Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs

TL;DR

This paper analyzes the coloring problem on random graphs, deriving threshold values and stability conditions for the 1RSB ansatz, and provides asymptotic formulas for large q that match rigorous bounds.

Contribution

It establishes the validity of the 1RSB solution for threshold determination and derives asymptotic thresholds for large q, aligning with mathematical bounds.

Findings

1RSB solution yields exact phase transition thresholds.

Asymptotic thresholds for large q are c_q = 2qlog(q)-log(q)-1+o(1).

Global phase diagram of the coloring problem is provided.

Abstract

We consider the problem of coloring Erdos-Renyi and regular random graphs of finite connectivity using q colors. It has been studied so far using the cavity approach within the so-called one-step replica symmetry breaking (1RSB) ansatz. We derive a general criterion for the validity of this ansatz and, applying it to the ground state, we provide evidence that the 1RSB solution gives exact threshold values c_q for the q-COL/UNCOL phase transition. We also study the asymptotic thresholds for q >> 1 finding c_q = 2qlog(q)-log(q)-1+o(1) in perfect agreement with rigorous mathematical bounds, as well as the nature of excited states, and give a global phase diagram of the problem.