no. 3
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.

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)).

DOI : 10.1214/07-AIHP124
Classification : 60E15, 60K35
Mots clés : modified Poincaré inequality, concentration inequality, hypercontractivity, first passage percolation 
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     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},
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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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