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1 LM-Orsay - Laboratoire de Mathématiques d-Orsay

Abstract : In this thesis I study an infinite measure on loops naturally associated to a wide range of Markovian processes and the Poisson point processes of intensity proportional to this measure intensity parameter alpha>0. This Poissson point processes are called Poisson ensembles of Markov loops or loop soups. The measure on loops is covariant with some transformation on Markovian processes, for instance the change of time. In the setting of Brownian loop soups inside a proper open simply connected domain of C it was shown that the outer boundaries of outermost clusters of loops are, for alpha1-2, Conformal Loop Ensembles CLEkappa, kappa in 8-3,4. Besides, it was shown for a wide range of symmetric Markovian processes that for alpha=1-2 the occupation field of a loop soup the sum of times spent by loops over points is the square of the Gaussian free field. First I studied the loop soups associated to one-dimensional diffusions, and particularly the occupation field and its zeroes that delimit in this case the clusters of loops. Then I studied the loop soups on discrete graphs and metric graphs edges replaced by continuous lines. On a metric graph on one hand the loops have a non-trivial geometry and on the other hand one has the same property as in the setting of one-dimensional diffusions that the zeroes of the occupation field delimit the clusters of loops. By combing metric graphs and the isomorphism with the Gaussian free field I have shown that alpha=1-2 is the critical parameter for random walk loop soup percolation on the discrete half-plane Z*N existence or not of an infinite cluster of loops and that for alpha0. Ces processus ponctuels de Poisson portent le nom d-ensembles poissoniens de boucles markoviennes ou de soupes de boucles. La mesure sur les boucles est covariante par un certain nombre de transformations sur les processus de Markov, par exemple le changement de temps.Dans le cadre de soupe de boucles brownienne à l-intérieur d-un sous-domaine ouvert propre simplement connexe de C, il a été montré que les contours extérieurs des amas extérieurs de boucles sont, pour alpha



Autor: Titus Lupu -

Fuente: https://hal.archives-ouvertes.fr/



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