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1. CJM 2013 (vol 66 pp. 1050)

Holmes, Mark; Salisbury, Thomas S.
 Random Walks in Degenerate Random Environments We study the asymptotic behaviour of random walks in i.i.d. random environments on $\mathbb{Z}^d$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience, and in 2-dimensions the existence of a deterministic limiting velocity. Keywords:random walk, non-elliptic random environment, zero-one law, couplingCategory:60K37

2. CJM 2011 (vol 64 pp. 961)

Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim
 Densities of Short Uniform Random Walks We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed. Keywords:random walks, hypergeometric functions, Mahler measureCategories:60G50, 33C20, 34M25, 44A10

3. CJM 2011 (vol 64 pp. 805)

Chapon, François; Defosseux, Manon
 Quantum Random Walks and Minors of Hermitian Brownian Motion Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion. Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor processCategories:46L53, 60B20, 14L24

4. CJM 2007 (vol 59 pp. 828)

Ortner, Ronald; Woess, Wolfgang
 Non-Backtracking Random Walks and Cogrowth of Graphs Let $X$ be a locally finite, connected graph without vertices of degree $1$. Non-backtracking random walk moves at each step with equal probability to one of the forward'' neighbours of the actual state, \emph{i.e.,} it does not go back along the preceding edge to the preceding state. This is not a Markov chain, but can be turned into a Markov chain whose state space is the set of oriented edges of $X$. Thus we obtain for infinite $X$ that the $n$-step non-backtracking transition probabilities tend to zero, and we can also compute their limit when $X$ is finite. This provides a short proof of old results concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when $X$ is non-regular, but \emph{small cycles are dense in} $X$, we show that the graph $X$ is non-amenable if and only if the non-backtracking $n$-step transition probabilities decay exponentially fast. This is a partial generalization of the cogrowth criterion for regular graphs which comprises the original cogrowth criterion for finitely generated groups of Grigorchuk and Cohen. Keywords:graph, oriented line grap, covering tree, random walk, cogrowth, amenabilityCategories:05C75, 60G50, 20F69