Fluctuations induce transitions in frustrated sparse networks

TL;DR

This paper uses statistical mechanics to analyze phase transitions in a sparse, frustrated network model, revealing a single ergodicity-breaking transition driven by fluctuations and correlations among order parameters.

Contribution

It introduces a novel mathematical technique showing that a single transition line governs all order parameters in the model, contrary to naive expectations.

Findings

No multiple transition lines, only one ergodicity-breaking transition.

The first order parameter influences all others through strong correlations.

Fluctuations at the critical point drive the transition.

Abstract

We analyze, by means of statistical mechanics, a sparse network with random competitive interactions among dichotomic variables pasted on the nodes, namely a Viana-Bray model. The model is described by an infinite series of order parameters (the multi-overlaps) and has two tunable degrees of freedom: the noise level and the connectivity (the averaged number of links). We show that there are no multiple transition lines, one for every order parameter, as a naive approach would suggest, but just one corresponding to ergodicity breaking. We explain this scenario within a novel and simple mathematical technique via a driving mechanism such that, as the first order parameter (the two replica overlap) becomes different from zero due to a real second order phase transition (with properly associated diverging rescaled fluctuations), it enforces all the other multi-overlaps toward positive values thanks to the strong correlations which develop among themselves and the two replica overlap at the critical line.