238 research outputs found
On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization
For general spin systems, we prove that a contractive coupling for any local
Markov chain implies optimal bounds on the mixing time and the modified
log-Sobolev constant for a large class of Markov chains including the Glauber
dynamics, arbitrary heat-bath block dynamics, and the Swendsen-Wang dynamics.
This reveals a novel connection between probabilistic techniques for bounding
the convergence to stationarity and analytic tools for analyzing the decay of
relative entropy. As a corollary of our general results, we obtain
mixing time and modified log-Sobolev constant of
the Glauber dynamics for sampling random -colorings of an -vertex graph
with constant maximum degree when for
some fixed . We also obtain mixing time and
modified log-Sobolev constant of the Swendsen-Wang dynamics for the
ferromagnetic Ising model on an -vertex graph of constant maximum degree
when the parameters of the system lie in the tree uniqueness region. At the
heart of our results are new techniques for establishing spectral independence
of the spin system and block factorization of the relative entropy. On one hand
we prove that a contractive coupling of a local Markov chain implies spectral
independence of the Gibbs distribution. On the other hand we show that spectral
independence implies factorization of entropy for arbitrary blocks,
establishing optimal bounds on the modified log-Sobolev constant of the
corresponding block dynamics
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 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 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 at all
temperatures.Comment: 20 page
Rapid mixing of Swendsen-Wang dynamics in two dimensions
We prove comparison results for the Swendsen-Wang (SW) dynamics, the
heat-bath (HB) dynamics for the Potts model and the single-bond (SB) dynamics
for the random-cluster model on arbitrary graphs. In particular, we prove that
rapid mixing of HB implies rapid mixing of SW on graphs with bounded maximum
degree and that rapid mixing of SW and rapid mixing of SB are equivalent.
Additionally, the spectral gap of SW and SB on planar graphs is bounded from
above and from below by the spectral gap of these dynamics on the corresponding
dual graph with suitably changed temperature.
As a consequence we obtain rapid mixing of the Swendsen-Wang dynamics for the
Potts model on the two-dimensional square lattice at all non-critical
temperatures as well as rapid mixing for the two-dimensional Ising model at all
temperatures. Furthermore, we obtain new results for general graphs at high or
low enough temperatures.Comment: Ph.D. thesis, 66 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
Random-Cluster Dynamics in Z^2: Rapid Mixing with General Boundary Conditions
The random-cluster (FK) model is a key tool for the study of phase transitions and for the design of efficient Markov chain Monte Carlo (MCMC) sampling algorithms for the Ising/Potts model. It is well-known that in the high-temperature region beta<beta_c(q) of the q-state Ising/Potts model on an n x n box Lambda_n of the integer lattice Z^2, spin correlations decay exponentially fast; this property holds even arbitrarily close to the boundary of Lambda_n and uniformly over all boundary conditions. A direct consequence of this property is that the corresponding single-site update Markov chain, known as the Glauber dynamics, mixes in optimal O(n^2 log{n}) steps on Lambda_{n} for all choices of boundary conditions. We study the effect of boundary conditions on the FK-dynamics, the analogous Glauber dynamics for the random-cluster model.
On Lambda_n the random-cluster model with parameters (p,q) has a sharp phase transition at p = p_c(q). Unlike the Ising/Potts model, the random-cluster model has non-local interactions which can be forced by boundary conditions: external wirings of boundary vertices of Lambda_n. We consider the broad and natural class of boundary conditions that are realizable as a configuration on Z^2 Lambda_n. Such boundary conditions can have many macroscopic wirings and impose long-range correlations even at very high temperatures (p 1 and p != p_c(q) the mixing time of the FK-dynamics is polynomial in n for every realizable boundary condition. Previously, for boundary conditions that do not carry long-range information (namely wired and free), Blanca and Sinclair (2017) had proved that the FK-dynamics in the same setting mixes in optimal O(n^2 log n) time. To illustrate the difficulties introduced by general boundary conditions, we also construct a class of non-realizable boundary conditions that induce slow (stretched-exponential) convergence at high temperatures
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