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Convergence to equilibrium for a directed (1+d)-dimensional polymer
Pietro Caputo; Julien Sohier
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, p. 289-318

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.

Nous considérons une dynamique de flips pour des chemins de longueur L sur le réseau 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.

Published online : 2017-04-13
DOI : https://doi.org/10.5802/afst.1534
Classification:  60K35,  82C20,  82C41
Keywords: exclusion process, adjacent transpositions, logarithmic Sobolev inequality, mixing time
@article{AFST_2017_6_26_2_289_0,
     author = {Pietro Caputo and Julien Sohier},
     title = {Convergence to equilibrium for a directed $(1+d)-$dimensional polymer},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
     number = {2},
     year = {2017},
     pages = {289-318},
     doi = {10.5802/afst.1534},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2017_6_26_2_289_0}
}
Caputo, Pietro; Sohier, Julien. Convergence to equilibrium for a directed $(1+d)-$dimensional polymer. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 2, pp. 289-318. doi : 10.5802/afst.1534. afst.centre-mersenne.org/item/AFST_2017_6_26_2_289_0/

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