HEIGHT OF WALKS WITH RESETS, THE MORAN MODEL, AND THE DISCRETE GUMBEL DISTRIBUTION

Journal Article

2024

Published in: https://hal.science/hal-04303806/document

I

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(h3) 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.