Large random correlations in individual mean field spin glass samples

TL;DR

This paper demonstrates that mean field spin glass models exhibit broad, long-range correlations across samples, with a uniform distribution of spin-spin correlations at low temperatures, highlighting complex system behavior.

Contribution

It shows that in mean field spin glasses, correlations are broadly distributed and become uniform at low temperatures, revealing intrinsic long-range correlations in complex systems.

Findings

Correlation distribution broadens with decreasing temperature.

At low temperatures, correlation distribution becomes nearly uniform.

Single phase space valleys have narrower correlation distributions.

Abstract

We argue that complex systems must possess long range correlations and illustrate this idea on the example of the mean field spin glass model. Defined on the complete graph, this model has no genuine concept of distance, but the long range character of correlations is translated into a broad distribution of the spin-spin correlation coefficients for almost all realizations of the random couplings. When we sample the whole phase space we find that this distribution is so broad indeed that at low temperatures it essentially becomes uniform, with all possible correlation values appearing with the same probability. The distribution of correlations inside a single phase space valley is also studied and found to be much narrower.