Related papers: The Evolution of the Mixing Rate

It is well-known that the $G(n,p)$ model of random graphs undergoes a dramatic change around $p=\frac 1n$. It is here that the random graph is, almost surely, no longer a forest, and here it first acquires a giant (i.e., order $\Omega(n)$)…

Following the recent work of Sznitman (arXiv:0805.4516), we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N x Z, where G_N is a large finite connected weighted graph, and relate it to…

The independence number of a sparse random graph G(n,m) of average degree d=2m/n is well-known to be \alpha(G(n,m))~2n ln(d)/d with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1+o(1)) n…

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest eigenvalue of the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random subgraph of $G_n$ obtained…

We show that in three different critical regimes, the masses of the connected components of rank-2 multiplicative random graph converge to lengths of excursions of a thinned L\'{e}vy process, perhaps with random coefficients. The three…

In this paper we show that the random degree constrained process (a time-evolving random graph model with degree constraints) has a local weak limit, provided that the underlying host graphs are high degree almost regular. We, moreover,…

Recently, it has been shown that one-dimensional quantum walks can mix more quickly than classical random walks, suggesting that quantum Monte Carlo algorithms can outperform their classical counterparts. We study two quantum walks on the…

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be a positive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a (random) graph on $n$ vertices obtained by drawing uniformly at random a $d$-regular (simple)…

We derive an annealed large deviation principle (LDP) for the normalised and rescaled local times of a continuous-time random walk among random conductances (RWRC) in a time-dependent, growing box in $\Z^d$. We work in the interesting case…

We consider a random walk in dimension $d\geq 1$ in a dynamic random environment evolving as an interchange process with rate $\gamma>0$. We only assume that the annealed drift is non-zero. We prove that the empirical velocity of the walker…

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…

In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree…

We consider two random walks evolving synchronously on a random out-regular graph of $n$ vertices with bounded out-degree $r\ge 2$, also known as a random Deterministic Finite Automaton (DFA). We show that, with high probability with…

We show that the classical Pollard rho algorithm for discrete logarithms produces a collision in expected time O(sqrt(n)(log n)^3). This is the first nontrivial rigorous estimate for the collision probability for the unaltered Pollard rho…

We give a randomized algorithm that samples a nearly uniform Eulerian tour of a directed Eulerian multigraph with $m$ arcs in $\widetilde O(m^{3/2})$ time. The guarantee is worst-case, applies to arbitrary directed Eulerian multigraphs, and…

The classical result of Erdos and Renyi shows that the random graph G(n,p) experiences sharp phase transition around p=1/n - for any \epsilon>0 and p=(1-\epsilon)/n, all connected components of G(n,p) are typically of size O(log n), while…

This work focuses on the quantum mixing time, which is crucial for efficient quantum sampling and algorithm performance. We extend Richter's previous analysis of continuous time quantum walks on the periodic lattice $\mathbb{Z}_{n_1}\times…

Sampling permutations from S_n is a fundamental problem from probability theory. The nearest neighbor transposition chain \cal{M}}_{nn} is known to converge in time \Theta(n^3 \log n) in the uniform case and time \Theta(n^2) in the constant…

We study the asymptotics of large, moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes. We obtain the logarithmic asymptotics of large deviations of the joint…

Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running $k$ multiple random walks independently and in parallel. Although the cover…