Averaged large deviations for random walk in a random environment

Yilmaz, Atilla

Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 3, pp. 853-868.

Dans son article de 2003, Varadhan démontre un principe de grandes déviations pour la loi moyennée de la vitesse d'une particule suivant une marche aléatoire au plus proche voisin dans un environnement i.i.d. elliptique sur ℤd avec d≥1, et donne une formule variationnelle pour la fonction de taux correspondante Ia. Sous la condition (T) de transience de Sznitman, nous montrons que Ia est strictement convexe et analytique dans un ouvert non vide , et que la vraie vitesse de la particule est un élément de (resp. un élément de la frontière de ) quand la marche est “non-nichée” (resp. nichée). Nous identifions alors l'unique minimisant de la formule variationnelle de Varadhan pour toute vélocité de .

In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on ℤd with d≥1, and gives a variational formula for the corresponding rate function Ia. Under Sznitman's transience condition (T), we show that Ia is strictly convex and analytic on a non-empty open set , and that the true velocity of the particle is an element (resp. in the boundary) of when the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan's variational formula at any velocity in .

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