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Some models on the interface of probability and combinatorics : particle systems and maps.

Abstract : This thesis consists in several works exploring some models belonging to two branches of probability theory: interacting particle systems and random planar maps. A first work concerns algebraic aspects of interacting particle systems invariant measures. We obtain some necessary and sufficient conditions for some continuous time particle systems with discrete local state space, to have a simple invariant measure. In a second work we investigate the effect on survival and coexistence of introducing forest fire epidemics to a certain two-species spatial competition model. Our main results show that, for the two-type model, there are explicit parameter regions where either one species dominates or there is coexistence; contrary to the same model without forest fires, for which the fittest species alwaysdominates. The third and fourth works are related to tree-decorated planar maps. In the third work we present a bijection between the set of tree-decorated maps and the Cartesian product between the set of trees and the set of maps with a simple boundary. We obtain some counting results and some tools to study random decorated map models. In the fourth work we prove that uniform tree-decorated triangulations and quadrangulations with f faces, boundary of length p and decorated by a tree of size a converge weakly for the local topology to different limits, depending on the finite or infinite behavior of f, p and a.
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Submitted on : Tuesday, October 22, 2019 - 11:28:12 AM
Last modification on : Thursday, March 5, 2020 - 3:33:38 PM
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  • HAL Id : tel-02325423, version 1


Luis Fredes Carrasco. Some models on the interface of probability and combinatorics : particle systems and maps.. Probability [math.PR]. Université de Bordeaux, 2019. English. ⟨NNT : 2019BORD0142⟩. ⟨tel-02325423⟩



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