HEIGHT OF WALKS WITH RESETS, THE MORAN MODEL, AND THE DISCRETE GUMBEL DISTRIBUTION
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.
In this paper, we consider a two-dimension symmetric random walk with reset. We give, in the first part, some results about the distribution of every component. In the second part, we give some…
In this paper, we give some results about a multi-drawing urn with random
addition matrix. The process that we study is described as: at stage n ≥ 1,
we pick out at random…
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…