It is believed that in the long time limit, the limiting behavior of the discrete-time quantum random walk will cross from quantum to classical if we take into account of the decoherence. The computer simulation has already shown that for the discrete-time one-dimensional Hadamard quantum random walk with coin decoherence such that the measurement operators on the coin space are defined by A0 = Ic √1 − p, A1 = |R > < R| √p and A2 = |L > < L > < L| √p is diffusive when 0 < p ≤ 1 and it is ballistic when P = 0. In this thesis, we are going to let p to be dynamical depending on the step t, that is, we consider p = 1/tß, ß ≥ 0 and we found that it has sub-ballistic behavior for 0 < ß < 1. Furthermore, we study not only the coin decoherence, but the total decoherence, that means the measurement operators apply on the Hilbert space H = Hp ⊗ Hc instead of the coin space only. We find that the results are both sub-ballistic for the coin and total decoherence when 0 < ß < 1. Moreover, according to the model given in [T. A. Brun, H. A. Carteret, and A. Ambainis, Phys. Rev. A 67, 032304 (2003)], we know that if the walker has chance to hop to the second nearest neighbor lattice in one step, the long-time behavior is also sub-ballistic and it is similar to that the walker can hop to the third nearest neighbor lattice in one step. By the way, we also find that if we combine the classical part of the model given in [Jing Zhao and Peiqing Tong. One-dimensional quantum walks subject to next nearest neighbor hopping decoherence, Nanjing Normal University, preprint (2014)] with different step length, then this decoherence will also cross from quantum to classical. Finally, we define the quantum γ-stable walk and obtain the quantum γ-stable law with decoherence. By this decoherence, we can see that the limiting behavior of the quantum stable walk will also cross from quantum to classical and we shows that it spreads out faster than the classical stable walk. / Mathematics

Les marches aléatoires en milieux aléatoires constituent un modèle permettant de décrire des phénomènes de diffusion en milieux inhomogènes, possédant des propriétés de régularité à grande échelle. La thèse comportent 6 chapitres. Les trois premiers sont introductifs : le chapitre 1 est une courte introduction générale, le chapitre 2 donne une présentation des modèles considérés par la suite et le chapitre 3 un bref aperçu des résultats obtenus. Les preuves sont renvoyées aux chapitres 4, 5 et 6. Le contenu du chapitre 4 porte sur les théorèmes limites pour une marche aléatoire avec biais sur un arbre de Galton-Watson avec des feuilles dans un régime transient sous-balistique. Le chapitre 5 porte sur le comportement de la vitesse d'une marche aléatoire avec biais sur un amas de percolation quand le paramètre de percolation se rapproche de 1. Un développement asymptotique de la vitesse en fonction du paramètre de percolation est obtenu. On en déduit que la vitesse est croissante en $p=1$. Finalement le chapitre 6 porte sur des estimées de déviations modérées pour une marche aléatoire en milieu aléatoire unidimensionnel. / Random walks in random environments is a suitable model to describe diffusions in inhomogeneous media that have regularity properties on a macroscopic scale. The three first chapters are introductive : chapter 1 is a short general introduction, chapter 2 presents the models considered afterwards and chapter 3 is a brief overview of the results obtained. The proofs are postponed to the chapters4, 5 and 6.The content of chapter 4sheds light on limit theorems for a biased random walk on a Galton-Watson tree with leaves in the transient and sub-ballistic regime. Next, chapter 5 deals with the behaviour of the speed of a biased random walk on a percolation cluster as the percolation parameter goes to 1. An expansion of the speed in function of the percolation parameter is obtained. It can be deduced from this that the speed is increasing in $p=1$. Finally, chapter 6 tackles the problem of moderate deviations for random walks in random environments in dimension $1$.