A random hike between combinatorics and statistical mechanics

Abstract : This thesis is at the interface between combinatorics and probability,and contributes to the study of a few models stemming from statisticalmechanics: polymers, self-interacting random walks and random walks inrandom environment, random maps.bigskipThe first model that we investigate is a one-parameter family ofprobability measures on self-avoiding paths of infinite length on aregular lattice, constructed from biased random walks on the tree offinite self-avoiding paths. These measures, initially introduced byBeretti and Sokal, exist for every bias larger than the inverseconnectivity constant, and their limit at the critical bias would beaamong the natural definitions of the uniform self-avoiding walk ofinfinite length. The aim of our work, in collaboration with VincentBeffara, is to understand the link between this limit, if it indeedexists, and other random infinite paths such as Kesten's measure(which is the weak limit of uniformly random finite self-avoidingwalks in the half-plane) and critical Bernoulli percolationinterfaces; the model can be seen as an interpolation between thesetwo.In a second part, we consider random walks with random conductances ona tree, in the case when the law of the conductances has heavy tail.Our aim, in collabration with Andrea Collevecchio and Daniel Kious, isto show a phase transition in the tail parameter; we express thecritical point as an explicit function of the underlying tree.In parallel, we study excited random walks on trees and their phasetransitions: we extend a conjecture of Volkov's and generalize resultsby Basdevant and Singh.Finally, a third part in collaboration with Vincent Beffara andBenjamin Lévêque contributes to the study of random maps of highergenus: we show the existence of subsequential scaling limits foruniformly random simple triangulations of the torus, extending to thatsetup fromer results by Adario-Berri and Albenque (on simpletriangulations of the sphere) and by Bettinelli (on quadrangulationsof the torus). The question of uniqueness and universality of thelimit remain open, but we obtain partial results in that direction.
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Cong Bang Huynh. A random hike between combinatorics and statistical mechanics. Combinatorics [math.CO]. Université Grenoble Alpes, 2019. English. ⟨NNT : 2019GREAM026⟩. ⟨tel-02303529⟩

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