The Thermodynamic Limit in Mean Field Spin Glass Models

By Francesco Guerra, Fabio Lucio Toninelli. 2002

408 citations recorded by the source corpus.

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 N1 and N2 sites, respectively, with N1+N2=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.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/90b09cb54343f589913918012231626fb074ebb6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

MAX k-CUT and approximating the chromatic number of random graphs (2006, methodology): “Upper bounds αUB(K) on αc have been found using first moment methods [8, 17] and variational interpolation methods [10, 14], and lower bounds αLB(K) have been found using either explicit analysis of some algorithms [1], or some second moment methods [2].”
Optimal errors and phase transitions in high-dimensional generalized linear models (2017, methodology): “Here we formulate the method as a direct evolution of the Guerra and Toninelli interpolation method developed in the context of spin glasses [72].”
Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model (2002, background): “A careful combination of all information given by Lemma 2 and Lemma 3, finally leads to the proof of Theorem 4. On the other hand, the main Theorem 3 follows easily from Theorem 5, and the results of [ 9 ].”
Gaussian multiplicative chaos and applications: A review (2013, background): “In an important work, the authors of [80] rederived (2.”
Statistical physics of inference: thresholds and algorithms (2015, background): “In fact, the self-averaging hypothesis for the free energy has been proven now rigorously in many cases, in particular for all lattices in finite dimension [229] and for some mean-field models [93].”
The adaptive interpolation method: a simple scheme to prove replica formulas in Bayesian inference (2018, methodology): “These proofs usually combine Guerra-Toninelli interpolation bounds with some other non-trivial idea or method, namely algorithmic approaches involving so-called spatially coupled models [28, 22, 17, 18], information theoretic methods [29, 30] or rigorous versions of the cavity method [31, 32, 33, 34”