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arxiv: 2604.20693 · v1 · submitted 2026-04-22 · 🧮 math.PR

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Uniqueness and Mixing in the Low-Temperature Random-Cluster Model on Trees and Random Graphs

Antonio Blanca , Reza Gheissari , Heehyun Park , Xusheng Zhang
Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:03 UTC · model grok-4.3

classification 🧮 math.PR
keywords random-cluster modeluniqueness thresholdGlauber dynamicsmixing timewired treerandom regular graphsspatial mixingPotts model
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The pith

The random-cluster model on the infinite Δ-regular wired tree has a uniqueness transition at p_s(q,Δ) for every clustering weight q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that the random-cluster model undergoes a transition from non-unique to unique infinite-volume measures on the infinite Δ-regular tree with wired boundaries exactly at the threshold p_s(q,Δ). This holds for all q ≥ 1 and resolves a 1996 conjecture. The proof proceeds by showing weak spatial mixing above the threshold under wired boundaries, which then implies that Glauber dynamics mix in near-linear time on finite trees. The same spatial mixing is lifted to show rapid mixing on random regular graphs when q is at least logarithmic in the degree, yielding efficient sampling for the model and its associated Potts systems.

Core claim

On the infinite Δ-regular wired tree, for all q ≥ 1, the random-cluster model with edge probability p and clustering weight q has a unique infinite-volume measure precisely when p exceeds p_s(q,Δ). This is shown by proving weak spatial mixing under sufficiently wired boundary conditions. The uniqueness yields a near-optimal mixing time of n^{1+o(1)} for the random-cluster Glauber dynamics on the n-vertex wired tree whenever p > p_s(q,Δ) and q > 1. The spatial and temporal mixing results extend from the tree to treelike geometries, producing rapid mixing of Glauber dynamics on the random Δ-regular graph for all p > p_s(q,Δ) provided q ≥ C log Δ.

What carries the argument

The threshold p_s(q,Δ) on the wired tree, which separates non-uniqueness from uniqueness and is used to establish weak spatial mixing under wired boundary conditions.

If this is right

  • Glauber dynamics for the random-cluster model mix in time n^{1+o(1)} on the n-vertex wired tree whenever p > p_s(q,Δ) and q > 1.
  • The same dynamics mix rapidly on the random Δ-regular graph for every p > p_s(q,Δ) as long as q ≥ C log Δ.
  • Efficient sampling algorithms exist for both the random-cluster model and the associated Potts model on random regular graphs in the uniqueness regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The tree uniqueness may serve as a building block for mixing-time bounds on other sparse, locally tree-like graphs.
  • When q is smaller than logarithmic in Δ, rapid mixing on random graphs may still hold but would need a separate argument.
  • The decay of correlations established on the tree could be used to bound the hardness of sampling below the threshold.

Load-bearing premise

Rapid mixing on random regular graphs is proven only when the clustering weight q is at least proportional to the logarithm of the degree Δ.

What would settle it

An explicit construction or numerical check on a large finite Δ-regular wired tree that exhibits two distinct infinite-volume measures for some p strictly larger than p_s(q,Δ) and some q > 2.

Figures

Figures reproduced from arXiv: 2604.20693 by Antonio Blanca, Heehyun Park, Reza Gheissari, Xusheng Zhang.

Figure 1
Figure 1. Figure 1: Plots of g(y) in the various regimes of p for q > 2 and ∆ ≥ 3. This depicts the transition in the number of fixed points as p passes through pu and ps as shown in Theorem 2.6. Theorem 2.6. Let ∆ ≥ 3, q ≥ 1, and p ∈ (0, 1). Then, the function g has a fixed point at y = 1. Furthermore, the fixed points of g in [1,∞) satisfy: (i) If p < pu, g has the unique fixed point at y = 1; (ii) If q > 2 and p ∈ (pu, ps)… view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the function P(x) and Q(x) with the locations of their roots when q < C0(a) log d vs. when q ≥ C0(a) log d. Lemma 5.8. Fix d ≥ 2, q > 1 and p ≥ ps . Then, g ′ (y ∗ (p)) attains its maximum at p = ps . We are ready to prove Theorem 5.7. Proof of Theorem 5.7. For the regime p > ps , Lemma 5.8 implies that g ′ (y ∗ ) is decreasing in p. Thus, by the continuity of g ′ , it is enough to show that g ′ (… view at source ↗
read the original abstract

