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Determining the total variation mixing time of Kac's random walk on the special orthogonal group $\mathrm{SO}(n)$ has been a long-standing open problem. In this paper, we construct a novel non-Markovian coupling for bounding this mixing…

The random walk process underlies the description of a large number of real world phenomena. Here we provide the study of random walk processes in time varying networks in the regime of time-scale mixing; i.e. when the network connectivity…

Establishing cutoff, an abrupt transition from "not mixed" to "well mixed", is a classical topic in the theory of mixing times for Markov chains. Interest has grown recently in determining not only the existence of cutoff and the order of…

A comparison technique for finite random walks on finite graphs is introduced, using the well-known interlacing method. It yields improved return probability bounds. A key feature is the incorporation of parts of the spectrum of the…

We study a natural random walk over the upper triangular matrices, with entries in the field $\Z_2$, generated by steps which add row $i+1$ to row $i$. We show that the mixing time of the lazy random walk is $O(n^2)$ which is optimal up to…

Reflecting boundary conditions cause two one-dimensional random walks to synchronize if a common direction is chosen in each step. The mean synchronization time and its standard deviation are calculated analytically. Both quantities are…

We extend the conductance and canonical paths methods to the setting of general finite Markov chains, including non-reversible non-lazy walks. The new path method is used to show that a known bound for mixing time of a lazy walk on a Cayley…

Let $\mathcal{S}_n$ be the permutation group on $n$ elements, and consider a random walk on $\mathcal{S}_n$ whose step distribution is uniform on $k$-cycles. We prove a well-known conjecture that the mixing time of this process is…

This paper explores the mixing time of the random transposition walk on permutations with one-sided interval restrictions. In particular, we're interested in the notion of cutoff, a phenomenon which occurs when mixing occurs in a window of…

Consider the random walk on the $n \times n$ upper triangular matrices with ones on the diagonal and elements over $\mathbb{F}_p$ where we pick a row at random and either add it or subtract it from the row directly above it. The main result…

We study an approximation by time-discretized geodesic random walks of a diffusion process associated with a family of time-dependent metrics on manifolds. The condition we assume on the metrics is a natural time-inhomogeneous extension of…

We establish bounds on the mixing times of conjugacy-invariant random walks on finite nilpotent groups in terms of the mixing times of their projections onto the abelianization. This comparison framework shows that, in several natural cases…

We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive `step' or a multiplicative `jump'. When the probability of making a jump tends to zero as an appropriate power of $n$ we prove the…

This paper considers non-backtracking random walks on random graphs generated according to the configuration model. The quantity of interest is the scaling of the mixing time of the random walk as the number of vertices of the random graph…

We consider a variant of random walks on finite groups. At each step, we choose an element from a set of generators ("directions") uniformly, and an integer from a power law ("speed") distribution associated with the chosen direction. We…

We prove a law of large numbers for a class of multidimensional random walks in random environments where the environment satisfies appropriate mixing conditions, which hold when the environment is a weak mixing field in the sense of…

Motivated by novel results in the theory of complex adaptive systems, we analyze the dynamics of random walks in which the jumping probabilities are {\it time-dependent}. We determine the survival probability in the presence of an absorbing…

We study random walks on the integers mod $G_n$ that are determined by an integer sequence $\{ G_n \}_{n \geq 1}$ generated by a linear recurrence relation. Fourier analysis provides explicit formulas to compute the eigenvalues of the…

We identify a fundamental phenomenon of heterogeneous one dimensional random walks: the escape (traversal) time is maximized when the heterogeneity in transition probabilities forms a pyramid-like potential barrier. This barrier corresponds…

We evaluate the mixing time of certain random walks on large unitary groups