Height of walks with resets, the Moran model, and the discrete Gumbel distribution

TL;DR

This paper analyzes random walks with resets, establishing their height distribution, proving algebraic generating functions with an efficient algorithm, and showing that Moran model walks asymptotically follow a discrete Gumbel distribution, with applications to multidimensional models.

Contribution

It introduces an efficient algorithm for generating functions of bounded height walks and proves the asymptotic Gumbel distribution for Moran model walk heights.

Findings

Generated functions have algebraic form with O(1) computation cost.

Moran model walk heights follow a discrete Gumbel distribution asymptotically.

Explicit fluctuation descriptions for generic reset probability q.

Abstract

In this article, we consider several models of random walks in one or several dimensions, additionally allowing, at any unit of time, a reset (or "catastrophe") of the walk with probability $q$. We establish the distribution of the final altitude. We prove algebraicity of the generating functions of walks of bounded height $h$ (showing in passing the equivalence between Lagrange interpolation and the kernel method). To get these generating functions, our approach offers an algorithm of cost $O(1)$, instead of cost $O(h^3)$ if a Markov chain approach would be used. The simplest nontrivial model corresponds to famous dynamics in population genetics: the Moran model. We prove that the height of these Moran walks asymptotically follows a discrete Gumbel distribution. For $q=1/2$, this generalizes a model of carry propagation over binary numbers considered e.g. by von Neumann and Knuth. For generic $q$, using a Mellin transform approach, we show that the asymptotic height exhibits fluctuations for which we get an explicit description (and, in passing, new bounds for the digamma function). We end by showing how to solve multidimensional generalizations of these walks (where any subset of particles is attributed a different probability of dying) and we give an application to the soliton wave model.