Nous considérons une dynamique de flips pour des chemins de longueur sur le réseau . Il est naturel d’interpréter ce modèle comme une généralisation multidimensionnelle du processus d’exclusion simple, qui correspond au cas . Nous montrons que le temps de mélange de la chaîne de Markov associée se comporte comme à des constantes multiplicatives près, qui dépendent de la dimension . L’idée clef de la preuve pour la borne supérieure est de montrer une inégalité de Sobolev logarithmique pour une constante d’ordre ; 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 dimensional lattice paths with length . The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter corresponding to the case . We prove that the mixing time of the associated Markov chain scales like up to a –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 for every fixed , 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.
Mots clés : exclusion process, adjacent transpositions, logarithmic Sobolev inequality, mixing time
Pietro Caputo 1 ; Julien Sohier 2
CC-BY 4.0
@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},
pages = {289--318},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 26},
number = {2},
year = {2017},
doi = {10.5802/afst.1534},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1534/}
}
TY - JOUR AU - Pietro Caputo AU - Julien Sohier TI - Convergence to equilibrium for a directed $(1+d)-$dimensional polymer JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2017 SP - 289 EP - 318 VL - 26 IS - 2 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1534/ DO - 10.5802/afst.1534 LA - en ID - AFST_2017_6_26_2_289_0 ER -
%0 Journal Article %A Pietro Caputo %A Julien Sohier %T Convergence to equilibrium for a directed $(1+d)-$dimensional polymer %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2017 %P 289-318 %V 26 %N 2 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1534/ %R 10.5802/afst.1534 %G en %F AFST_2017_6_26_2_289_0
Pietro Caputo; Julien Sohier. Convergence to equilibrium for a directed $(1+d)-$dimensional polymer. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 2, pp. 289-318. doi : 10.5802/afst.1534. https://afst.centre-mersenne.org/articles/10.5802/afst.1534/
[1] The logarithmic Sobolev constant of Kawasaki dynamics under a mixing condition revisitedq, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 38 (2002) no. 4, pp. 385-436 | DOI
[2] Spectral gap inequalities in product spaces with conservation laws, Stochastic analysis on large scale interacting systems (Studies in Pure Mathematics), Volume 39, Mathematical Society of Japan, 2004, p. 53-8
[3] Polymer dynamics in the depinned phase: metastability with logarithmic barriers, Probab. Theory Relat. Fields, Volume 153 (2012) no. 3–4, pp. 587-641 | DOI
[4] Proof of Aldous’ spectral gap conjecture, J. Am. Math. Soc., Volume 23 (2010) no. 3, pp. 831-851 | DOI
[5] On the approach to equilibrium for a polymer with adsorption and repulsion, Electron. J. Probab., Volume 13 (2008), pp. 213-258 (electronic only) | DOI
[6] Logarithmic Sobolev inequalities for finite Markov chains, Ann. Appl. Probab., Volume 6 (1996) no. 3, pp. 695-750 | DOI
[7] Random polymer models, Imperial College Press, 2007, xvi+242 pages
[8] Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion (https://arxiv.org/abs/1309.3873v2, to appear in Ann. Probab.)
[9] Random walk: a modern introduction, Cambridge Studies in Advances Mathematics, 123, Cambridge University Press, 2010, xii+364 pages
[10] Logarithmic Sobolev inequality for some models of random walks, Ann. Probab., Volume 26 (1998) no. 4, pp. 1855-1873
[11] Markov chains and mixing times, American Mathematical Society, 2009, xvii+371 pages (With a chapter by James G. Propp and David B. Wilson)
[12] An example of application of discrete Hardy’s inequalities, Markov Process. Relat. Fields, Volume 5 (1999) no. 3, pp. 319-330
[13] Mixing times of Lozenge tiling and card shuffling Markov chains, Ann. Appl. Probab., Volume 14 (2004) no. 1, pp. 274-325 | DOI
[14] Logarithmic Sobolev inequality for generalized simple exclusion processes, Probab. Theory Relat. Fields, Volume 109 (1997) no. 4, pp. 507-538 | DOI
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