Super-Arrhenius relaxation of the triangular plaquette model in any dimension

Laurent Bartholdi, Ivailo Hartarsky, Ivan Mitrofanov

Jun 15, 2026arXiv:2606.16259v1

math.PRcond-mat.stat-mechmath.COmath.DSmath.GR

#6of 223·math.PR

#6 of 223 · math.PR

Tournament Score

1572±39

11001700

81%

Win Rate

Wins

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Matches

Rating

8.5/ 10

Significance9

Rigor9

Novelty8.5

Clarity8

Abstract

Consider the following plaquette model from statistical physics: a lamp lies at every vertex of the triangular lattice and a switch lies at every even vertex of the (bipartite) dual hexagonal lattice. Each switch toggles the three lamps on its face. The energy of a configuration is the number of ON lamps. For the Glauber dynamics associated with the Gibbs measure defined by this Hamiltonian at any inverse temperature $β > 0$ , we show that, in any dimension $d\ge 2$ , the infinite volume relaxation time satisfies $e^{β^2/C}/C \le T_{\mathrm{rel}}\le Ce^{e^{Cβ}}$ for some $C > 0$ . Our result entails that the Gibbs measure is unique. The $e^{β^2}$ scaling was conjectured by Newman and Moore in 1999 and matches the behaviour of supercritical rooted kinetically constrained models such as the East model, thus recovering fragile glass phenomenology in the absence of kinetic constraints. More precisely, we show that, on a torus of side length $2^{k}$ , when $β\to\infty$ and $k/β\to0$ , we have $T_{\mathrm{rel}}=e^{2βk(1+o(1))}$ . Quite surprisingly, however, we also prove that, on non-periodic finite domains of size $n\le e^{β/C}$ for large $C > 0$ , we have the much larger asymptotics $\ln T_{\mathrm{rel}}=βn^{Θ(1)}$ . The main ingredients of the proofs are new results in extremal and enumerative combinatorics and rely on renormalisation ideas for the dynamics and its groundstates also known as the Ledrappier subshift. We note consequences of our results to geometric group theory (more precisely to the complexity of the word problem for the Baumslag finitely presented group) and to ergodic theory.

AI Impact Assessments

(1 models)

Scientific Impact Assessment

Core Contribution

This paper resolves a 25-year-old conjecture by Newman and Moore (1999) concerning the super-Arrhenius relaxation time of the triangular plaquette model (TPM), a statistical mechanics model with three-body interactions on the triangular lattice. The main results establish that in any dimension $d \geq 2$ , the infinite-volume relaxation time satisfies $e^{\beta^2/C}/C \leq T_{\text{rel}} \leq Ce^{e^{C\beta}}$ , confirming the predicted $e^{\beta^2}$ scaling. The paper also proves sharp asymptotics $\ln T_{\text{rel}} \sim 2k\beta$ on dyadic tori of side $2^{k}$ (when $k/\beta \to 0$ ), and discovers a surprising phenomenon: on non-periodic finite domains, the relaxation time is exponentially larger ( $\ln T_{\text{rel}} = \beta n^{\Theta(1)}$ ), suggesting a remarkable boundary-condition-dependent speedup in larger volumes.

Methodological Rigor

The paper is technically formidable, combining tools from at least four distinct mathematical areas:

1. Extremal combinatorics (Section 2.1): A renormalization-based proof that moving a single ON lamp out of a ball of radius $n$ requires at least $\log_2 n$ simultaneously ON lamps. The inductive argument through nested simplices and the classification of plaquette-inverted plaquette intersections is clean and sharp (tight up to additive constants in $d = 2$ ).

2. Entangled configurations (Section 2.2): Construction of admissible configurations requiring $n^{\Theta(1)}$ additional ON lamps to annihilate, using a novel metric $\rho$ based on "long vectors" (edges of large plaquettes) and an exponential decay lemma (Lemma 2.17) proved by induction on dimension with careful parity-class arguments.

