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The asymmetric simple exclusion process and its analysis by mode coupling theory (MCT) is reviewed. To treat the weakly asymmetric case at large space scale $x\varepsilon^{-1}$, %(corresponding to small Fourier momentum at scale $p\varepsilon$), large time scale $t \varepsilon^{-\chi}$ and weak hopping bias $b \varepsilon^{\kappa}$ in the limit $\varepsilon \to 0$ we develop a mesoscale MCT that allows for studying the crossover at $\kappa=1/2$ and $\chi=2$ from Kardar-Parisi-Zhang (KPZ) to Edwards-Wilkinson (EW) universality. The dynamical structure function is shown to satisfy for all $\kappa$ an integral equation that is independent of the microscopic model parameters and has a solution that yields a scale-invariant function with the KPZ dynamical exponent $z=3/2$ at scale $\chi=3/2+\kappa$ for $0\leq\kappa1/2$. At the crossover point it is a function of both scaling variables which converges at macroscopic scale to the conventional MCT approximation of KPZ universality for $\kappa1/2$ despite the numerous uncontrolled approximations on which MCT is based., Comment: 29 pages, 1 figure

For the supercritical contact process on the hyper-cubic lattice started from a single infection at the origin and conditioned on survival, we establish two uniformity results for the hitting times $t(x)$, defined for each site $x$ as the first time at which it becomes infected. First, the family of random variables $(t(x)-t(y))/|x-y|$, indexed by $x \neq y$ in $\mathbb{Z}^d$, is stochastically tight. Second, for each $\varepsilon >0$ there exists $x$ such that, for infinitely many integers $n$, $t(nx) < t((n+1)x)$ with probability larger than $1-\varepsilon$. A key ingredient in our proofs is a tightness result concerning the essential hitting times of the supercritical contact process introduced by Garet and Marchand (Ann.\ Appl.\ Probab., 2012)., 11 pages

We consider the KPZ fixed point starting from a general class of initial data. In this article, we study the growth of the large peaks of the KPZ fixed point at a spatial point $0$ when time $t$ goes to $\infty$ and when $t$ approaches $1$. We prove that for a very broad class of initial data, as $t\to \infty$, the limsup of the KPZ fixed point height function when scaled by $t^{1/3}(\log\log t)^{2/3}$ almost surely equals a constant. The value of the constant is $(3/4)^{2/3}$ or $(3/2)^{2/3}$ depending on the initial data being non-random or Brownian respectively. Furthermore, we show that the increments of the KPZ fixed point near $t=1$ admits a short time law of iterated logarithm. More precisely, as the time increments $\Delta t :=t-1$ goes down to $0$, for a large class of initial data including the Brownian data initial data, we show that limsup of the height increments the KPZ fixed point near time $1$ when scaled by $(\Delta t)^{1/3}(\log\log (\Delta t)^{-1})^{2/3}$ almost surely equals $(3/2)^{2/3}$., Comment: 35 pages, no figures

We consider the equilibrium perturbations for two stochastic systems: the $d$-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order $N^{-\alpha}$ to the equilibrium profile and speed up the process by $N^{1+\kappa}$ for parameters $0, Comment: 30 pages, accepted version

We consider a one-dimensional discrete-space birth process with a bounded number of particle per site. Under the assumptions of the finite range of interaction, translation invariance, and non-degeneracy, we prove a shape theorem. We also derive a limiting estimate and an exponential estimate on the fluctuations of the position of the rightmost particle., 17 pages; a short section devoted to numerical simulations is added; other small changes and improvements

We introduce and study the mutating contact process, a variant of the multitype contact process, where one type mutates at a constant rate to the other type. We prove that on $\mathbb{Z}$ a single mutant cannot survive while on $\mathbb{T}_{d}$ there are distinct weak survival and extinction of a single mutant phases, yet the limiting distribution concentrates on configurations with no mutants of the first type for any values of the parameters., 17 pages, 1 figure

We identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior, respectively (including zero-range, misanthrope and many other models). The initial condition is a step profile which, in some classical cases of asymmetric models, gives rise to a rarefaction fan scenario. We also point out a model with non-concave, non-convex hydrodynamics, where the rescaled second class particle distribution has both continuous and discrete counterparts. The results follow from a substantial generalization of P.A. Ferrari and C. Kipnis' arguments ("Second class particles in the rarefaction fan", Ann. Inst. H. Poincar\'e, 31, 1995) for the totally asymmetric simple exclusion process. The main novelty is the introduction of a signed coupling measure as initial data, which nevertheless results in a proper probability initial distribution for the site of the second class particle and makes the extension possible. We also reveal in full generality a very interesting invariance property of the one-site marginal distribution of the process underneath the second class particle which in particular proves the intrinsicality of our choice for the initial distribution. Finally, we give a lower estimate on the probability of survival of a second class particle-antiparticle pair., Comment: 25 pages, 3 figures. Improved exposition after referee's comments. Accepted for publication in the Annals of Probability

