Asymptotic direction of random walks in Dirichlet environment

Tournier, Laurent

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 2, pp. 716-726.

On démontre que, dans ${\mathbb{Z}}^d$, 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 $\frac{X_n}{\parallel X_n\parallel }$ converge vers $\frac{E_o\left[X_1\right]}{\parallel E_o\left[X_1\right]\parallel }$ quand $n\to \infty$, 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.

We prove that, on ${\mathbb{Z}}^d$, 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. $\frac{X_n}{\parallel X_n\parallel }$ converges to $\frac{E_o\left[X_1\right]}{\parallel E_o\left[X_1\right]\parallel }$ as $n\to \infty$, 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.

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