Tree-uniqueness threshold for the antiferromagnetic Potts model

Determine whether the value \(B_c = \max\{0,1-q/\Delta\}\) is the tree-uniqueness threshold for the \(q\)-state antiferromagnetic Potts model on \(\Delta\)-regular trees for all fixed \(q\ge 3\) and sufficiently large \(\Delta\).

Background

The paper studies the antiferromagnetic qq-state Potts model on graphs of maximum degree Δ\Delta, using the smallest adjacency-matrix eigenvalue to establish ultra-log-concavity and rapid mixing of Glauber dynamics. For uniformly random Δ\Delta-regular graphs, the resulting mixing regime reaches parameters below the conjectured tree-uniqueness threshold.

The threshold is stated as Bc=max⁡{0,1−q/Δ}B_c=\max\{0,1-q/\Delta\}. The paper notes that this formula has been established for q=3,4q=3,4, and for fixed q≥5q\ge5 when Δ\Delta is sufficiently large, leaving the general validity of the conjectured threshold unresolved.

References

For each fixed q, this regime includes values B < B_c for all sufficiently large \Delta, where the conjectured tree-uniqueness threshold B_c is

B_c := \max#1{0, 1 - \frac{q}{\Delta}.

This formula for the threshold has been established for q = 3, 4 and for fixed q \ge 5 when \Delta is sufficiently large .

— High-Dimensional Ultra-Log-Concave Distributions  (2609.23994 - Chen et al., 21 Sep 2026) in Section 6, Subsection 6.2, immediately following Corollary \ref{cor:q-potts-random-regular-mixing}