We consider a class of infinite critical tree-indexed random walks on $\mathbb Z$, where the motion of particles is subject to vertex reinforcement. We mainly focus on the strong reinforcement regime, where we expect the process to localize almost surely on two sites. Part of our analysis includes the study of a time-dependent generalized P\'olya urn process, where the number of draws at each step is prescribed by a sequence $(\sigma_n)_{n\ge 1}$ of arbitrary positive integers, and the probability to pick a ball of a given color is proportional to a function of the {\it number} of balls of that color. In particular for bounded sequences $(\sigma_n)_{n\ge 1}$, we recover Rubin's characterization for the fixation of one color.
A vertex-reinforced random walk steps to a neighbour with probability proportional to $1+\beta n^{a}$, where $n$ counts previous visits to that neighbour and $a\in(0,1)$ sets the memory strength. On the rooted $b$-ary tree the exponential growth of the vertex set drives the walk outward while the reinforcement pulls it...
We study the linear growth rate of the range of size-conditioned Branching Random Walks (BRW) when the offspring distribution $\mu$ is critical and attracted to an $\alpha$-stable law. This is done via the infinite invariant BRW introduced by Le Gall&Lin and a new criterion which relates this growth rate of the range t...
We study the superdiffusive behavior of random walks on the randomly oriented Manhattan lattice, i.e., the $d$-dimensional integer lattice $\mathbb{Z}^d$ where each axis-aligned line is independently assigned a random direction (forward or backward) with equal probability. The walker takes nearest-neighbor steps, choos...
In this paper, we study the random walk on the symmetric group $\mathfrak{S}_n$ generated by the conjugacy class of $k$-cycles, where $2\le k=o(n/(\log n)^4)$. We prove that the walk exhibits hitting-time mixing: at the first time when every card has been touched, the distribution is already close to equilibrium. For o...
Chen Shang, Jiahe Shen, Jiyue Zeng et al.· 1 citation
We consider a competition between two independent random walks on a cycle of length $N$. Each vertex is claimed by the walker that visits it first, and remains claimed thereafter. We prove that if the initial distance between the walkers is $d$, then the expected number of edges whose endpoints are claimed by different...
S. Chatterjee, Nadya Nabahi, Grigory Terlov· 1 citation
Mixing time bounds for Markov chains play a central role in characterizing the sample complexity of learning and inference from correlated data. While the mixing behavior of symmetric random walks on standard graph structures such as cycles, tori, and hypercubes is well understood, the impact of transition asymmetry re...
Mrudula A Mahindrakar, Hrushikesh A Kant, Avhishek Chatterjee· International Conference on...· 0 citations
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