Results 1  10
of
21
Broadcasting vs. mixing and information dissemination on Cayley graphs
 In 24th Int. Symp. on Theor. Aspects of Computer Science (STACS
, 2007
"... Abstract. One frequently studied problem in the context of information dissemination in communication networks is the broadcasting problem. In this paper, we study the following randomized broadcasting protocol: At some time t an information r is placed at one of the nodes of a graph G. In the succe ..."
Abstract

Cited by 20 (6 self)
 Add to MetaCart
(Show Context)
Abstract. One frequently studied problem in the context of information dissemination in communication networks is the broadcasting problem. In this paper, we study the following randomized broadcasting protocol: At some time t an information r is placed at one of the nodes of a graph G. In the succeeding steps, each informed node chooses one neighbor, independently and uniformly at random, and informs this neighbor by sending a copy of r to it. First, we consider the relationship between randomized broadcasting and random walks on graphs. In particular, we prove that the runtime of the algorithm described above is upper bounded by the corresponding mixing time, up to a logarithmic factor. One key ingredient of our proofs is the analysis of a continuoustype version of the afore mentioned algorithm, which might be of independent interest. Then, we introduce a general class of Cayley graphs, including (among others) Star graphs, Transposition graphs, and Pancake graphs. We show that randomized broadcasting has optimal runtime on all graphs belonging to this class. Finally, we develop a new proof technique by combining martingale tail estimates with combinatorial methods. Using this approach, we show the optimality of our algorithm on another Cayley graph and obtain new knowledge about the runtime distribution on several Cayley graphs. 1
The stationary measure of a 2type totally asymmetric exclusion process
 J. Combin. Theory Ser. A
, 2006
"... We give a combinatorial description of the stationary measure for a totally asymmetric exclusion process (TASEP) with second class particles, on either�or on the cycle�N. The measure is the image by a simple operation of the uniform measure on some larger finite state space. This reveals a combinato ..."
Abstract

Cited by 18 (1 self)
 Add to MetaCart
(Show Context)
We give a combinatorial description of the stationary measure for a totally asymmetric exclusion process (TASEP) with second class particles, on either�or on the cycle�N. The measure is the image by a simple operation of the uniform measure on some larger finite state space. This reveals a combinatorial structure at work behind several results on the TASEP with second class particles. 1
Sampling Biased Lattice Configurations using Exponential Metrics
"... Monotonic surfaces spanning finite regions of Zd arise in many contexts, including DNAbased selfassembly, cardshuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing w ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
Monotonic surfaces spanning finite regions of Zd arise in many contexts, including DNAbased selfassembly, cardshuffling and lozenge tilings. We explore how we can sample these surfaces when the distribution is biased to favor higher surfaces. We show that a natural local chain is rapidly mixing with any bias for regions in Z², and for bias λ> d² in Z^d, when d > 2. Moreover, our bounds on the mixing time are optimal on ddimensional hypercubic regions. The proof uses a geometric distance function and introduces a variant of path coupling in order to handle distances that are exponentially large.
Spectral gap for the interchange process in a box
 ELECTRON. COMMUN. PROBAB
, 2008
"... We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a ddimensional box of side length L is asymptotic to π 2 /L 2. This gives more evidence in favor of Aldous’s conjecture that in any graph the spectral gap for the interchange process is the same as th ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
We show that the spectral gap for the interchange process (and the symmetric exclusion process) in a ddimensional box of side length L is asymptotic to π 2 /L 2. This gives more evidence in favor of Aldous’s conjecture that in any graph the spectral gap for the interchange process is the same as the spectral gap for a corresponding continuoustime random walk. Our proof uses a technique that is similar to that used by Handjani and Jungreis, who proved that Aldous’s conjecture holds when the graph is a tree.
