Related papers: Green geometry, Martin boundary and random walk as…

A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq \alpha\leq\beta\leq1$, there is a group $\Gamma$ with measure $\mu$ equidistributed on a finite…

Consider a closed surface $S$ with negative Euler characteristic, and an admissible probability measure on the fundamental group of $S$ with finite first moment with respect to some hyperbolic metric on $S$. Corresponding to each point in…

Let $\left\{ S_{n},n\geq 0\right\} $ be a random walk whose increment distribution belongs without centering to the domain of attraction of an $% \alpha $-stable law, i.e., there are some scaling constants $a_{n}$ such that the sequence…

The main aim of the present set of notes is to give new, short and essentially self-contained proofs of some classical, as well as more recent, results about random walks on groups. For instance, we shall see that the drift characterization…

We investigate the Martin boundary of the space-time Markov chain associated to a finitely supported random walk $(\Gamma, \mu)$ with spectral radius $\rho$ and relate it to several classical compactifications of $\Gamma$. Assuming the…

Let $\mu$ and $\nu$ be probability measures on a group \Gamma and let G_\mu and G_\nu denote Green's function with respect to \mu and \nu . The group \Gamma is said to admit instability of Green's function if there are symmetric, finitely…

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an…

A classical construction associates to a transient random walk on a discrete group $\Gamma$ a compact $\Gamma$-space $\partial_M \Gamma$ known as the Martin boundary. The resulting crossed product $C^*$-algebra $C(\partial_M \Gamma)…

We establish an invariance principle for a one-dimensional random walk in a dynamical random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite…

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is…

We consider a run-and-tumble particle whose speed and tumbling rate are space-dependent on an infinite line. Unlike most of the previous work on such models, here we make the physical assumption that at large distances, these rates saturate…

We consider super-diffusive L\'evy walks in $d \geqslant 2$ dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean.…

Let ${\cal T}$ be a rooted Galton-Watson tree with offspring distribution $\{p_k\}$ that has $p_0=0$, mean $m=\sum kp_k>1$ and exponential tails. Consider the $\lambda$-biased random walk $\{X_n\}_{n\geq 0}$ on ${\cal T}$; this is the…

Let $\mathcal G$ be an infinite graph -- not necessarily one-ended -- on which the simple random walk is transient. We define a variant of the continuous-time random walk on $\mathcal G$ which reaches $\infty$ in finite time and "reflects…

For a finitely generated group G and a banach space X let \alpha^*_X(G) (respectively \alpha^#_X(G)) be the supremum over all \alpha\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such…

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin…

The discrete Green's function (without boundary) $\mathbb{G}$ is a pseudo-inverse of the combinatorial Laplace operator of a graph $G=(V,E)$. We reveal the intimate connection between Green's function and the theory of exact stopping rules…

A complete representation of the Martin boundary of killed random walks on the quadrant ${\mathbb{N}}^*\times{\mathbb{N}}^*$ is obtained. It is proved that the corresponding full Martin compactification of the quadrant…

We consider the limit behavior of a one-dimensional random walk with unit jumps whose transition probabilities are modified every time the walk hits zero. The invariance principle is proved in the scheme of series where the size of…

In [BEI] we introduced a Levy process on a hierarchical lattice which is four dimensional, in the sense that the Green's function for the process equals 1/x^2. If the process is modified so as to be weakly self-repelling, it was shown that…