A mean field spin glass with short-range interactions

Abstract

Abstract. We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties. We establish that there is a spin glass transition temperature above which the single-site magnetizations vanish, and below which the Edwards-Anderson order parameter is strictly positive. In a neighborhood below the transition temperature, we use bifurcation theory to establish the existence of a nontrivial distribution of single-site magnetizations. Two properties of this distribution are studied: the leading perturbative correction to the Gaussian scaling form at the transition, and the (nonperturbative) behavior of the tails.

Statistical mechanics: entropy, order parameters, and complexity

“…Can we go from one to the other smoothly, without passing through a phase transition? 28 You cannot mix oil and water, but you can mix oil and alcohol, and certainly can mix alcohol and water. Changing the concentrations smoothly starting from oil, going through pure alcohol, and ending at water demonstrates that these two fluids are part of the same phase (see Fig.…”

“…But we could have done an analogous calculation for a system with several gases of different masses; our momentum sphere would become an ellipsoid, but the momentum distribution is given by the same formula. More surprising, we shall see (using the canonical ensemble in Section 6.2) that interactions do not matter either, as long as the system is classical: 28 the probability densities for the momenta are still given by 28 Relativistic effects, magnetic fields, and quantum mechanics will change the velocity distribution. Equation 3.19 will be reasonably accurate for all gases at reasonable temperatures, all liquids but helium, and many solids that are not too cold.…”

“…That is, they are subject to no external potential, apart from being confined in a box of volume L 3 = V with periodic boundary conditions. 28 The single-particle quantum eigenstates 28 That is, the value of ψ at the walls need not be zero (as for an infinite square well), but rather must agree on opposite sides, so ψ(0, y, z) ≡ ψ(L, y, z), ψ(x, 0, z) ≡ ψ(x, L, z), and ψ(x, y, 0) ≡ ψ(x, y, L). Periodic boundary conditions are not usually seen in experiments, but are much more natural to compute with, and the results are unchanged for large systems.…”

Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree

“…If one considers for instance the model with h = 0 on the finite tree T with i.i.d. Bernoulli random boundary data η with p = 1/2 at the leaves of T , then the distribution of the magnetization at the root (as a function of η) becomes non trivial only if β > β 1 , see [9]. In particular, as → ∞, for β β 1 the Gibbs measure on T with the above random boundary η, converges (weakly) a.s. to the free measure µ free .…”

“…µ + = µ − as soon as β β 0 ). Then, in sharp contrast to the model on Z d , there is a second critical point b−1 which is often referred to as the "spin-glass critical point" [9] and has different interpretations. If one considers for instance the model with h = 0 on the finite tree T with i.i.d.…”