Related papers: Cutoff and mixing time for transient random walks …

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…

We study the mixing time of the Rook's Walk Markov chain on a $d$-dimensional chess board of side length $n\geq 3$, where a rook moves by first selecting an axis uniformly at random and then selecting a new position along that axis…

We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d.\ random field of obstacles. When the particle hits an obstacle, it disappears with a positive probability. We obtain quenched and annealed bounds on the tails of the…

The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often…

We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction…

Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the environment is frozen between a fixed sequence of times (called the cooling map) where it is resampled. Naturally the…

We prove that the mixing time of driven-dissipative activated random walk on an interval of length $n$ with uniform or central driving exhibits cutoff at $n$ times the critical density for activated random walk on the integers. The proof…

We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive `step' or a multiplicative `jump'. When the probability of making a jump tends to zero as an appropriate power of $n$ we prove the…

We study the asymptotic behaviour of occupation times of a transient random walk in quenched random environment on a strip in a sub-diffusive regime. The asymptotic behaviour of hitting times, which is a more traditional object of study, is…

We prove a cutoff for the random walk on random $n$-lifts of finite weighted graphs, even when the random walk on the base graph $\mathcal{G}$ of the lift is not reversible. The mixing time is w.h.p. $t_{mix}=h^{-1}\log n$, where $h$ is a…

We examine the mixing time for random walks on graphs. In particular we are interested on investigating graphs with bottlenecks. Furthermore, the cutoff phenomenon is examined.

We consider a random walk in a random environment (RWRE) on the strip of finite width $\mathbb{Z} \times \{1,2,\ldots,d\}$. We prove both quenched and averaged large deviation principles for the position and the hitting times of the RWRE.…

We study convergence to equilibrium for a large class of Markov chains in random environment. The chains are sparse in the sense that in every row of the transition matrix $P$ the mass is essentially concentrated on few entries. Moreover,…

It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparing the walk…

This paper considers non-backtracking random walks on random graphs generated according to the configuration model. The quantity of interest is the scaling of the mixing time of the random walk as the number of vertices of the random graph…

We study the simple random walk on trees and give estimates on the mixing and relaxation time. Relying on a recent characterization by Basu, Hermon and Peres, we give geometric criteria, which are easy to verify and allow to determine…

For birth and death chains, we derive bounds on the spectral gap and mixing time in terms of birth and death rates. Together with the results of Ding et al. in 2010, this provides a criterion for the existence of a cutoff in terms of the…

We investigate the mixing properties of a model of reversible Markov chains in random environment, which notably contains the simple random walk on the superposition of a deterministic graph and a second graph whose vertex set has been…

In this thesis, we study the diffusive and ballistic behaviors of random walk in random environment (RWRE) in an integer lattice with dimension at least 2. Our contributions are in three directions: a conditional law of large numbers and…

We consider dynamical percolation on the complete graph $K_n$, where each edge refreshes its state at rate $\mu \ll 1/n$, and is then declared open with probability $p = \lambda/n$ where $\lambda > 1$. We study a random walk on this…