Abstract:
Given a finite graph G, a vertex of the lamplighter graph consists of a zero-one labeling of the vertices of G, and a marked vertex of G. For transitive graphs G, we show that, up to constants, the relaxation time for simple random walk in corresponding lamplighter graph is the maximal hitting time for simple random walk in G, while the mixing time in total variation on the lamplighter graph is the expected cover time on G. The mixing time in the uniform metric on the lamplighter graph admits a sharp threshold, and equals |G| multiplied by the relaxation time on G, up to a factor of log |G|. For the lamplighter group over the discrete two dimensional torus of sidelength n, the relaxation time is of order n^2 log n, the total variation mixing time is of order n^2 log^2 n, and the uniform mixing time is of order n^4. In dimension d>2, the relaxation time is of order n^d, the total variation mixing time is of order n^d log n, and the uniform mixing time is of order n^{d+2}. These are the first examples we know of of finite transitive graphs with uniformly bounded degrees where these three mixing time parameters are of different orders of magnitude.

Abstract:
We study the set of probability distributions visited by a continuous-time quantum walk on graphs. An edge-weighted graph G is universal mixing if the instantaneous or average probability distribution of the quantum walk on G ranges over all probability distributions on the vertices as the weights are varied over non-negative reals. The graph is uniform mixing if it visits the uniform distribution. Our results include the following: (a) All weighted complete multipartite graphs are instantaneous universal mixing. This is in contrast to the fact that no unweighted complete multipartite graphs are uniform mixing (except for the four-cycle). (b) The weighted claw or star graph is a minimally connected instantaneous universal mixing graph. In fact, as a corollary, the unweighted claw is instantaneous uniform mixing. This adds a new family of uniform mixing graphs to a list that so far contains only the hypercubes. (c) Any weighted graph is average almost-uniform mixing unless its spectral type is sublinear in the size of the graph. This provides a nearly tight characterization for average uniform mixing on circulant graphs. (d) No weighted graphs are average universal mixing. This shows that weights do not help to achieve average universal mixing, unlike the instantaneous case. Our proofs exploit the spectra of the underlying weighted graphs and path collapsing arguments.

Abstract:
A classical lazy random walk on cycles is known to mix to the uniform distribution. In contrast, we show that a continuous-time quantum walk on cycles exhibit strong non-uniform mixing properties. Our results include the following: - The instantaneous distribution of a quantum walk on most even-length cycles is never uniform. - The average distribution of a quantum walk on any Abelian circulant graph is never uniform. As a corollary, the average distribution of a quantum walk on any standard circulant graph, such as the cycles, complete graphs, and even hypercubes, is never uniform.

Abstract:
We investigate two constructions - the replacement and the zig-zag product of graphs - describing several fascinating connections with Combinatorics, via the notion of expander graph, Group Theory, via the notion of semidirect product and Cayley graph, andwith Markov chains, via the Lamplighter random walk. Many examples are provided.

Abstract:
Continuous-time quantum walks on graphs is a generalization of continuous-time Markov chains on discrete structures. Moore and Russell proved that the continuous-time quantum walk on the $n$-cube is instantaneous exactly uniform mixing but has no average mixing property. On complete (circulant) graphs $K_{n}$, the continuous-time quantum walk is neither instantaneous (except for $n=2,3,4$) nor average uniform mixing (except for $n=2$). We explore two natural {\em group-theoretic} generalizations of the $n$-cube as a $G$-circulant and as a bunkbed $G \rtimes \Int_{2}$, where $G$ is a finite group. Analyses of these classes suggest that the $n$-cube might be special in having instantaneous uniform mixing and that non-uniform average mixing is pervasive, i.e., no memoryless property for the average limiting distribution; an implication of these graphs having zero spectral gap. But on the bunkbeds, we note a memoryless property with respect to the two partitions. We also analyze average mixing on complete paths, where the spectral gaps are nonzero.

Abstract:
We provide new examples of Cayley graphs on which the quantum walks reach uniform mixing. Our first result is a complete characterization of all $2(d+2)$-regular Cayley graphs over $\mathbb{Z}_3^d$ that admit uniform mixing at time $2\pi/9$. Our second result shows that for every integer $k\ge 3$, we can construct Cayley graphs over $\mathbb{Z}_q^d$ that admit uniform mixing at time $2\pi/q^k$, where $q=3, 4$. We also find the first family of irregular graphs, the Cartesian powers of the star $K_{1,3}$, that admit uniform mixing.

Abstract:
Suppose that G and H are finite, connected graphs, G regular, X is a lazy random walk on G and Z is a reversible ergodic Markov chain on H. The generalized lamplighter chain X* associated with X and Z is the random walk on the wreath product H\wr G, the graph whose vertices consist of pairs (f,x) where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H and x is a vertex in G. In each step, X* moves from a configuration (f,x) by updating x to y using the transition rule of X and then independently updating both f_x and f_y according to the transition probabilities on H; f_z for z different of x,y remains unchanged. We estimate the mixing time of X* in terms of the parameters of H and G. Further, we show that the relaxation time of X* is the same order as the maximal expected hitting time of G plus |G| times the relaxation time of the chain on H.

Abstract:
We settle an open problem, raised by Y. Peres and D. Revelle, concerning the $L^2$ mixing time of the random walk on the lamplighter graph. We also provide general bounds relating the entropy decay of a Markov chain to the separation distance of the chain, and show that the lamplighter graphs once again provide examples of tightness of our results.

Abstract:
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time quantum walks on other well-behaved graphs do not exhibit this uniform mixing. We prove that the only graphs amongst balanced complete multipartite graphs that have the instantaneous uniform mixing property are the complete graphs on two, three and four vertices, and the cycle graph on four vertices. Our proof exploits the circulant structure of these graphs. Furthermore, we conjecture that most complete cycles and Cayley graphs lack this mixing property as well.

Abstract:
We show that the measure on markings of $\mathbf {Z}_n^d$, $d\geq3$, with elements of ${0,1}$ given by i.i.d. fair coin flips on the range $\mathcal {R}$ of a random walk $X$ run until time $T$ and 0 otherwise becomes indistinguishable from the uniform measure on such markings at the threshold $T=1/2T_{{\mathrm {cov}}}(\mathbf {Z}_n^d)$. As a consequence of our methods, we show that the total variation mixing time of the random walk on the lamplighter graph $\mathbf {Z}_2\wr \mathbf {Z}_n^d$, $d\geq3$, has a cutoff with threshold $1/2T_{{\mathrm {cov}}}(\mathbf {Z}_n^d)$. We give a general criterion under which both of these results hold; other examples for which this applies include bounded degree expander families, the intersection of an infinite supercritical percolation cluster with an increasing family of balls, the hypercube and the Caley graph of the symmetric group generated by transpositions. The proof also yields precise asymptotics for the decay of correlation in the uncovered set.