Can one hear the shape of a random walk?

TL;DR

The paper addresses whether the return-probability spectrum $I_n$ determines the underlying finitely supported unbiased random walk on ${\mathbb Z}$. It combines Laplace-method asymptotics with a Galois-theoretic, monodromy-based analysis to show that, for primitive unbiased walks of degree $n$ with $n \neq 10$ and $n$ not a square, the spectrum determines the walk up to equivalence; in generic cases the data suffices to recover the generating function $\chi(t)$ uniquely. Key steps include deriving Puiseux expansions via $L(s)$ and its real-argument variant, and translating the problem into a primitive Galois-group setting where the action on conjugates yields identification of the halves $\chi^-$ and $\chi^+$ up to scale. The results illuminate when inverse-spectral type questions succeed and connect to representation-theoretic asymptotics, while also acknowledging known exceptional degrees where non-uniqueness can arise.

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.

Theorems & Definitions (41)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• Proposition 3.1
• proof
• Lemma 3.2
• proof