3. Enumerative combinatorics (Section 2.3): A recursive bound on the generating function of cycles (groundstate configurations) via a dimension-reduction scheme, yielding polynomial decay estimates that control partition function ratios.

4. Probabilistic arguments (Section 3): Standard but carefully executed Glauber dynamics techniques — spectral gap bounds, canonical paths, path coupling with block dynamics — are combined with the combinatorial inputs.

The proofs are self-contained, carefully structured, and appear correct. The paper handles the delicate interplay between periodic and non-periodic boundary conditions with precision. The upper bound proof via path coupling at scale $e^{C\beta}$ (Section 3.4) is particularly elegant, using the cycle generating function estimates to establish that boundary conditions have negligible influence above a critical scale.

Potential Impact

Statistical physics and glassy dynamics: This is arguably the first rigorous confirmation of fragile glass phenomenology ( $T_{\text{rel}} \approx e^{\beta^2}$ ) in a model without artificially imposed kinetic constraints. Previous results of this type required kinetically constrained models (East model, etc.) where constraints are imposed ad hoc. The TPM achieves the same super-Arrhenius behavior purely from energetic considerations, providing a more physically motivated explanation.

Geometric group theory: The connection to Baumslag's metabelian group is genuinely novel. Corollary 4.1 gives a $\log_2(n)$ lower bound on the $a$ -filling length function, and Corollary 4.2 produces words requiring polynomially many extra generators to reduce — both are new results about the word problem complexity.

Ergodic theory: Theorem 5.2 resolves a question from Arenas-Carmona, Berend, and Bergelson (2008) by showing the function $h (n)$ (minimal number of large plaquettes to represent admissible sets) grows superlinearly: $h(n) \geq n^{1+\eta}$ .

Combinatorics: The classification of small admissible configurations (Lemma 2.16), the metric $\rho$ and its properties, and the recursive cycle-counting estimates are independently interesting combinatorial results.

Timeliness & Relevance

The paper addresses a long-standing open problem at the intersection of mathematical physics and probability. The recent surge of interest in kinetically constrained models (with mathematical breakthroughs by Martinelli, Morris, Toninelli, and others in the 2010s-2020s) makes this particularly timely. The fact that plaquette models can reproduce the same universality class behavior without kinetic constraints addresses a fundamental criticism of the KCM paradigm. The connections to quantum plaquette models (currently active in physics) add further relevance.

Strengths

Breadth of impact: Results spanning statistical mechanics, combinatorics, group theory, and ergodic theory from a unified framework.

Resolution of a specific conjecture with a 25-year pedigree.

Discovery of unexpected phenomena: The boundary-condition dependence of relaxation time (polynomial vs. exponential in

\beta

on tori vs. simplices) is surprising and physically meaningful.

Generality: Results hold in all dimensions

d \geq 2

, not just

d = 2

Self-contained and well-structured presentation despite the technical complexity.

Limitations

The gap between lower bound

e^{\beta^2/C}

and upper bound

e^{e^{C\beta}}

for infinite-volume

T_{\text{rel}}

remains significant. The authors expect the lower bound is closer to truth, but do not close this gap.

The upper bound for tori (Theorem 1.2) is restricted to

d = 2

, while the lower bound works for even

d

Some constants (

C

\alpha_d

a_{d}

) are not explicit, limiting quantitative applicability.

The connection between the finite-volume "entangled configuration" phenomenon and infinite-volume behavior deserves further exploration.

Overall Assessment

This is an outstanding paper that solves a well-known conjecture through a creative synthesis of combinatorial, algebraic, and probabilistic techniques. The discovery of boundary-condition-sensitive relaxation regimes and the cross-disciplinary applications significantly amplify its impact beyond the immediate problem.

Rating:8.5/ 10

Significance 9Rigor 9Novelty 8.5Clarity 8

Generated Jun 16, 2026

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#6of 223·math.PR

#6 of 223 · math.PR

Tournament Score

1572±39

11001700

81%

Win Rate

Wins

Losses

Matches

Rating

8.5/ 10

Significance9

Rigor9

Novelty8.5

Clarity8