Can one hear the shape of a random walk?

2024/12/27 by Larsen, Michael J.
#60G50 (Primary) 58J53 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2412.19762

Abstract

To what extent is the underlying distribution of a finitely supported unbiased random walk on ℤ determined by the sequence of times at which the walk returns to the origin? The main result of this paper is that, in various senses, most unbiased random walks on ℤ are determined up to equivalence by the sequence I₁,I₂,I₃,…, where I_n denotes the probability of being at the origin after n steps. We also give an application to an inverse problem from asymptotic representation theory. The proof uses Laplace's method and a delicate Galois-theoretic analysis which ultimately depends on the classification of finite simple groups.

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