# Charting the replica symmetric phase

## TL;DR

This paper rigorously confirms the physicists' predictions about the replica symmetric phase in diluted mean-field models, including models like Potts antiferromagnet, k-XORSAT, and stochastic block models, clarifying phase transitions and detection thresholds.

## Contribution

It provides a rigorous mathematical validation of the replica symmetric phase and phase transition predictions for a broad class of diluted mean-field models, previously based on non-rigorous methods.

## Key findings

- Confirmed the existence of a replica symmetry breaking phase transition.
- Validated the detailed evolution of the Gibbs measure within the replica symmetric phase.
- Proved a conjecture on the detection problem in the stochastic block model.

## Abstract

Diluted mean-field models are spin systems whose geometry of interactions is induced by a sparse random graph or hypergraph. Such models play an eminent role in the statistical mechanics of disordered systems as well as in combinatorics and computer science. In a path-breaking paper based on the non-rigorous `cavity method', physicists predicted not only the existence of a replica symmetry breaking phase transition in such models but also sketched a detailed picture of the evolution of the Gibbs measure within the replica symmetric phase and its impact on important problems in combinatorics, computer science and physics [Krzakala et al.: PNAS 2007]. In this paper we rigorise this picture completely for a broad class of models, encompassing the Potts antiferromagnet on the random graph, the $k$-XORSAT model and the diluted $k$-spin model for even $k$. We also prove a conjecture about the detection problem in the stochastic block model that has received considerable attention [Decelle et al.: Phys. Rev. E 2011].