This book focuses on percolation on high-dimensional lattices. We give a general
introduction to percolation, stating the main results and defining the central objects. We …

Abstract Discovered by Aldous, Diaconis and Shahshahani in the context of card shuffling,
the cutoff phenomenon has since then been established for a variety of Markov chains …

A finite ergodic Markov chain exhibits cutoff if its distance to stationarity remains close to 1
over a certain number of iterations and then abruptly drops to near 0 on a much shorter time …

We study convergence to equilibrium for a 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 …

A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its
initial value over a certain number of iterations and then abruptly drops to near 0 on a much …

The voter model is a classical interacting particle system modelling how consensus is
formed across a network. We analyze the time to consensus for the voter model when the …

It is a fact simple to establish that the mixing time of the simple random walk on a d-regular
graph with n vertices is asymptotically bounded from below by. Such a bound is obtained by …

It is well known that the behaviour of a random subgraph of ad-dimensional hypercube,
where we include each edge independently with probability p, undergoes a phase transition …

Consider the random Cayley graph of a finite group $ G $ with respect to $ k $ generators
chosen uniformly at random, with $1\ll\log k\ll\log| G| $; denote it $ G_k $. A conjecture of …

We establish universality of cutoff for simple random walk on a class of random graphs
defined as follows. Given a finite graph G=(V, E) with| V| even we define a random graph …