Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

Abstract

By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric Sherrington-Kirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site.Comment: 16 page

Localization Transition for Polymers in Poissonian Medium

“…The similar remark also holds for bulk disorder with long range correlation [6]. Unrelated disordered systems, including the Sherrington-Kirkpatrick model, have seen the emergence of smart interpolation techniques [38,18,19], allowing to compare the free energy with that of an auxiliary, simpler model.…”

Universality in Sherrington–Kirkpatrick's spin glass model

“…Toninelli [1,2] gave a rigorous proof, at the mathematical level, of the convergence of free energy to a deterministic limit, in a Gaussian environment, 1 N log Z N (β, g) → α ∞ (β) a.s. and in average. Talagrand [4] then proved that one can replace the Gaussian environment by a Bernoulli environment η ij , P (η ij = ±1) = 1 2 , and obtain the same limit: α ∞ (β).…”

Spin glasses and Stein’s method

“…averaged over disorder) distribution for the local field at site 1, essentially using Gaussian integration by parts. To prove the result for the quenched distribution, it does not suffice to show (18), but rather, we have to show…”