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A Finitevolume Version of AizenmanHiguchi Theorem for the 2d Ising model
 Probab. Theory Relat. Fields
, 2012
"... In the late 1970s, in two celebrated papers, Aizenman and Higuchi independantly established that all infinitevolume Gibbs measures of the twodimensional ferromagnetic nearestneighbor Ising model at inverse temperature β ≥ 0 are of the form αµ+β + (1 − α)µ−β, where µ+β and µ−β are the two pure ph ..."
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In the late 1970s, in two celebrated papers, Aizenman and Higuchi independantly established that all infinitevolume Gibbs measures of the twodimensional ferromagnetic nearestneighbor Ising model at inverse temperature β ≥ 0 are of the form αµ+β + (1 − α)µ−β, where µ+β and µ−β are the two pure phases and 0 ≤ α ≤ 1. We present here a new approach to this result, with a number of advantages: 1. We obtain a finitevolume, quantitative analogue (implying the classical claim); 2. the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; 3. this new approach seems substantially more robust. Known results for Gibbs measures and the 2d Ising model • The set Gβ of infinite volume Gibbs measures at inverse temperature β is a Choquet simplex. In particular, any µ ∈ Gβ has a unique decomposition onto a set of extremal measures Gexβ. • µ±β (infinite volume limits with + or boundary conditions) are extremal measures. • 1975 [MiracleSole, Messager]: If µ ∈ Gβ is translation invariant, then µ = αµ+β +(1−α)µ−β. • 1980 [Aizenman, Higuchi (independantly)] Any µ ∈ Gβ is translation invariant. The theorem Let β> βc. For every ξ < 1/2 and 0 < δ < 1/2 − ξ, as n tends to infinity, there exists a constant αn,ω(β) ∈ [0, 1] such that, µωΛn,β(f) = α n,ωµ+β (f) + (1 − αn,ω)µ−β (f) + O(‖f‖ ∞ n−δ) where theO notation is uniform in the boundary condition ω and in function f having support in Λnξ Key tools • FKG inequality: Increasing functions are positively correlated. •KramersWannier duality: Strict positivity of the surface tension at inverse temperature β ⇔ Strict positivity of the inverse correlation length at dual inverse temperature β?. (tanh β? = e−2β)
Algebraic Topology of Spin Glasses
, 805
"... We study topology of frustrations in ddimensional Ising spin glasses with nearestneighbor interactions. We prove the following. (i) For any given spin configuration, the domain walls on the unfrustration network are all transverse to the frustrated loops in the unfrustration network, where a domai ..."
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We study topology of frustrations in ddimensional Ising spin glasses with nearestneighbor interactions. We prove the following. (i) For any given spin configuration, the domain walls on the unfrustration network are all transverse to the frustrated loops in the unfrustration network, where a domain wall is given by a (d − 1)dimensional hypersurface whose (d − 1) cells are dual to bonds having an unfavorable energy, and the unfrustration network is the collection of all the unfrustrated plaquettes. (ii) For a groundstate spin configuration, the rest of the domain walls are all confined into a neighborhood of the frustration network which is the collection of all the frustrated plaquettes. Relying on these results, we conjecture the following. In three and higher dimensions, the domain walls are stable against thermal fluctuation. As a result, there appears long range order of the spins on the unfrustration network having infinite volume at low temperatures, while the spins on the frustration network exhibit disorder. But the domain walls are not stable in two dimensions. Namely the thermal fluctuation of the domain
Research Article The Stochastic Ising Model with the Mixed Boundary Conditions
"... We estimate the spectral gap of the twodimensional stochastic Ising model for four classes of mixed boundary conditions. On a finite square, in the absence of an external field, twosided estimates on the spectral gap for the first class of weak positive boundary conditions are given. Further, at ..."
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We estimate the spectral gap of the twodimensional stochastic Ising model for four classes of mixed boundary conditions. On a finite square, in the absence of an external field, twosided estimates on the spectral gap for the first class of weak positive boundary conditions are given. Further, at inverse temperatures β> βc, we will show lower bounds of the spectral gap of the Ising model for the other three classes mixed boundary conditions. Copyright q 2009 Jun Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Examples of DLR states which are not weak limits of finite volume Gibbs measures with deterministic boundary conditions
, 2014
"... We prove that the mixture 12 (µ ± + µ∓) of two reflectionsymmetric Dobrushin states of the 3dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finitevolume measures with deterministic boundary conditions. Furthermore, we discuss what is known about the stru ..."
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We prove that the mixture 12 (µ ± + µ∓) of two reflectionsymmetric Dobrushin states of the 3dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finitevolume measures with deterministic boundary conditions. Furthermore, we discuss what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and give a few conjectures. 1