Abstract
We estimate the spectral gap of the two-dimensional stochastic Ising model for four classes of mixed boundary conditions. On a finite square, in the absence of an external field, two-sided estimates on the spectral gap for the first class of (weak positive) boundary conditions are given. Further, at inverse temperatures , we will show lower bounds of the spectral gap of the Ising model for the other three classes mixed boundary conditions.
Highlights
Introduction and DefinitionsWe consider the most popular ferromagnetic model of statistical physics, which is the Ising model, see 1–5
We estimate the spectral gap of the two-dimensional stochastic Ising model for four classes of mixed boundary conditions
The object of the present paper is to study the spectral gap of the Ising model; the rate at which the Ising model converges to the equilibrium and the spectral gap of the model are closed linked, see 1, Chapter 9 for more details
Summary
We consider the most popular ferromagnetic model of statistical physics, which is the Ising model, see 1–5. The cases of free, plus, and minus boundary conditions for finite-volume Gibbs measures have been studied, see; 1–10 for more details. We study the Ising model with four classes mixed boundary conditions in a finite square of side L 1 in the absence of an external field. We are interested in the case where β is greater than the critical value βc In this case, the Gibbs measures μβΛ, and μβΛ,− corresponding to and − boundary conditions respectively, will converge to different limits μ and μ− as Λ expands to the whole plane Z2, and the famous. Let μβΛ,∅ denote the Gibbs measure with free boundary conditions, it is known that the free boundary condition state converges to the symmetric mixture of the plus and minus states. 1.14 where EβΛ,τ f, f is the Dirichlet form associated with the generator LβΛ,τ , and VarβΛ,τ is the variance relative to the probability measure μβΛ,τ
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