Mixing in time and space for lattice spin systems: A combinatorial view

Abstract

ABSTRACT:The paper considers spin systems on the d-dimensional integer lattice ‫ޚ‬ d with nearest-neighbor interactions. A sharp equivalence is proved between decay with distance of spin correlations (a spatial property of the equilibrium state) and rapid mixing of the Glauber dynamics (a temporal property of a Markov chain Monte Carlo algorithm). Specifically, we show that if the mixing time of the Glauber dynamics is O(n log n) then spin correlations decay exponentially fast with distance. We also prove the converse implication for monotone systems, and for general systems we prove that exponential decay of correlations implies O(n log n) mixing time of a dynamics that updates sufficiently large blocks (rather than single sites). While the above equivalence was already known to hold in various forms, we give proofs that are purely combinatorial and avoid the functional analysis machinery employed in previous proofs.

On the Dynamics of the Glass Transition on Bethe Lattices

“…Proof: The proof is an application of some probabilistic arguments developed in [43,44,45]. For greater convenience of the reader, we shall provide a self contained presentation of most of these ideas.…”

Spatial mixing and nonlocal Markov chains

“…We observe that there are no restrictions on the geometry of the blocks A i in this theorem, other than V = ∪ i A i . This bound on spectral gap, which was known before only for certain special cases (see, e.g., ), is easily seen to be optimal for many natural choices of blocks. Moreover, previous analytic methods used to establish this type of result (for specific collections of blocks) apparently do not apply to the general setting.…”

“…We use the path coupling method of Bubley and Dyer to establish our results for the tiled (heat‐bath) block dynamics in Theorem ; see Section 3. Our proof of this theorem is a generalization of the methods in . We then develop a novel comparison methodology, consisting of several new comparison inequalities concerning various block dynamics, that together with this result allow us to establish Theorems and .…”

“…It has been well known since pioneering work in mathematical physics from the late 1980s (see, e.g., ) that SSM implies that the mixing time (i.e., rate of convergence) of the Glauber dynamics is , and hence optimal ; indeed, the reverse implication is also true, so the phase transition is manifested in the mixing time of the dynamics (see, e.g., ). The above implication was established using sophisticated functional analytic techniques, though more recently a simple combinatorial proof was given in for the special case of monotone systems (where the edge potential favors pairs of equal spins—see Section 6 for a precise definition).…”

Randomly coloring planar graphs with fewer colors than the maximum degree

“…Considerable attention has been paid to the Glauber dynamics, which is of particular interest for its simplicity and intimate connections to properties of infinite-volume Gibbs distributions (e.g., see [20,7,18]). In the Glauber dynamics, at each step, a random vertex is recolored with a color chosen randomly from those colors not appearing in its neighborhood.…”