Open Quantum Random Walks, as developed in [2], are a quantum generalization of Markov chains on finite graphs or on lattices. These random walks are typically quantum in their behavior, step by step, but they seem to show up a rather classical asymptotic behavior, as opposed to the quantum random walks usually considered in Quantum Information Theory (such as the well-known Hadamard random walk). Typically, in the case of Open Quantum Random Walks on lattices, their distribution seems to always converge to a Gaussian distribution or a mixture of Gaussian distributions. In the case of nearest neighbors homogeneous Open Quantum Random Walks on Z d we prove such a Central Limit Theorem, in the case where only one Gaussian distribution appears in the limit. Through the quantum trajectory point of view on quantum master equations, we transform the problem into studying a certain functional of a Markov chain on Z d times the Banach space of quantum states. The main difficulty is that we know nothing about the invariant measures of this Markov chain, even their existence. Surprisingly enough, we are able to produce a Central Limit Theorem with explicit drift and explicit covariance matrix. The interesting point which appears with our construction and result is that it applies actually to a wider setup: it provides a Central Limit Theorem for the sequence of recordings of the quantum trajectories associated to any completely positive map. This is what we show and develop as an application of our result.In a second step we are able to extend our Central Limit Theorem to the case of several asymptotic Gaussians, in the case where the operator coefficients of the quantum walk are block-diagonal in a common basis.

To cite this version:Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pene, Bruno Schapira. A local limit theorem for random walks in random scenery and on randomly oriented lattices. Annals of Probability, Institute of Mathematical Statistics, 2011, pp.Vol. 39, No 6, 2079-2118.1214/10-AOP606>. A LOCAL LIMIT THEOREM FOR RANDOM WALKS IN RANDOM SCENERY AND ON RANDOMLY ORIENTED LATTICESFABIENNE CASTELL, NADINE GUILLOTIN-PLANTARD, FRANÇ OISE PÈNE, AND BRUNO SCHAPIRA Abstract. Random walks in random scenery are processes defined by Zn := n k=1 ξX 1 +...+X k , where (X k , k ≥ 1) and (ξy, y ∈ Z) are two independent sequences of i.i.d. random variables. We assume here that their distributions belong to the normal domain of attraction of stable laws with index α ∈ (0, 2] and β ∈ (0, 2] respectively. These processes were first studied by H. Kesten and F. Spitzer, who proved the convergence in distribution when α = 1 and as n → ∞, of n −δ Zn, for some suitable δ > 0 depending on α and β. Here we are interested in the convergence, as n → ∞, of n δ P(Zn = ⌊n δ x⌋), when x ∈ R is fixed. We also consider the case of random walks on randomly oriented lattices for which we obtain similar results.

31 pagesInternational audienceThe aim of this paper is to present a result of discrete approximation of some class of stable self-similar stationary increments processes. The properties of such processes were intensively investigated, but little is known on the context in which such processes can arise. To our knowledge, discretisation and convergence theorems are available only in the case of stable Lévy motions and fractional Brownian motions. This paper yields new results in this direction. Our main result is the convergence of the random rewards schema, which was firstly introduced by Cohen and Samorodnitsky, and that we consider in a more general setting. Strong relationships with Kesten and Spitzer's random walk in random sceneries are evidenced. Finally, we study some path properties of the limit process

We study the persistence probability for processes with stationary increments. Our results apply to a number of examples: sums of stationary correlated random variables whose scaling limit is fractional Brownian motion; random walks in random sceneries; random processes in Brownian scenery; and the Matheron-de Marsily model in Z 2 with random orientations of the horizontal layers. Using a new approach, strongly related to the study of the range, we obtain an upper bound of the optimal order in general and improved lower bounds (compared to previous literature) for many specific processes.2000 Mathematics Subject Classification. 60F05; 60G52.