Related papers: Cutoff phenomenon for asymmetric zero range proces…
We consider the zero-range process with arbitrary bounded monotone rates on the complete graph, in the regime where the number of sites diverges while the density of particles per site converges. We determine the asymptotics of the mixing…
We survey recent results concerning the total-variation mixing time of the simple exclusion process on the segment (symmetric and asymmetric) and a continuum analog, the simple random walk on the simplex with an emphasis on cutoff results.…
We study the mixing time of the unit-rate zero-range process on the complete graph, in the regime where the number $n$ of sites tends to infinity while the density of particles per site stabilizes to some limit $\rho>0$. We prove that the…
We consider the simple exclusion process with $k$ particles on a segment of length $N$ performing random walks with transition $p>1/2$ to the right and $q=1-p$ to the left. We focus on the case where the asymmetry in the jump rates…
We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times: our result show that these two continuous-time Markov chains display cutoff. Our…
We consider the mean-field Zero-Range process in the regime where the potential function $r$ is increasing to infinity at sublinear speed, and the density of particles is bounded. We determine the mixing time of the system, and establish…
We investigate the asymptotic in $N$ of the mixing times of a Markov dynamics on $N-1$ ordered particles in an interval. This dynamics consists in resampling at independent Poisson times each particle according to a probability measure on…
We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial…
We study mixing times of the symmetric and asymmetric simple exclusion process on the segment where particles are allowed to enter and exit at the endpoints. We consider different regimes depending on the entering and exiting rates as well…
Given a sequence $(\mathfrak{X}_i, \mathscr{K}_i)_{i=1}^\infty$ of Markov chains, the cut-off phenomenon describes a period of transition to stationarity which is asymptotically lower order than the mixing time. We study mixing times and…
In this article, we provide detailed analysis of the long-time behavior of the underdamped Langevin dynamics. We first provide a necessary condition guaranteeing that the zero-noise dynamical system converges to its unique attractor. We…
In this article, we prove the cutoff phenomenon for a general class of the discrete-time nonlinear recombination models. This system models the evolution of a probability measure on a finite product space $S^n$ representing the state of…
This paper summarizes results and some open problems about the large-scale and long-time behavior of asymmetric, disordered exclusion and zero-range processes. These processes have randomly chosen jump rates at the sites of the underlying…
The cutoff phenomenon is an abrupt transition from out of equilibrium to equilibrium undergone by certain Markov processes in the limit where the size of the state space tends to infinity: instead of decaying gradually over time, their…
We prove a fluid limit for the coarsening phase of the condensing zero-range process on a finite number of sites. When time and occupation per site are linearly rescaled by the total number of particles, the evolution of the process is…
It has been recently suggested that a totally asymmetric exclusion process with two species on an open chain could exhibit spontaneous symmetry breaking in some range of the parameters defining its dynamics. The symmetry breaking is…
We study asymmetric zero-range processes on Z with nearest-neighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. For any given environment satisfying…
We study the totally asymmetric simple exclusion process with open boundaries in the high density and the low density phase. In the bulk of the two phases, we show that the process on a segment of length $N$ exhibits cutoff at order $N$,…
We consider the contact process on a countable-infinite and connected graph of bounded degree. For this process started from the upper invariant measure, we prove certain uniform mixing properties under the assumption that the infection…
We investigate the mixing properties of a finite Markov chain in random environment defined as a mixture of a deterministic chain and a chain whose state space has been permuted uniformly at random. This work is the counterpart of a…