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We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $mathbb{Z}times Gamma$, with $Gamma$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.
We study the asymptotic behaviour of once-reinforced biased random walk (ORbRW) on Galton-Watson trees. Here the underlying (unreinforced) random walk has a bias towards or away from the root. We prove that in the setting of multiplicative once-reinf
In this paper, we reveal the branching structure for a non-homogeneous random walk with bounded jumps. The ladder time $T_1,$ the first hitting time of $[1,infty)$ by the walk starting from $0,$ could be expressed in terms of a non-homogeneous multit
We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is an
We consider random walk on dynamical percolation on the discrete torus $mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation en
Consider an arbitrary transient random walk on $Z^d$ with $dinN$. Pick $alphain[0,infty)$ and let $L_n(alpha)$ be the spatial sum of the $alpha$-th power of the $n$-step local times of the walk. Hence, $L_n(0)$ is the range, $L_n(1)=n+1$, and for int