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Person# Thomas Mountford

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Random walk

In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the rando

Random graph

In mathematics, random graph is the general term to refer to probability distributions over graphs. Random graphs may be described simply by a probability distribution, or by a random process which

Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
As an illustration, suppose that we are interested in the properties of a func

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Courses taught by this person (2)

MATH-101(en): Analysis I (English)

We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable.

MATH-332: Stochastic processes

The course follows the text of Norris and the polycopie (which will be distributed chapter by chapter).

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In this article, we consider an anisotropic finite-range bond percolation model on Z(2). On each horizontal layer {(x, i): x is an element of Z} we have edges for 1

We consider one-dimensional excited random walks (ERWs) with i.i.d. Markovian cookie stacks in the non-boundary recurrent regime. We prove that under diffusive scaling such an ERW converges in the standard Skorokhod topology to a multiple of Brownian motion perturbed at its extrema (BMPE). All parameters of the limiting process are given explicitly in terms of those of the cookie Markov chain at a single site. While our results extend the results in Dolgopyat and Kosygina (Electron Commun Probab 17:1-14, 2012) (ERWs with boundedly many cookies per stack) and Kosygina and Peterson (Electron J Probab 21:1-24, 2016) (ERWs with periodic cookie stacks), the approach taken is very different and involves coarse graining of both the ERW and the random environment changed by the walk. Through a careful analysis of the environment left by the walk after each "mesoscopic" step, we are able to construct a coupling of the ERW at this "mesoscopic" scale with a suitable discretization of the limiting BMPE. The analysis is based on generalized Ray-Knight theorems for the directed edge local times of the ERW stopped at certain stopping times and evolving in both the original random cookie environment and (which is much more challenging) in the environment created by the walk after each "mesoscopic" step.

2021Thomas Mountford, Jacques Saliba

In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of verticesntends to infinity, in the case where the weight distributionGhas an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.