The Thermodynamic Limit in Mean Field Spin Glass Models

TL;DR

This paper introduces a straightforward method to prove the existence and uniqueness of the thermodynamic limit for mean field spin glass models, using subadditivity and concentration of measure arguments.

Contribution

It provides a simple, general proof technique for the thermodynamic limit in mean field disordered models like SK and p-spin models.

Findings

Proves the free energy per site converges in the thermodynamic limit.

Establishes almost sure convergence of free energy and ground state energy.

Applies to a broad class of mean field disordered models.

Abstract

We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the Sherrington-Kirkpatrick model, and the Derrida p-spin model. The main argument is based on a smooth interpolation between a large system, made of N spin sites, and two similar but independent subsystems, made of N_1 and N_2 sites, respectively, with N_1+N_2=N. The quenched average of the free energy turns out to be subadditive with respect to the size of the system. This gives immediately convergence of the free energy per site, in the infinite volume limit. Moreover, a simple argument, based on concentration of measure, gives the almost sure convergence, with respect to the external noise. Similar results hold also for the ground state energy per site.