Related papers: Three steps mixing for general random walks on the…
We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense…
The step-reinforced random walk (SRRW), where each step may replicate a randomly chosen past step, exhibits complex dependencies on the history. This paper introduces a generalized SRRW on groups, incorporating arbitrary transformations of…
We recently demonstrated that standard fixed-time lattice random-walk models cannot be modified to properly represent biased diffusion processes in more than two dimensions. The origin of this fundamental limitation appears to be the fact…
We implement the proof of principle for the quantum walk of one ion in a linear ion trap. With a single-step fidelity exceeding 0.99, we perform three steps of an asymmetric walk on the line. We clearly reveal the differences to its…
We investigate random walks on a lattice with imperfect traps. In one dimension, we perturbatively compute the survival probability by reducing the problem to a particle diffusing on a closed ring containing just one single trap. Numerical…
Activated Random Walks, on $\mathbb{Z}^d$ for any $d\geqslant 1$, is an interacting particle system, where particles can be in either of two states: active or frozen. Each active particle performs a continuous-time simple random walk during…
We consider the set of points visited by the random walk on the discrete torus $(\mathbb{Z}/N\mathbb{Z})^d$, for $d \geq 3$, at times of order $uN^d$, for a parameter $u>0$ in the large-$N$ limit. We prove that the vacant set left by the…
A random walk scheme, consisting of alternating phases of regular Brownian motion and L\'evy walks, is proposed as a model for run-and-tumble bacterial motion. Within the continuous-time random walk approach we obtain the long-time and…
We are interested in the Guivarc'h inequality for admissible random walks on finitely generated relatively hyperbolic groups, endowed with a word metric. We show that for random walks with finite super-exponential moment, if this inequality…
We study the random walk on the symmetric group $S_n$ generated by the conjugacy class of cycles of length $k$. We show that the convergence to uniform measure of this walk has a cut-off in total variation distance after $\frac{n}{k} log n$…
We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate \mu\ while at the same time a random walker moves on G at rate 1 but only along…
Hitting times for discrete quantum walks on graphs give an average time before the walk reaches an ending condition. To be analogous to the hitting time for a classical walk, the quantum hitting time must involve repeated measurements as…
We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph G is universal mixing if the instantaneous or average probability distribution of the quantum walk on G ranges over…
For $d\ge 3$ we construct a new coupling of the trace left by a random walk on a large $d$-dimensional discrete torus with the random interlacements on $\mathbb Z^d$. This coupling has the advantage of working up to macroscopic subsets of…
In the last twenty years network science has proven its strength in modelling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Yet, in many relevant cases, interactions are not pairwise but…
We study the neighborhoods of a typical point $Z_n$ visited at $n$-th step of a random walk, determined by the condition that the transition probabilities stay close to $\mu^{*n}(Z_n)$. If such neighborhood contains a ball of radius $C…
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…
In random walk theory, it is customary to assume that a given walk is irreducible and/or aperiodic. While these prevailing assumptions make particularly tractable the analysis of random walks and help to highlight their diffusive nature,…
Among random sampling methods, Markov Chain Monte Carlo algorithms are foremost. Using a combination of analytical and numerical approaches, we study their convergence properties towards the steady state, within a random walk Metropolis…
Hit-and-Run is known to be one of the best random sampling algorithms, its mixing time is polynomial in dimension. Nevertheless, in practice the number of steps required to achieve uniformly distributed samples is rather high. We propose…