Related papers: Random Walks on Small World Networks
The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how fast this "deterministic random walk" covers all…
A second-order random walk on a graph or network is a random walk where transition probabilities depend not only on the present node but also on the previous one. A notable example is the non-backtracking random walk, where the walker is…
We develop a framework to determine the complete statistical behavior of a fundamental quantity in the theory of random walks, namely, the probability that $n_1$, $n_2$, $n_3$, . . . distinct sites are visited at times $t_1$, $t_2$, $t_3$,…
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…
This paper analyzes the meeting time between a pair of pursuer and evader performing random walks on digraphs. The existing bounds on the meeting time usually work only for certain classes of walks and cannot be used to formulate…
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 study the behavior of the random walk in a continuum independent long-range percolation model, in which two given vertices $x$ and $y$ are connected with probability that asymptotically behaves like $|x-y|^{-\alpha}$ with $\alpha>d$,…
We study a random walk in random environment on the non-negative integers. The random environment is not homogeneous in law, but is a mixture of two kinds of site, one in asymptotically vanishing proportion. The two kinds of site are (i)…
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…
The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these…
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…
Suppose that $X$ is a simple random walk on $\Z_n^d$ for $d \geq 3$ and, for each $t$, we let $\U(t)$ consist of those $x \in \Z_n^d$ which have not been visited by $X$ by time $t$. Let $\tcov$ be the expected amount of time that it takes…
We show that the total variation mixing time of the simple random walk on the giant component of supercritical Erdos-Renyi graphs is log^2 n. This statement was only recently proved, independently, by Fountoulakis and Reed. Our proof…
We study the mixing time of the averaging process on a large random $d$-regular graph, $d\ge 3$, and prove an $L^2$-cutoff with an explicit cutoff time. Somewhat surprisingly, we uncover a phase transition at the finite, fixed degree…
In this paper we study the cover time of the simple random walk on the giant component of supercritical $d$-dimensional random geometric graphs on $\mathrm{Poi}(n)$ vertices. We show that the cover time undergoes a jump at the connectivity…
Quantum random walk finds application in efficient quantum algorithms as well as in quantum network theory. Here we study the mixing time of a discrete quantum walk over a square lattice in presence percolation and decoherence. We consider…
We examine the mixing time for random walks on graphs. In particular we are interested on investigating graphs with bottlenecks. Furthermore, the cutoff phenomenon is examined.
We show that the order of the $L_{\infty}$-mixing time of simple random walks on a sequence of uniformly bounded degree graphs of size $n$ may increase by an optimal factor of $\Theta( \log \log n)$ as a result of a bounded perturbation of…
We study the mixing time of the Rook's Walk Markov chain on a $d$-dimensional chess board of side length $n\geq 3$, where a rook moves by first selecting an axis uniformly at random and then selecting a new position along that axis…
We present a general framework, applicable to a broad class of random walks on complex networks, which provides a rigorous lower bound for the mean first-passage time of a random walker to a target site averaged over its starting position,…