This paper investigates the non-equilibrium hydrodynamic behavior of a simple totally asymmetric interacting particle system of particles, antiparticles and holes on $\mathbb{Z}$. Rigorous hydrodynamic results apply to our model with a hydrodynamic flux that is exactly calculated and shown to change convexity in some region of the model parameters. We then characterize the entropy solutions of the hydrodynamic equation with step initial condition in this scenario which include various mixtures of rarefaction fans and shock waves. We highlight how the phase diagram of the model changes by varying the model parameters., 9 pages, 4 figures in J. Stat. Phys. (2016)

We introduce the Mass Migration Process (MMP), a conservative particle system on ${\mathbb N}^{{\mathbb Z}^d}$. It consists in jumps of $k$ particles ($k\ge 1$) between sites, with a jump rate depending only on the state of the system at the departure and arrival sites of the jump. It generalizes misanthropes processes, hence in particular zero range and target processes. After the construction of MMP, our main focus is on its invariant measures. We obtain necessary and sufficient conditions for the existence of translation-invariant and invariant product probability measures. In the particular cases of asymmetric mass migration zero range and mass migration target dynamics, these conditions yield explicit solutions. If these processes are moreover attractive, we obtain a full characterization of all translation-invariant, invariant probability measures. We also consider attractiveness properties (through couplings) and condensation phenomena for MMP. We illustrate our results on many examples; in particular, we prove the coexistence of both condensation and attractiveness in one of them., 46 pages

We obtain estimates on the exponential rate of decay of the relative entropy from equilibrium for Markov processes with a non-local infinitesimal generator. We adapt some of the ideas coming from the Bakry-Emery approach to this setting. In particular, we obtain volume- independent lower bounds for the Glauber dynamics of interacting point particles and for various classes of hardcore models.

Zero-range processes with decreasing jump rates exhibit a condensation transition, where a positive\ud fraction of all particles condenses on a single lattice site when the total density exceeds a critical\ud value. We study the onset of condensation, i.e. the behaviour of the maximum occupation number\ud after adding or subtracting a subextensive excess mass of particles at the critical density. We establish\ud a law of large numbers for the excess mass fraction in the maximum, which turns out to jump\ud from zero to a positive value at a critical scale. Our results also include distributional limits for the\ud fluctuations of the maximum, which change from standard extreme value statistics to Gaussian when\ud the density crosses the critical point. Fluctuations in the bulk are also covered, showing that the mass\ud outside the maximum is distributed homogeneously. In summary, we identify the detailed behaviour\ud at the critical scale including sub-leading terms, which provides a full understanding of the crossover\ud from sub- to supercritical behaviour.

We give a full description for the shape of the classical (2+1)\Dim Solid-On-Solid model above a wall, introduced by Temperley (1952). On an $L\times L$ box at a large inverse-temperature $\beta$ the height of most sites concentrates on a single level $h = \lfloor (1/4\beta)\log L\rfloor$ for most values of $L$. For a sequence of diverging boxes the ensemble of level lines of heights $(h,h-1,...)$ has a scaling limit in Hausdorff distance iff the fractional parts of $(1/4\beta)\log L$ converge to a noncritical value. The scaling limit is explicitly given by nested distinct loops formed via translates of Wulff shapes. Finally, the $h$-level lines feature $L^{1/3+o(1)}$ fluctuations from the side boundaries., Comment: 5 pages

This paper is the continuation of our earlier paper, where we proved t^{1/3}-order of current fluctuations across the characteristics in a class of one dimensional interacting systems with one conserved quantity. We also claimed two models with concave hydrodynamic flux which satisfied the assumptions which made our proof work. In the present note we show that the totally asymmetric exponential bricklayers process also satisfies these assumptions. Hence this is the first example with convex hydrodynamics of a model with t^{1/3}-order current fluctuations across the characteristics. As such, it further supports the idea of universality regarding this scaling., 34 pages, revised version after referees comments, accepted at JSP. A few explanations added, some proofs shortened in the Appendix

We consider the hydrodynamic behavior of some conservative particle systems with degenerate jump rates without exclusive constraints. More precisely, we study the particle systems without restrictions on the total number of particles per site with nearest neighbor exchange rates which vanish for certain configurations. Due to the degeneracy of the rates, there exists blocked configurations which do not evolve under the dynamics and all of the hyperplanes of configurations with a fixed number particles can be decomposed into different irreducible sets. We show that, for initial profiles smooth enough and bounded away from zero, the macroscopic density profile evolves under the diffusive time scaling according to a nonlinear diffusion equation (which we call the modified porous medium equation). The proof is based on the Relative Entropy method but it cannot be straightforwardly applied because of the degeneracy., 17 pages

