Convergence to equilibrium for a directed $(1+d)-$dimensional polymer

Caputo, Pietro; Sohier, Julien

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 289-318.

Nous considérons une dynamique de flips pour des chemins de longueur $L$ sur le réseau ${\mathbb{Z}}^d$. Il est naturel d’interpréter ce modèle comme une généralisation multidimensionnelle du processus d’exclusion simple, qui correspond au cas $d=1$. Nous montrons que le temps de mélange de la chaîne de Markov associée se comporte comme $L^{2}logL$ à des constantes multiplicatives près, qui dépendent de la dimension $d$. L’idée clef de la preuve pour la borne supérieure est de montrer une inégalité de Sobolev logarithmique pour une constante d’ordre $L^2$ ; pour ce faire, nous combinons une récurrence sur la dimension et une estimée pour des transpositions adjacentes. Nous montrons la borne inférieure en utilisant une version de l’inégalité de Wilson [13] pour le cas unidimensionnel.

We consider a flip dynamics for directed $(1+d)-$dimensional lattice paths with length $L$. The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter corresponding to the case $d=1$. We prove that the mixing time of the associated Markov chain scales like $L^{2}logL$ up to a $d$–dependent multiplicative constant. The key step in the proof of the upper bound is to show that the system satisfies a logarithmic Sobolev inequality on the diffusive scale $L^2$ for every fixed $d$, which we achieve by a suitable induction over the dimension together with an estimate for adjacent transpositions. The lower bound is obtained with a version of Wilson’s argument [13] for the one-dimensional case.

Classification : 60K35, 82C20, 82C41
Mots clés : exclusion process, adjacent transpositions, logarithmic Sobolev inequality, mixing time

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