The mixing time for simple exclusion
, 2004
"... We obtain a tight bound of O(L2 log k) for the mixing time of the exclusion process in Zd/LZd with k < = 12 Ld particles. Previously the best bound, based on the log Sobolev constantdetermined by Yau, was not tight for small k. When dependence on the dimension d is considered, our bounds are an ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
We obtain a tight bound of O(L2 log k) for the mixing time of the exclusion process in Zd/LZd with k < = 12 Ld particles. Previously the best bound, based on the log Sobolev constantdetermined 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 thatare lowerorder in d than previous estimates: our bound of O(L2 log d) improves on the earlierbound O(L2 d), obtained by Quastel. Our proof is based on an auxiliary Markov chain we callthe chameleon process, which may be of independent interest. Introduction Let G = (V, E) be a finite, connected graph and define a configuration as follows. In a configuration, each vertex in V contains either a black ball or a white ball (where balls of the same color are indistinguishable), and the number of black balls is at most V /2. The exclusion process on G is the following continuoustime Markov process on configurations. For each edge e at rate 1: switch the balls at the endpoints of e. Note that since the exclusion process is irreducible and has symmetric transition rates, the uniform distribution is stationary. Let C denote the space of configurations, and for probability distributions u, * on C, let
THERMODYNAMIC LIMIT FOR THE MALLOWS MODEL ON Sn
, 904
"... Abstract. The Mallows model on Sn is a probability distribution on permutations, q d(π,e) /Pn(q), where d(π, e) is the distance between π and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs (i, j) where 1 ≤ i < j ≤ n, but πi> πj. An ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
Abstract. The Mallows model on Sn is a probability distribution on permutations, q d(π,e) /Pn(q), where d(π, e) is the distance between π and the identity element, relative to the Coxeter generators. Equivalently, it is the number of inversions: pairs (i, j) where 1 ≤ i < j ≤ n, but πi> πj. Analyzing the normalization Pn(q), Diaconis and Ram calculated the mean and variance of d(π, e) in the Mallows model, which suggests the appropriate n → ∞ limit has qn scaling as 1 − β/n. We calculate the distribution of the empirical measure in this limit, u(x, y) dx dy = limn→ ∞ 1 Pn n i=1 δ (i,πi). Treating it as a meanfield problem, analogous to the CurieWeiss model, the selfconsistent meanfield equations are ∂2 ln u(x, y) = 2βu(x, y), which is an integrable PDE, known as the ∂x∂y hyperbolic Liouville equation. The explicit solution also gives a new proof of formulas for the blocking measures in the weakly asymmetric exclusion process, and the ground state of the Uq(sl2)symmetric XXZ ferromagnet.
COMPARISON INEQUALITIES AND FASTESTMIXING MARKOV CHAINS
, 2011
"... We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution π on a given finite partially ordered state space X. When K ≼ L in this partial order we say that K and L satisfy a comparison inequality. We establish that if K1,...,Kt and ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
We introduce a new partial order on the class of stochastically monotone Markov kernels having a given stationary distribution π on a given finite partially ordered state space X. When K ≼ L in this partial order we say that K and L satisfy a comparison inequality. We establish that if K1,...,Kt and L1,...,Lt are reversible and Ks ≼ Ls for s = 1,...,t, then K1···Kt ≼ L1···Lt. In particular, in the timehomogeneous case we have K t ≼ L t for every t if K and L are reversible and K ≼ L, and using this we show that (for suitable common initial distributions) the Markov chain Y with kernel K mixes faster than the chain Z with kernel L, in the strong sense that at every time t the discrepancy—measured by total variation distance or separation or L 2distance—between the law of Yt and π is smaller than that between the law of Zt and π. Using comparison inequalities together with specialized arguments to remove the stochastic monotonicity restriction, we answer a question of Persi Diaconis by showing that, among all symmetric birthanddeath kernels on the path X = {0,...,n}, the one (we call it the uniform chain) that produces fastest convergence from initial state 0 to the uniform distribution has transition probability 1/2 in each direction along each edge of the path, with holding probability 1/2 at each endpoint. We also use comparison inequalities (i) to identify, when π is a given logconcave distribution on the path, the fastestmixing stochastically monotone birthanddeath chain started at 0, and (ii) to recover and extend a result of Peres and Winkler that extra updates do not delay mixing for monotone spin systems. Among the fastestmixing chains in (i), we show that the chain for uniform π is slowest in the sense of maximizing separation at every time.
The twosided infinite extension of the Mallows model for random permutations
 Advances in Applied Mathematics
"... ar ..."
(Show Context)
The oriented swap process
, 2008
"... Particles labelled 1 · · · n are arranged initially in increasing order. Subsequently, each pair of neighbouring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behaviour of this process as n → ∞. We prove that the spacetim ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Particles labelled 1 · · · n are arranged initially in increasing order. Subsequently, each pair of neighbouring particles that is currently in increasing order swaps according to a Poisson process of rate 1. We analyze the asymptotic behaviour of this process as n → ∞. We prove that the spacetime trajectories of individual particles converge (when suitably scaled) to a certain family of random curves with two points of nondifferentiability, and that the permutation matrix at a given time converges to a certain deterministic measure with absolutely continuous and singular parts. The absorbing state (where all particles are in decreasing order) is reached at time (2+o(1))n. The finishing times of individual particles converge to deterministic limits with fluctuations asymptotically governed by the TracyWidom distribution.
Mixing Times of Markov Chains for SelfOrganizing Lists and Biased Permutations ∗
"... We study the mixing time of a Markov chainMnn on biased permutations, a problem arising in the context of selforganizing lists. In each step,Mnn chooses two adjacent elements k, and ` and exchanges their positions with probability p`,k. Here we define two general classes and give the first proofs t ..."
Abstract
 Add to MetaCart
(Show Context)
We study the mixing time of a Markov chainMnn on biased permutations, a problem arising in the context of selforganizing lists. In each step,Mnn chooses two adjacent elements k, and ` and exchanges their positions with probability p`,k. Here we define two general classes and give the first proofs that the chain is rapidly mixing for both. We also demonstrate that the chain is not always rapidly mixing.