We consider tilings of a closed region of the Kagome lattice (partition of the plane into regular hexagons and equilateral triangles such that each edge is shared by one triangle and one hexagon). We are interested in the rate of convergence to the stationarity of a natural Markov chain defined on the set of Kagome tilings. The rate of convergence can be represented by the mixing time which mes...

Let C1 be the largest component of the Erdős–Rényi random graph G(n,p). The mixing time of random walk on C1 in the strictly supercritical regime, p = c/n with fixed c > 1, was shown to have order log n by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald. In the critical window, p = (1 + ε)/n where λ= εn is bounded, Nachmias and Peres proved that the mixing time on C1 is...

It is convenient to transform G to a directed graph ←→ G , and look at absorbences of G. From G, we form ←→ G by replacing each edge (u, v) by a pair of anti-parallel edges −→ uv and −→ vu. An absorbence rooted at a vertex r of a directed graph is a subset S of edges, such that |S| = |V | − 1 and every vertex, except r, has exactly one edge directed away from it, thus S forms a tree rooted at r...

We obtain a tight bound of O(L log k) for the mixing time of the exclusion process in Z/LZ with k ≤ 1 2 L particles. Previously the best bound, based on the log Sobolev constant determined by Yau, was not tight for small k. When dependence on the dimension d is considered, our bounds are an improvement for all k. We also get bounds for the relaxation time that are lower-order in d than previous...

We extend a technique for lower-bounding the mixing time of card-shuffling Markov chains, and use it to bound the mixing time of the Rudvalis Markov chain, as well as two variants considered by Diaconis and Saloff-Coste. We show that in each case Θ(n log n) shuffles are required for the permutation to randomize, which matches (up to constants) previously known upper bounds. In contrast, for the...

This paper describes the evolution of localised ensembles of initial conditions in two-and three-dimensional time-independent potentials which admit a coexistence of regular and chaotic orbits. The coarse-grained approach towards an invariant, or near-invariant, distribution was probed by tracking (1) moments hx i y j z k p l x p m y p n z i for i + j + k + l + m + n 4 and (2) binned representa...

to study the feasibility of anaerobic filter in raw sewage and slaughter house wastewater treatment, slaughter house wastewater treatment, a batch flow pilot plant was designed. in phase one, with a detention time of 3-days and no mixing, the maximum reduction in cod was 23%. in the second phase, with a detention time of 2-days and complete mixing, glucose was added to the raw sewage, the reduc...

We prove that random walks in random environments, that are exponentially mixing in space and time, are almost surely diffusive, in the sense that their scaling limit is given by the Wiener measure. 1 The results Random walks in random environments are walks where the transition probabilities are themselves random variables (see [22, 23] for recent reviews of the literature). The environments c...