We study the random-cluster model on trees and treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p\in (0,1)$ and a clustering weight $q\in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, approximately sampling from this model on graphs of degree at most $\Delta$ is computationally hard. At parameter $p$ below the tree uniqueness threshold $p_{\mathsf{u}}(q,\Delta)$, it is expected that sampling is easy and local Markov chains mix rapidly on all bounded degree graphs. On typical graphs (e.g., random regular graphs), the same is predicted at $p > p_{\mathsf{s}}(q,\Delta)$, where $p_{\mathsf{s}}(q,\Delta)$ is a second uniqueness transition point on the $\Delta$-regular wired tree. Our first result establishes this non-uniqueness/uniqueness phase transition at $p_{\mathsf{s}}(q,\Delta)$ for all $q$ on the infinite $\Delta$-regular wired tree, resolving a conjecture of H{\"a}ggstr{\"o}m (1996). For this, we establish weak spatial mixing at $p>p_{\mathsf{s}}(q,\Delta)$ under sufficiently wired boundary conditions. We use this understanding of decay of correlations to show that on the wired tree on $n$ vertices, whenever $q>1$ and $p>p_{\mathsf{s}}(q,\Delta)$, the mixing time of random-cluster Glauber dynamics is a near-optimal $n^{1+o(1)}$. We then extend these results on spatial and temporal mixing from the tree to treelike geometries with mostly wired boundaries and use them to show that the random-cluster Glauber dynamics mix rapidly on the random $\Delta$-regular graph for all $p>p_{\mathsf{s}}(q,\Delta)$ as long as $q \ge C \log \Delta$, providing an efficient sampling algorithm for both the random-cluster and Potts models in this context.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims to resolve Häggström's 1996 conjecture by establishing the non-uniqueness/uniqueness phase transition for the random-cluster model at the threshold p_s(q,Δ) on the infinite Δ-regular wired tree for every q ≥ 1. It proves weak spatial mixing above this threshold under sufficiently wired boundary conditions, derives a near-optimal n^{1+o(1)} mixing time for random-cluster Glauber dynamics on finite wired trees whenever q > 1 and p > p_s(q,Δ), and extends the spatial and temporal mixing results to show rapid mixing of the Glauber dynamics on random Δ-regular graphs for all p > p_s(q,Δ) provided q ≥ C log Δ, yielding efficient sampling algorithms for both the random-cluster and Potts models in this regime.

Significance. If the proofs hold, the work is significant for resolving a long-standing conjecture on the random-cluster model and for supplying the first efficient sampling results for the model (and related Potts models) on random regular graphs in the low-temperature regime. The tree results hold for all q > 1 without additional restrictions, which is a strong feature. The derivation of temporal mixing from spatial mixing via standard coupling arguments is cleanly executed and provides explicit, near-optimal bounds.

minor comments (3)
  1. [Abstract and final section on random graphs] The condition q ≥ C log Δ for the random-graph mixing result is stated without an explicit value or bound on C (or its dependence on Δ). While the restriction is acknowledged, supplying a concrete bound would clarify the range of applicability.
  2. [Introduction] The thresholds p_u(q,Δ) and p_s(q,Δ) are referenced throughout but would benefit from an explicit early definition or comparison table in the introduction to aid readers new to the model.
  3. A few instances of notation for boundary conditions (e.g., wired vs. free) could be made more uniform across sections to improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The referee summary accurately reflects our resolution of Häggström's conjecture on the uniqueness transition at p_s(q,Δ) for the random-cluster model on wired Δ-regular trees, along with the spatial mixing, near-optimal Glauber mixing times on trees, and the extension to rapid mixing on random regular graphs when q ≥ C log Δ. No specific major comments were enumerated in the report.

Circularity Check

0 steps flagged

Derivation self-contained; no circular steps

full rationale

The paper resolves Häggström's 1996 conjecture on the uniqueness threshold p_s(q,Δ) for the random-cluster model on the wired Δ-regular tree by establishing weak spatial mixing above the threshold under wired boundaries, using the model's definition, standard couplings, and correlation decay arguments. Subsequent mixing-time bounds on finite trees and random regular graphs follow from these spatial-mixing estimates via standard Glauber dynamics analysis, with the q ≥ C log Δ restriction on graphs stated explicitly as a technical limitation rather than a hidden assumption. No quantity is defined in terms of another, no fitted parameter is relabeled as a prediction, and the central claims do not reduce to self-citations or ansatzes imported from the authors' prior work. The derivation chain is therefore independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The work rests on the standard definition of the random-cluster measure and the Glauber dynamics; no new free parameters, invented entities, or non-standard axioms are introduced in the abstract.

pith-pipeline@v0.9.0 · 5703 in / 1252 out tokens · 28566 ms · 2026-05-09T23:03:35.952414+00:00 · methodology

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Reference graph

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