Height of walks with resets, the Moran model, and the discrete Gumbel distribution.
CoRR(2023)
摘要
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.
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