Random Currents and Continuity of Ising Model’s Spontaneous Magnetization

Abstract

The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on Z d in d = 3 dimensions. The analysis applies also to higher dimensions, for which the result is already known, and to systems with interactions of power law decay. The proof employs in an essential way an extension of Ising model's random current representation to the model's infinite volume limit. Using it, we relate the continuity of the magnetization to the vanishing of the free boundary condition Gibbs state's Long Range Order parameter. For reflection positive models the resulting criterion for continuity may be established through the infrared bound for all but the borderline lower dimensional cases. The exclusion applies to the one dimensional model with 1 r 2 interaction for which the spontaneous magnetization is known to be discontinuous at T c . x∈Λ hσ x − {x,y}⊂Λ∶x≠y

Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for $${d\ge d_{\rm c}}$$

“…also exhibits a phase transition at p c . (Unlike the case for percolation, the continuity of θ p in p has been proven for all dimensions, as long as J o,x satisfies a strong symmetry condition called the reflection positivity [4].) Furthermore, it is known [1,5] that, if the bubble condition (1.24) holds for the critical Ising model, then…”

Absence of Infinite Cluster for Critical Bernoulli Percolation on Slabs

“…The best results (for q > 1) in this direction are mostly restricted to integer values of q, for which the model is related to the Potts model. On the one hand, the fact that the phase transition is continuous for q = 2 (corresponding to the Ising model) is known for any d ≥ 3 [ADCS13]. On the other hand for any q ≥ 3, the random-cluster model undergoes a discontinuous phase transition above some dimension d c (q) [BCC06].…”

“…Bernoulli percolation has offered mathematicians a setup to develop techniques to prove either continuity or discontinuity of the phase transition, which in the case of continuity corresponds to the absence of an infinite cluster at criticality. Harris [16] proved that the nearestneighbor bond percolation model with parameter 1 2 on Z 2 does not contain an infinite cluster almost surely. Viewed together with Kesten's result that p c Ä 1 2 [17], it provided the first proof of such type of statement.…”

Sharp Asymptotics for the Truncated Two-Point Function of the Ising Model with a Positive Field

“…Let Λ = [−N, N ] d be a large square box in Z d . Then, there exists K 0 ≥ 0 (not depending on N ) such that for any K ≥ K 0 , for any ∆ ⊂∆ ⊂ Λ with d(∆, ∂ int∆ ) ≥ log(K) 3 and any u, v ∈ ∆ with v ∈ ∂ ext B K (u),…”