Reconstruction of Random Colourings

TL;DR

This paper refines the understanding of the reconstruction problem for random k-colorings on large-degree trees, establishing tighter thresholds for when reconstruction is possible or impossible.

Contribution

It improves existing bounds by precisely determining the thresholds for reconstruction and non-reconstruction in the k-coloring problem on trees.

Findings

Non-reconstruction when Δ ≤ k[log k + log log k + 1 - ln 2 - o(1)]

Reconstruction when Δ ≥ k[log k + log log k + 1 + o(1)]

Tightened thresholds for phase transition in coloring reconstruction

Abstract

Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $\Delta$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq \frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k\log k + o(k\log k)$. We tighten this result and show non-reconstruction when $\Delta \leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $\Delta \geq k[\log k + \log \log k + 1+o(1)]$.