Rapid mixing of Swendsen-Wang and single-bond dynamics in two dimensions

We prove that the spectral gap of the Swendsen-Wang dynamics for the random-cluster model on arbitrary graphs with m edges is bounded above by 16 m log m times the spectral gap of the single-bond (or heat-bath) dynamics. This and the corresponding lower bound imply that rapid mixing of these two dynamics is equivalent. Using the known lower bound on the spectral gap of the Swendsen-Wang dynamics for the two dimensional square lattice $Z_L^2$ of side length L at high temperatures and a result for the single-bond dynamics on dual graphs, we obtain rapid mixing of both dynamics on $\Z_L^2$ at all non-critical temperatures. In particular this implies, as far as we know, the first proof of rapid mixing of a classical Markov chain for the Ising model on $\Z_L^2$ at all temperatures.Comment: 20 page

Comparison of Swendsen-Wang and Heat-Bath Dynamics

We prove that the spectral gap of the Swendsen-Wang process for the Potts model on graphs with bounded degree is bounded from below by some constant times the spectral gap of any single-spin dynamics. This implies rapid mixing of the Swendsen-Wang process for the two-dimensional Potts model at all temperatures above the critical one, as well as rapid mixing at the critical temperature for the Ising model. After this we introduce a modified version of the Swendsen-Wang algorithm for planar graphs and prove rapid mixing for the two-dimensional Potts models at all non-critical temperatures.Comment: 22 pages, 1 figur

A class of stochastic games with infinitely many interacting agents related to Glauber dynamics on random graphs

We introduce and study a class of infinite-horizon nonzero-sum non-cooperative stochastic games with infinitely many interacting agents using ideas of statistical mechanics. First we show, in the general case of asymmetric interactions, the existence of a strategy that allows any player to eliminate losses after a finite random time. In the special case of symmetric interactions, we also prove that, as time goes to infinity, the game converges to a Nash equilibrium. Moreover, assuming that all agents adopt the same strategy, using arguments related to those leading to perfect simulation algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity allows us to prove "fixation", i.e. that players will adopt a constant strategy after a finite time. The resulting dynamics is related to zerotemperature Glauber dynamics on random graphs of possibly infinite volume

A graph polynomial for independent sets of bipartite graphs

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings and the number of perfect matchings. Most importantly, for bipartite graphs the polynomial encodes the number of independent sets (#BIS). We analyze the complexity of exact evaluation of the polynomial at rational points and show that for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial. We conjecture that a natural Markov chain can be used to approximately evaluate the polynomial for a range of parameters. The conjecture, if true, would imply an approximate counting algorithm for #BIS, a problem shown, by [Dyer et al. 2004], to be complete (with respect to, so called, AP-reductions) for a rich logically defined sub-class of #P. We give a mild support for our conjecture by proving that the Markov chain is rapidly mixing on trees. As a by-product we show that the "single bond flip" Markov chain for the random cluster model is rapidly mixing on constant tree-width graphs

Tunneling behavior of Ising and Potts models in the low-temperature regime

We consider the ferromagnetic $q$-state Potts model with zero external field in a finite volume and assume that the stochastic evolution of this system is described by a Glauber-type dynamics parametrized by the inverse temperature $\beta$. Our analysis concerns the low-temperature regime $\beta \to \infty$, in which this multi-spin system has $q$ stable equilibria, corresponding to the configurations where all spins are equal. Focusing on grid graphs with various boundary conditions, we study the tunneling phenomena of the $q$-state Potts model. More specifically, we describe the asymptotic behavior of the first hitting times between stable equilibria as $\beta \to \infty$ in probability, in expectation, and in distribution and obtain tight bounds on the mixing time as side-result. In the special case $q=2$, our results characterize the tunneling behavior of the Ising model on grid graphs.Comment: 13 figure

Spatial Mixing and Non-local Markov chains

We consider spin systems with nearest-neighbor interactions on an $n$-vertex $d$-dimensional cube of the integer lattice graph $\mathbb{Z}^d$. We study the effects that exponential decay with distance of spin correlations, specifically the strong spatial mixing condition (SSM), has on the rate of convergence to equilibrium distribution of non-local Markov chains. We prove that SSM implies $O(\log n)$ mixing of a block dynamics whose steps can be implemented efficiently. We then develop a methodology, consisting of several new comparison inequalities concerning various block dynamics, that allow us to extend this result to other non-local dynamics. As a first application of our method we prove that, if SSM holds, then the relaxation time (i.e., the inverse spectral gap) of general block dynamics is $O(r)$, where $r$ is the number of blocks. A second application of our technology concerns the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models. We show that SSM implies an $O(1)$ bound for the relaxation time. As a by-product of this implication we observe that the relaxation time of the Swendsen-Wang dynamics in square boxes of $\mathbb{Z}^2$ is $O(1)$ throughout the subcritical regime of the $q$-state Potts model, for all $q \ge 2$. We also prove that for monotone spin systems SSM implies that the mixing time of systematic scan dynamics is $O(\log n (\log \log n)^2)$. Systematic scan dynamics are widely employed in practice but have proved hard to analyze. Our proofs use a variety of techniques for the analysis of Markov chains including coupling, functional analysis and linear algebra