On obtient une nouvelle inégalité de concentration exponentielle pour la percolation de premier passage, valable pour une large classe de distributions des temps d'arêtes. Ceci améliore et étend un résultat de Benjamini, Kalai et Schramm (Ann. Probab. 31 (2003)) qui donnait une borne sur la variance pour des temps d'arêtes suivant une loi de Bernoulli. Notre approche se fonde sur des inégalités fonctionnelles étendant les travaux de Rossignol (Ann. Probab. 35 (2006)), Falik et Samorodnitsky (Combin. Probab. Comput. 16 (2007)).
We provide a new exponential concentration inequality for first passage percolation valid for a wide class of edge times distributions. This improves and extends a result by Benjamini, Kalai and Schramm (Ann. Probab. 31 (2003)) which gave a variance bound for Bernoulli edge times. Our approach is based on some functional inequalities extending the work of Rossignol (Ann. Probab. 35 (2006)), Falik and Samorodnitsky (Combin. Probab. Comput. 16 (2007)).
Mots clés : modified Poincaré inequality, concentration inequality, hypercontractivity, first passage percolation
@article{AIHPB_2008__44_3_544_0, author = {Bena{\"\i}m, Michel and Rossignol, Rapha\"el}, title = {Exponential concentration for first passage percolation through modified {Poincar\'e} inequalities}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {544--573}, publisher = {Gauthier-Villars}, volume = {44}, number = {3}, year = {2008}, doi = {10.1214/07-AIHP124}, mrnumber = {2451057}, zbl = {1186.60102}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP124/} }
TY - JOUR AU - Benaïm, Michel AU - Rossignol, Raphaël TI - Exponential concentration for first passage percolation through modified Poincaré inequalities JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2008 SP - 544 EP - 573 VL - 44 IS - 3 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP124/ DO - 10.1214/07-AIHP124 LA - en ID - AIHPB_2008__44_3_544_0 ER -
%0 Journal Article %A Benaïm, Michel %A Rossignol, Raphaël %T Exponential concentration for first passage percolation through modified Poincaré inequalities %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 544-573 %V 44 %N 3 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP124/ %R 10.1214/07-AIHP124 %G en %F AIHPB_2008__44_3_544_0
Benaïm, Michel; Rossignol, Raphaël. Exponential concentration for first passage percolation through modified Poincaré inequalities. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 3, pp. 544-573. doi : 10.1214/07-AIHP124. http://www.numdam.org/articles/10.1214/07-AIHP124/
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