Related papers: Mixing time for random walk on supercritical dynam…

We prove the sharpness of the phase transition for speed in the biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least 2, and for any supercritical parameter p > p_c, we prove the existence of a…

We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices $\{u,v\}$ with distance $d>1$ is added as a "long-range" edge with probability…

We consider large deviations of the cover time of the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, $d \geq 3$ by simple random walk. We prove a lower bound on the probability that the cover time is smaller than $\gamma\in (0,1)$ times its…

We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and H\"aggstr\"om and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli…

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we investigate what happens when the random…

How does removal of sites by a random walk lead to blockage of percolation? To study this problem of correlated site percolation, we consider a random walk (RW) of $N=uL^d$ steps on a $d$-dimensional hypercubic lattice of size $L^d$ (with…

We study biased random walks on dynamical percolation in $\mathbb{Z}^d$, which were recently introduced by Andres et al. We provide a second order expansion for the asymptotic speed and show for $d \ge 2$ that the speed of the biased random…

Let $\mathcal{T}$ be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean $\mu >1$, conditioned to survive. Let $\varphi_{\mathcal{T}}$ be a random embedding of $\mathcal{T}$ into $\mathbb{Z}^d$ according…

Suppose that $\CG$ is a finite, connected graph and $X$ is a lazy random walk on $\CG$. The lamplighter chain $X^\diamond$ associated with $X$ is the random walk on the wreath product $\CG^\diamond = \Z_2 \wr \CG$, the graph whose vertices…

We consider a random geometric graph obtained by placing a Poisson point process of intensity 1 in the d-dimensional torus of side length n^(1/d) and connecting two points by an edge if their distance is at most r. We consider the case of…

Random walks on expanders play a crucial role in Markov Chain Monte Carlo algorithms, derandomization, graph theory, and distributed computing. A desirable property is that they are rapidly mixing, which is equivalent to having a spectral…

We consider the random walk on a simple point process on $\Bbb{R}^d$, $d\geq2$, whose jump rates decay exponentially in the $\alpha$-power of jump length. The case $\alpha =1$ corresponds to the phonon-induced variable-range hopping in…

Let $\mathcal{C}_1$ be the largest component of the Erd\H{o}s--R\'{e}nyi random graph $\mathcal{G}(n,p)$. The mixing time of random walk on $\mathcal {C}_1$ in the strictly supercritical regime, $p=c/n$ with fixed $c>1$, was shown to have…

We investigate the relation between the local picture left by the trajectory of a simple random walk on the torus (Z/NZ)^d, d >= 3, until u N^d time steps, u > 0, and the model of random interlacements recently introduced by Sznitman. In…

We study a natural random walk on the $n \times n$ upper triangular matrices, with entries in $\mathbb{Z}/m \mathbb{Z}$, generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this…

Consider the random walk on the $n \times n$ upper triangular matrices with ones on the diagonal and elements over $\mathbb{F}_p$ where we pick a row at random and either add it or subtract it from the row directly above it. The main result…

We study the percolation time of the $r$-neighbour bootstrap percolation model on the discrete torus $(\Z/n\Z)^d$. For $t$ at most a polylog function of $n$ and initial infection probabilities within certain ranges depending on $t$, we…

We study the mixing time of the Dikin walk in a polytope - a random walk based on the log-barrier from the interior point method literature. This walk, and a close variant, were studied by Narayanan (2016) and Kannan-Narayanan (2012).…

Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $\Z^d$ up to time $t$. This is the $p$-norm of the vector of the walker's local times, $\ell_t$.…

We analyse the mixing profile of a random walk on a dynamic random permutation, focusing on the regime where the walk evolves much faster than the permutation. Two types of dynamics generated by random transpositions are considered: one…