We study collision times of many independent simple random walks on $\mathbb Z^2$ through the joint moment generating function of their pairwise collision local times. For a fixed number of walks, these collision times are known to be asymptotically independent after a suitable logarithmic normalisation. We investigate the extent to which this independence persists when the number of walks grows. For $N$ being the walk length, our results identify a threshold $\asymp (\log N)^{\frac{1}{3}}$, on which the transition from independence to dependence happens. The proofs combine chaos expansion techniques and a correlation inequality, which is the result of a local limit theorem.
We study the scaling limit of the collision measure of two i.i.d. discrete-time random walks on $\mathbb Z^2$, which records their collision sites and times. This is a critical regime for collisions: the walks collide infinitely often, whereas the limiting Brownian motions do not collide at positive times. Hence, exist...
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
We study the limit shape of the range of the random walk excited to the center on $\mathbb Z^d$, with $d\geq2$. A cookie is placed at each nonzero vertex. On its first visit to such a vertex, the walker consumes the cookie and takes its next nearest-neighbor step with a bias toward the origin. On a visit to the origin,...
The probability that several independent random walks avoid one another decays algebraically with the number of steps $N\to\infty$ in two and three dimensions, $P_N\sim N^{-\xi/2}$, where the intersection exponent $\xi$ depends on the number of random walks. More generally, one may consider several groups of independen...
Qi-Yuan Shi, Run-Sheng Liu, Xin-Yi Li et al.· 0 citations
We determine the limiting distribution of the cover time of simple random walk on the two-dimensional discrete torus. For the continuous-time walk with total jump rate one on $(\mathbb Z/N\mathbb Z)^2$, let $T_N$ denote its cover time. We prove that $\frac{T_N}{(2/\pi)N^2\log N}-2\log N+\log\log N \Longrightarrow G+\lo...
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...
Antoine Aurillard· 0 citations
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