1,721,421 research outputs found
APPROACH TO EQUILIBRIUM OF GLAUBER DYNAMICS IN THE ONE-PHASE REGION .1. THE ATTRACTIVE CASE
No full text
On the Mixing Time of the 2D Stochastic Ising Model with "Plus" Boundary Conditions at Low Temperature
No full textWe consider the Glauber dynamics for the 2D Ising model in a box of side
L, at inverse temperature β and random boundary conditions τ whose distribution P
either stochastically dominates the extremal plus phase (hence the quotation marks in
the title) or is stochastically dominated by the extremal minus phase. A particular case is
when P is concentrated on the homogeneous configuration identically equal to + (equal
to −). For β large enough we show that for any ε > 0 there exists c = c(β, ε) such
that the corresponding mixing time Tmix satisfies limL→∞ P (Tmix ≥ exp(cLε)) = 0. In
the non-random case τ ≡ + (or τ ≡ −), this implies that Tmix ≤ exp(cLε). The same
bound holds when the boundary conditions are all + on three sides and all − on the
remaining one. The result, although still very far from the expected Lifshitz behavior
Tmix = O(L2), considerably improves upon the previous known estimates of the form
Tmix ≤ exp(cL
12
+ε). The techniques are based on induction over length scales, combined
with a judicious use of the so-called “censoring inequality” of Y. Peres and P. Winkler,
which in a sense allows us to guide the dynamics to its equilibrium measure
Mixing time for the solid-on-solid model
No full textWe analyze the mixing time of a natural local Markov chain (the Glauber
dynamics) on configurations of the solid-on-solid model of statistical physics.
This model has been proposed, among other things, as an idealization of the
behavior of contours in the Ising model at low temperatures. Our main result
is an upper bound on the mixing time of ̃ O(n3.5), which is tight within a
factor of ̃ O(
√
n). The proof, which in addition gives some insight into the
actual evolution of the contours, requires the introduction of a number of novel analytical techniques that we conjecture will have other applications
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