1. CJM Online first
2. CJM Online first
 Handelman, David

Nearly approximate transitivity (AT) for circulant matrices
By previous work of Giordano and the author, ergodic
actions of $\mathbf Z$ (and other discrete groups) are completely classified
measuretheoretically by their dimension space, a construction
analogous to the dimension group used in C*algebras and topological
dynamics. Here we investigate how far from AT (approximately
transitive) can actions be which derive from circulant (and related)
matrices. It turns out not very: although nonAT actions can
arise from this method of construction, under very modest additional
conditions, ATness arises; in addition, if we drop the positivity
requirement in the isomorphism of dimension spaces, then all
these ergodic actions satisfy an analogue of AT. Many examples
are provided.
Keywords:approximately transitive, ergodic transformation, circulant matrix, hemicirculant matrix, dimension space, matrixvalued random walk Categories:37A05, 06F25, 28D05, 46B40, 60G50 

3. 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 2valued environment, and show that this does not hold for 3valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong, but nontrivial conditions on the distribution of the environment.
Our results include generalisations (to the nonelliptic setting) of 01 laws for directional transience, and in 2dimensions the existence of a deterministic limiting velocity.
Keywords:random walk, nonelliptic random environment, zeroone law, coupling Category:60K37 

4. 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 measure Categories:60G50, 33C20, 34M25, 44A10 

5. 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 discretetime
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 process Categories:46L53, 60B20, 14L24 

6. CJM 2007 (vol 59 pp. 828)
 Ortner, Ronald; Woess, Wolfgang

NonBacktracking Random Walks and Cogrowth of Graphs
Let $X$ be a locally finite, connected graph without vertices of
degree $1$. Nonbacktracking 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 nonbacktracking 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 nonregular, but
\emph{small cycles are dense in} $X$, we show that the graph $X$ is
nonamenable if and only if the nonbacktracking $n$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, amenability Categories:05C75, 60G50, 20F69 
