In the Pearson random walk the direction of the i-th step is a random variable uniformly distributed on the d-dimensional sphere, and its length is a non-negative random variable. In a general framework, we are going to study the asymptotic behaviour of the Pearson random walks when the dimension d goes to infinity. Further, we investigate the same convergence result in a more general framework, where both the number and the lengths of the steps depend on the dimension. The results in the present paper can be applied to approximate the distribution of some classical Pearson random walk, for example, in the Dirichlet case.
In the paper the asymptotic behavior of a pair of random walks on the hypercube is investigated. The joint distribution of the walks is specified by a Markov chain constructed optimally to minimize the mean time to their coalescence. Under various scenarios of the growth of the hypercube’s dimension and the initial dis...
A. O. Misnik, I. Nishchenko· Kibernetyka ta Systemnyi Ana...· 0 citations
In this paper, we prove that the fluctuations of the graph distance and the effective resistance on the trace of a random walk in four and five dimensions converge in distribution to a stable law. In previous work, the first and second authors proved that the corresponding fluctuations converge to a Gaussian distributi...
A. Adhikari, Izumi Okada, D. Shiraishi· 0 citations
For simple random walk on infinite clusters of Bernoulli bond percolation on Cayley graphs of hyperbolic groups, we prove that the drift and asymptotic entropy depend continuously on the percolation parameter throughout the supercritical phase. Together with the dimension formula, this implies continuity of the Hausdor...
We consider scaling limits of integer-valued random walks whose transition probabilities are translation-invariant everywhere except on a set that we will call a membrane or an interface. If the jumps outside the membrane are i.i.d. mean-zero random variables with finite variance, then possible examples of the correspo...
We determine the exact first-order asymptotics of the expected optimal cost in two-dimensional random bipartite matching, for every finite power cost $q \ge 1$, on the flat torus. In the endpoint case $q=1$, this answers a question by Talagrand, in the periodic case. The argument involves the closely related asymptotic...
Victor Armegioiu, Michael Goldman, Francesco Grotto et al.· 0 citations
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