We prove almost sure Euler hydrodynamics for a large class of attractive particle systems on $\Z$ starting from an arbitrary initial profile. We generalize earlier works by Sepp\"al\"ainen (1999) and Andjel et al. (2004). Our constructive approach requires new ideas since the subadditive ergodic theorem (central to previous works) is no longer effective in our setting., Comment: 55 pages, LaTeX2e, corrected misprints, new title, revised abstract and introduction, references added. Mistakes in section 3 of the previous version corrected: to do so a few new arguments have been added

We consider two-species exclusion processes on the d-dimensional discrete torus taking the effects of exchange, creation and annihilation into account. The model is, in general, of nongradient type. We prove that the (charged) particle density converges to the solution of a certain nonlinear diffusion equation under the diffusive rescaling in space and time. We also prove a lower bound on the spectral gap for the generator of the process confined in a finite volume., 28 pages

We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik, are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the so-called Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams., 29 pages

We define stabilizability of an infinite volume height configuration and of a probability measure on height configurations. We show that for high enough densities, a probability measure cannot be stabilized. We also show that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model (ASM) is a maximal element of the set of stabilizable measures. In that sense the self-organized critical behavior of the ASM can be understood in terms of an ordinary transition between stabilizable and non-stabilizable, 17 pages, appeared in Markov Processes and Related Fields 2005

We develop a general technique, based on a Bochner-type identity, to estimate spectral gaps of a class of Markov operator. We apply this technique to various interacting particle systems. In particular, we give a simple and short proof of the diffusive scaling of the spectral gap of the Kawasaki model at high temperature. Similar results are derived for Kawasaki-type dynamics in the lattice without exclusion, and in the continuum. New estimates for Glauber-type dynamics are also obtained.

An unbounded one-dimensional solid-on-solid model with integer heights is studied. Unbounded here means that there is no a priori restrictions on the discrete gradient of the interface. The interaction Hamiltonian of the interface is given by a finite range part, proportional to the sum of height differences, plus a part of exponentially decaying long range potentials. The evolution of the interface is a reversible Markov process. We prove that if this system is started in the center of a box of size $L$ after a time of order $L^3$ it reaches, with a very large probability, the top or the bottom of the box.

We consider the 2D stochastic Ising model evolving according to the Glauber dynamics at zero temperature. We compute the initial drift for droplets which are suitable approximations of smooth domains. A specific spatial average of the derivative at time~0 of the volume variation of a droplet close to a boundary point is equal to its curvature multiplied by a direction dependent coefficient. We compute the explicit value of this coefficient., 46 pages. A modified version thanks to Herbert Spohn comments

We prove that the logarithmic-Sobolev constant for Zero-Range Processes in a box of diameter $L$ may depend on $L$ but not on the number of particles. This is a first, but relevant and quite technical step, in the proof that this logarithmic-Sobolev constant grows as the square of $L$, that is presented in a forthcoming paper.

The authors in a previous paper proved the hydrodynamic incompressible limit in $d\ge 3$ for a thermal lattice gas, namely a law of large numbers for the density, velocity field and energy. In this paper the equilibrium fluctuations for this model are studied and a central limit theorem is proved for a suitable modification of the vector fluctuation field $\z(t)$, whose components are the density, velocity and energy fluctuations fields. We consider a modified fluctuation field $\xi^\e(t)=\exp \{-\ve^{-1}t E\}\z^\ve$, where $E$ is the linearized Euler operator around the equilibrium and prove that $\xi^\e(t)$ converges to a vector generalized Ornstein-Uhlenbeck process $\xi(t)$, which is formally solution of the stochastic differential equation $d \xi(t)=N\xi(t)dt+ B dW_t$, with $ BB^*=-2 NC$, where $C$ is the compressibility matrix, $N$ is a matrix whose entries are second order differential operators and $B$ is a mean zero Gaussian field. The relation $-2NC=BB^*$ is the fluctuation-dissipation relation., Comment: 32 pages

We provide two methods to construct zero-range processes with superlinear rates on ${\mathbb Z}^d$. In the first method these rates can grow very fast, if either the dynamics and the initial distribution are translation invariant or if only nearest neigbour translation invariant jumps are permitted, in the one-dimensional lattice. In the second method the rates cannot grow as fast but more general dynamics are allowed., Comment: 37 pages