Related papers: Note: Random-to-front shuffles on trees

We study Markov chains on $\mathbb Z^m$, $m\geq 2$, that behave like a standard symmetric random walk outside of the hyperplane (membrane) $H=\{0\}\times \mathbb Z^{m-1}$. The transition probabilities on the membrane $H$ are periodic and…

Consider the Aldous--Pitman fragmentation process [Ann Probab, 26(4):1703--1726, 1998] of a Brownian continuum random tree ${\cal T}^{\mathrm{br}}$. The associated cut tree cut$({\cal T}^{\mathrm{br}})$, introduced by Bertoin and Miermont…

We consider the problem of defining and fitting models of autoregressive time series of probability distributions on a compact interval of $\mathbb{R}$. An order-$1$ autoregressive model in this context is to be understood as a Markov…

Let $Q$ be a probability measure on a finite group $G$, and let $H$ be a subgroup of $G$. We show that a necessary and sufficient condition for the random walk driven by $Q$ on $G$ to induce a Markov chain on the double coset space…

We consider random walk on a finite group $G$ as follows. We can consider $G$ as a group of substitutions. Randomly (i.e. with probability $U(g)=|G|^{-1}$ ) we choose a substitution $g \in G$ and execute it twice in a row, i.e. execute a…

We give three different criteria for transience of a Branching Markov Chain. These conditions enable us to give a classification of Branching Random Walks in Random Environment (BRWRE) on Cayley Graphs in recurrence and transience. This…

A deck of $n$ cards is shuffled by repeatedly moving the top card to one of the bottom $k_n$ positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as $k_n$ ranges from a constant…

Recently, a phase transition phenomenon has been established for parking on random trees. We extend the results of Curien and H\'enard on general Galton--Watson trees and allow different car arrival distributions depending on the vertex…

We compute the eigenvalues and eigenspaces of random-to-random Markov chains. We use a family of maps which reveal a remarkable recursive structure of the eigenspaces, yielding an explicit and effective construction of all eigenbases…

We study various classes of random processes defined on the regular tree $T_d$ that are invariant under the automorphism group of $T_d$. Most important ones are factor of i.i.d. processes (randomized local algorithms), branching Markov…

We define an analog of Plancherel measure for the set of rooted unlabeled trees on n vertices, and a Markov chain which has this measure as its stationary distribution. Using the combinatorics of commutation relations, we show that order…

We consider a Brownian motion with linear drift that splits at fixed time points into a fixed number of branches, which may depend on the branching point. For this process, which we shall refer to as the Brownian decision tree, we…

We introduce an algorithm for generating a random sequence of fragmentation trees, which we call the ancestral branching algorithm. This algorithm builds on the recursive partitioning structure of a tree and gives rise to an associated…

This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a…

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 shorter time scale. Originally discovered in the…

We develop a new bidirectional algorithm for estimating Markov chain multi-step transition probabilities: given a Markov chain, we want to estimate the probability of hitting a given target state in $\ell$ steps after starting from a given…

The motivation for this paper is the study of the phase transition for recurrence/transience of a class of self-interacting random walks on trees, which includes the once-reinforced random walk. For this purpose, we define a quantity, that…

In this paper, we study a regular rooted coloured tree with random labels assigned to its edges, where the distribution of the label assigned to an edge depends on the colours of its endpoints. We obtain some new results relevant to this…

Locally Markov walks are natural generalizations of classical Markov chains, where instead of a particle moving independently of the past, it decides where to move next depending on the last action performed at the current location. We…

We consider continuous-space, discrete-time Markov chains on $\mathbb{R}^d$, that admit a finite number $N$ of metastable states. Our main motivation for investigating these processes is to analyse random Poincar\'e maps, which describe…