperature Potts model. The domination involves a combinatorial lemma. Joint work with Inés Armendariz and Nahuel Soprano-Loto.

In this talk I will discuss the height fluctuations of models in the KPZ (Kardar-Parisi-Zhang) universality class with stationary initial conditions. In particular, I will present the solution for the KPZ equation, where the stationary initial height profile is a two-sided Brownian motion. This is a joint work with Alexei Borodin, Ivan Corwin, and Bálint Vető (arXiv: 1407.6977).

I will present the (stochastic and non stochastic) Kuramoto model of disordered oscillators, the prototypical model for synchronization phenomena, from a non equilibrium statistical mechanics viewpoint. I will then focus on recent results on the relaxation dynamics below the synchronization threshold in the non stochastic set-up and. The link with Landau damping in plasma will be discussed.

We consider a system with a time dependent driving and we analyse thermodynamic transformations between equilibrium or non-equilibrium states. We then discuss the role the large deviation rate in the energy balance of a transformation.

I will explain a Renormalization Group approach to the study of existence and uniqueness of solutions to stochastic partial differential equations driven by space-time white noise. As an example I sketch a proof of well-posedness and independence of regularization for the $\phi^4$ model in three dimensions recently studied by Hairer and by Catellier and Chouk.

We consider the exclusion process on a circle with 2n sites and n particles. The particles jumps to neighboring sites independently with rate one, but are forced to respect the exclusion rule: there cannot be more than one particle per site. In this talk we will discuss the convergence to equilibrium of this system in terms of total variation distance to equilibrium, and in particular the cutoff phenomenon: around a time $C N^2\log N$ the distance to equilibrium drops abruptly from $1$ to $0$ in a time window of width $N^2$.

I will discuss some old and some recent work (joint with B. Pittel, D. Ruelle and E. Speer) concerning the relation between the zeros of the grand canonical partition function and different properties,e.g. (Local) Central Limit Theorems, for equilibrium systems. Many results extend to determinantal point processes, such as the distribution of eigenvalues of random matrices, and also to graph counting polynomials.

The use of simple lattice models with short range interactions, like the FPU chains or lattice gases, unveiled the presence of anomalous energy transport in low-dimensional systems. On the other hand, recent numerical studies seem to predict quite contradictory results, also with the predictions of fluctuating hydrodynamics, that provided a solid theoretical ground for the universality of KPZ-scaling characterizing the divergence of transport coefficients in 1d models. We present some new results obtained for gas-like models, performed by the multi-particle-collision (MPC) protocol, that allow to understand the reason of the discrepancy observed in lattice models. Moreover, we shall briefly report on new, partially unexpected scenarions of energy transport in systems with long-range interactions.

For systems of interacting particles, or for interacting diffusions in d dimensions, driven out-of- equilibrium by an external field, a spatial fluctuation relation for the generating function of the current is derived as a consequence of spatial symmetries. Those symmetries are in turn associated to spatial transformations on the physical space that leave invariant the path space measure without driving. This shows that in dimension d ≥ 2 new fluctuation relations arise beyond the Gallavotti- Cohen fluctuation theorem related the time-reversal symmetry.

Modern statistical mechanics offers a large class of driven-dissipative systems that naturally evolve to a critical state. Here we consider two infinite-volume systems: the activated random walks and the stochastic sandpile.
The main goal in this field is to describe the critical behavior, the scaling relations and critical exponents of these systems, and whether their critical density is the same as the long-time limit attained in their driven-dissipative finite-volume version. These questions are however far beyond the reach of current techniques. Due to strong non-locality of correlations and dynamic long-range effects, classical analytic and probabilistic tools fail in most cases of interest, making the rigorous analysis of such systems a major mathematical challenge.
In this talk we will report on the progress obtained in recent years. We will show some novel ideas and techniques which allowed some steps forward in understanding the phase transition in these systems, and discuss some of the open problems.

This talk is based on joint works with C. Bahadoran, T. Mountford, K. Ravishankar. We establish necessary and sufficient conditions for weak convergence to the upper invariant measure for asymmetric nearest neighbour zero range processes with non homogeneous jump rates.

We consider a system of Brownian motions in one-dimension in which the j-th particle is reflected by the (j+1)-th particle with weight p and also by the (j-1)-th particle with weight q. The system is defined more precisely using the local time for positions of neighboring particles.
The system with a symmetric (p=q=1/2) reflection, corresponding to independent Brownian motions with ordering maintained, was introduced by Harris in 1965 and has been studied by many authors since then.
In the totally asymmetric (q=1) case, a particle with smaller index has a priority to the one with a larger index; the latter is simply reflected by the former. The finite particle system for this special case was discussed by Warren and others. More recently, there have been some progress for more general initial conditions by by Ferrai, Spohn and Weiss.
In this presentation we consider generic asymmetric case where 0<p<q<1. The large time properties are expected to be similar to the totally asymmetric case, i.e., to belong to the KPZ universality class. But the techniques for the totally asymmetric case do not work for the general case. Our analysis is based on a duality (in fact a self-duality) property for the process, which allows us to obtain a few formulas for quantities related to current and discuss the asymptotics.
The presentation is based on a collaboration with H. Spohn[SS].
[SS] T. Sasamoto, T. Spohn: Point-interacting Brownian motions in the KPZ universality class, arXiv: 1411.3142

The scalar KPZ equation is essentially equivalent to a disordered system of equilibrium statistical mechanics, namely the directed polymer in a random medium. Interesting physical applications concern several components. Then the mapping seems to break down. I will discuss lattice gases with several species in one dimension and explain the dramatic differences when compared to the scalar case.

Pressure is the mechanical force per unit area that a confined system exerts on its container. In thermal equilibrium, the pressure depends only on bulk properties (density, temperature, etc.) through an equation of state. I will show that in active systems containing self-propelled particles, the pressure instead can depend on the precise interactions between the system's contents and its confining walls. Generic active fluids therefore have no equation of state. I will discuss how one is recovered in certain limiting cases, which include "active Brownian spheres", a much-studied simplified model of self-propelled particles. Even in these cases, the mechanical pressure can exhibit anomalous properties that defy the familiar thermodynamic description of passive fluid materials.

Kinetically Constrained Models (KCM) are interacting particle systemswhich have been introduced by physicists to model the liquid/glass transition and more generally the "glassy behavior" occurringin a large variety of systems. The key feature is that an elementary move is allowed only if the configurationverifies a local constraint which specifies the maximal number of particles in a proper neighborhood. The study of the non equilibrium behavior of KCM poses very challenging problems. On the one hand numerical simulations often do not give clear cut answers in this regime due to the extremely slow dynamics. On the other hand the standard mathematical tools cannot be used due to the the non attractiveness and the failure of classic coercive inequalities to analyze relaxation to equilibrium. In this seminar we will present some results as well as open problems on the non equilibrium behavior for these models, in particular the so called East model.

Stationary non equilibrium states are characterized by the presence of steady currents flowing through the system as a response to external forces. We model this process considering the simple exclusion process in one space dimension with appropriate boundary mechanisms which create particles on the one side and kill particles on the other. The system is designed to model Fick's law which relates the current to the density gradient. In this talk we review some results obtained in collaboration with Anna De Masi, Errico Presutti and Maria Eulalia Vares. For this process, the hydrodynamic limit is given by the linear heat equation with Dirichlet boundary conditions obtained by solving a non-linear equation which fixes the values of the density at the boundary. The rescaled limiting density profile of the (unique) invariant measure of the process coincides with the unique stationary solution of the hydrodynamic equation. Last, we show a spectral gap estimate for the (non equilibrium) stationary process uniformly on the system size.