Abstract:
Using the electric and coupling approaches, we derive a series of results concerning the mixing times for the stratified random walk on the d-cube, inspired in the results of Chung and Graham (1997) Stratified random walks on the n-cube.

Abstract:
The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally non-uniform) distribution pi(x) in time O(n) and the stationary distribution is determined by the initial state of the walk. An explicit expression for pi(x) is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered. We find that the probability distribution mixes to the uniform distribution as expected. However, the mixing time has a minimum at a critical decoherence rate $p \approx 0.1$. A similar effect was previously reported for the QW on the N-cycle with decoherence from repeated measurements of position. A controlled amount of decoherence helps to obtain--and preserve--a uniform distribution over the $2^n$ sites of the hypercube in the shortest possible time.

Abstract:
We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge. The main assumption is that the graphs, associated measures and heat kernels converge in a suitable Gromov-Hausdorff sense. With this result we are able to establish the convergence of the mixing times on the largest component of the Erdos-Renyi random graph in the critical window, sharpening previous results for this random graph model. Our results also enable us to establish convergence in a number of other examples, such as finitely ramified fractal graphs, Galton-Watson trees and the range of a high-dimensional random walk.

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 a natural notion of decoherence on quantum random walks over the hypercube. We prove that in this model there is a decoherence threshold beneath which the essential properties of the hypercubic quantum walk, such as linear mixing times, are preserved. Beyond the threshold, we prove that the walks behave like their classical counterparts.

Abstract:
We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph G are either open or closed and refresh their status at rate \mu\ while at the same time a random walker moves on G at rate 1 but only along edges which are open. On the d-dimensional torus with side length n, we prove that in the subcritical regime, the mixing times for both the full system and the random walker are n^2/\mu\ up to constants. We also obtain results concerning mean squared displacement and hitting times. Finally, we show that the usual recurrence transience dichotomy for the lattice Z^d holds for this model as well.

Abstract:
We prove analytical results showing that decoherence can be useful for mixing time in a continuous-time quantum walk on finite cycles. This complements the numerical observations by Kendon and Tregenna (Physical Review A 67 (2003), 042315) of a similar phenomenon for discrete-time quantum walks. Our analytical treatment of continuous-time quantum walks includes a continuous monitoring of all vertices that induces the decoherence process. We identify the dynamics of the probability distribution and observe how mixing times undergo the transition from quantum to classical behavior as our decoherence parameter grows from zero to infinity. Our results show that, for small rates of decoherence, the mixing time improves linearly with decoherence, whereas for large rates of decoherence, the mixing time deteriorates linearly towards the classical limit. In the middle region of decoherence rates, our numerical data confirms the existence of a unique optimal rate for which the mixing time is minimized.

Abstract:
We derive asymptotics for the probability of the origin to be an extremal point of a random walk in R^n. We show that in order for the probability to be roughly 1/2, the number of steps of the random walk should be between e^{c n / log n}$ and e^{C n log n}. As a result, we attain a bound for the ?pi/2-covering time of a spherical brownian motion.

Abstract:
In this paper, we study mixing and large decoherence in continuous-time quantum walks on one dimensional regular networks, which are constructed by connecting each node to its $2l$ nearest neighbors($l$ on either side). In our investigation, the nodes of network are represented by a set of identical tunnel-coupled quantum dots in which decoherence is induced by continuous monitoring of each quantum dot with nearby point contact detector. To formulate the decoherent CTQWs, we use Gurvitz model and then calculate probability distribution and the bounds of instantaneous and average mixing times. We show that the mixing times are linearly proportional to the decoherence rate. Moreover, adding links to cycle network, in appearance of large decoherence, decreases the mixing times.

Abstract:
Several inequalities are proved for the mixing time of discrete-time quantum walks on finite graphs. The mixing time is defined differently than in Aharonov, Ambainis, Kempe and Vazirani (2001) and it is found that for particular examples of walks on a cycle, a hypercube and a complete graph, quantum walks provide no speed-up in mixing over the classical counterparts. In addition, non-unitary quantum walks (i.e., walks with decoherence) are considered and a criterion for their convergence to the unique stationary distribution is derived.