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Publication# Non-Exponential Variations for Classical Results in First Passage Percolation

Résumé

We study in this thesis the asymptotic behavior of optimal paths on a random graph model, the configuration model, for which we assign continuous random positive weights on its edges. We start by describing the asymptotic behavior of the diameter and the flooding time on the graph for a set of light-tailed edge-weights, and we prove that it is the largest class of densities for which these precise asymptotic expressions for the diameter/flooding time hold. We then show how the weighted optimal path can be constructed using a particular positive recurrent Markov chain. We finally study, in the last chapter, the noise sensitivity of the model by replacing every edge-weight independently by a new realization of the same distribution, with probability epsilon. We show that the optimal paths between two vertices before and after this modification are asymptotically ''independent'' conditioning on the graph, as the number of vertices tends to infinity.

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Concepts associés

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Concepts associés (8)

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Asymptotic analysis

In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior.
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Diamètre

thumb|Diamètre d'un cercle.
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This work is a study of three interacting particle systems that are modified versions of the contact process. The contact process is a spin system defined on a graph and is commonly taken as a model for the spread of an infection in a population; transmission of the infection happens by proximity (contact). The first two models we consider – the grass-bushes-trees model and the multitype contact process – are models for competition between species in Ecology. The third model – annealed approximation to boolean networks – approximately describes the transmission of information among genes in a cell. We consider the grass-bushes-trees model on the set of integers, ℤ. Each point of ℤ is a region of space. In the continuous-time dynamics, at each instant each region can be either empty (state 0) or occupied by an individual of one of two existing species (states 1 and 2). Occupants of both species die at rate 1, leaving their regions empty, and send descendents to neighboring regions at rate λ. An individuals of type 1 may be born on a region previously occupied by an individual of type 2, but the converse is forbidden. We take the "heaviside" initial configuration in which all sites to the left of the origin are occupied by type 1 individuals and all sites to the right of the origin are occupied by type 2 individuals. If the birth of new individuals is allowed to occur at sites that are not adjacent to the parent, and if the rate λ is supercritical for the usual contact process on ℤ, we see the formation of an interface region in which both types coexist. Addressing a conjecture of Cox and Durrett (1995), we prove that the size of this region is stochastically tight. The multitype contact process on ℤ is a process identical to the grass-bushes-trees model in every respect except that no births can occur at previously occupied sites; in particular, the model is symmetric for both species. We again start the process from the heaviside configuration and prove that the size of the interface region is tight. In addition, we prove that the position of the interface, when properly rescaled, converges to Brownian motion. Finally, we give necessary and sufficient conditions on the initial configuration so that one of the two species becomes extinct with probability one and also so that both species are present at all times with positive probability. Lastly, we consider a model proposed by Derrida and Pomeau (1986) and recently studied by Chatterjee and Durrett (2009); it is defined as an approximation to S. Kauffman's boolean networks (1969). The model starts with the choice of a random directed graph on n vertices; each node has r input nodes pointing at it. A discrete time threshold contact process is then considered on this graph: at each instant, each site has probability q of choosing to receive input; if it does, and if at least one of its inputs were occupied by a 1 at the previous instant, then it is labeled with a 1; in all other cases, it is labeled with a 0. r and q are kept fixed and n is taken to infinity. Improving a result of Chatterjee and Durrett, we show that if qr > 1, then the time of persistence of the dynamics is exponential in n.

Thomas 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.