Asymptotic direction of random walks in Dirichlet environment
Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 716-726.

We prove that, on d , random walks in i.i.d. Dirichlet environment – or equivalently oriented-edge reinforced random walks – have almost surely an asymptotic direction equal to the direction of the initial drift, i.e. X n X n converges to E o [X 1 ] E o [X 1 ] as n, unless this drift is zero. This is obtained by generalizing the result of directional transience from (Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8). In addition, we identify the exact value or distribution of certain probabilities, answering and generalizing a conjecture of that paper.

On démontre que, dans d , les marches aléatoires en milieu aléatoire i.i.d. de Dirichlet – ou, de façon équivalente, les marches renforcées par arêtes orientées – ont presque sûrement une direction asymptotique égale à la direction de la dérive initiale, c’est-à-dire que X n X n converge vers E o [X 1 ] E o [X 1 ] quand n, à moins que cette dérive soit nulle. Ceci est obtenu en généralisant le résultat de transience directionnelle de (Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 1–8). De plus, on explicite la valeur ou la loi de certaines probabilités, ce qui démontre et généralise une conjecture de ce dernier article.

DOI: 10.1214/13-AIHP582
Classification: 60K37,  60K35
Keywords: random walk, random environment, Dirichlet distribution, reinforced random walk, asymptotic direction, time reversal
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Tournier, Laurent. Asymptotic direction of random walks in Dirichlet environment. Annales de l'I.H.P. Probabilités et statistiques, Volume 51 (2015) no. 2, pp. 716-726. doi : 10.1214/13-AIHP582. http://www.numdam.org/articles/10.1214/13-AIHP582/

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