Can one hear the shape of a random walk?

TL;DR

This paper investigates whether the distribution of a finitely supported unbiased random walk on integers can be uniquely identified by the sequence of return probabilities, revealing that most walks are determined by this sequence.

Contribution

It demonstrates that, in various senses, most unbiased random walks on integers are uniquely determined by their return probability sequence, linking probabilistic and algebraic analysis.

Findings

Most unbiased random walks are determined by their return probabilities.

Application to an inverse problem in asymptotic representation theory.

Uses advanced mathematical tools including Galois theory and classification of finite simple groups.

Abstract

To what extent is the underlying distribution of a finitely supported unbiased random walk on $\mathbb{Z}$ 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 $\mathbb{Z}$ are determined up to equivalence by the sequence $I_1,I_2,I_3,\ldots$, 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.