Related papers: A Coupling Argument for the Random Transposition W…

Several cycle lexicographical orders are found to describe the relative likelihood of elements of the random walks on the symmetric group generated by the conjugacy classes of transpositions, 3-cycles, and n-cycles. Spectral analysis finds…

The involution walk is the random walk on $S_n$ generated by involutions with a binomially distributed with parameter $1-p$ number of $2$-cycles. This is a parallelization of the transposition walk. The involution walk is shown in this…

We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and…

We use coupling to study the time taken until the distribution of a statistic on a Markov chain is close to its stationary distribution. Coupling is a common technique used to obtain upper bounds on mixing times of Markov chains, and we…

We study a natural random walk on the $n \times n$ upper triangular matrices, with entries in $\mathbb{Z}/m \mathbb{Z}$, generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this…

We study a random walk on $\mathbb{F}_p$ defined by $X_{n+1}=1/X_n+\varepsilon_{n+1}$ if $X_n\neq 0$, and $X_{n+1}=\varepsilon_{n+1}$ if $X_n=0$, where $\varepsilon_{n+1}$ are independent and identically distributed. This can be seen as a…

Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated…

We consider random walks in which the walk originates in one set of nodes and then continues until it reaches one or more nodes in a target set. The time required for the walk to reach the target set is of interest in understanding the…

"Small worlds" are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use…

We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices $\{u,v\}$ with distance $d>1$ is added as a "long-range" edge with probability…

The motivation of this work is to extend the techniques of higher order random walks on simplicial complexes to analyze mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second…

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we investigate what happens when the random…

We examine the question of whether a collection of random walks on a graph can be coupled so that they never collide. In particular, we show that on the complete graph on n vertices, with or without loops, there is a Markovian coupling…

The theory of rapid mixing random walks plays a fundamental role in the study of modern randomised algorithms. Usually, the mixing time is measured with respect to the worst initial position. It is well known that the presence of…

Random walk is one of the most classical and well-studied model in probability theory. For two correlated random walks on lattice, every step of the random walks has only two states, moving in the same direction or moving in the opposite…

We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate \mu between open and closed, following a Glauber dynamics for the…

We study the mixing time of the Dikin walk in a polytope - a random walk based on the log-barrier from the interior point method literature. This walk, and a close variant, were studied by Narayanan (2016) and Kannan-Narayanan (2012).…

We study random walk on complex networks with transition probabilities which depend on the current and previously visited nodes. By using an absorbing Markov chain we derive an exact expression for the mean first passage time between pairs…

Mixing of finite time-homogeneous Markov chains is well understood nowadays, with a rich set of techniques to estimate their mixing time. In this paper, we study the mixing time of random walks in dynamic random environments. To that end,…

We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing…