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

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 $β&gt;0$, 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 $C&gt;0$. Our result entails that the Gibbs measure is unique. The $e<sup>{β<sup>2}$ 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 $T_{\mathrm{rel}}=e<sup>{2βk(1+o(1))}$. Quite surprisingly, however, we also prove that, on non-periodic finite domains of size $n\le e<sup>{β/C}$ for large $C&gt;0$, we have the much larger asymptotics $\ln T_{\mathrm{rel}}=βn<sup>{Θ(1)}$. 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.

• The paper proves the Newman–Moore conjecture by establishing an infinite-volume relaxation lower bound of e^(β²/C) for every dimension d ≥ 2, while proving Gibbs-state uniqueness.
• The analysis shows that dyadic two-dimensional tori relax as τ = e^(2βk(1+o(1)) when k/β → 0, whereas small non-periodic simplices can exhibit substantially slower, polynomially amplified barriers.
• The paper develops renormalization, Laurent-polynomial, and cycle-counting methods that connect glassy dynamics with new results in Ledrappier-subshift combinatorics, geometric group theory, and ergodic theory.

The triangular plaquette (Newman–Moore) model is a classical candidate for reproducing fragile-glass phenomenology within equilibrium statistical mechanics: spins on the triangular lattice interact through three-body terms on upward-pointing triangles, and Glauber dynamics at inverse temperature $\beta$ was conjectured by Newman and Moore in 1999 to exhibit super-Arrhenius relaxation with relaxation time $\tau \approx e^{\beta^2}$. This paper, by Bartholdi, Hartarsky and Mitrofanov (2606.16259), provides the first rigorous proof of this scaling in any dimension $d \geq 2$, together with a strikingly different finite-volume behavior, new extremal and enumerative combinatorics for the associated Ledrappier subshift, and applications to geometric group theory and ergodic theory.

Model and main results

The model is defined on $\mathbb{Z}^d$ via the simplex plaquette $T_d = \{0, e_1, \dots, e_d\}$: the Hamiltonian counts frustrated plaquettes, $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$, and Glauber dynamics updates single spins according to the Boltzmann measure. An equivalent "lamps and switches" picture maps spin configurations to lamp configurations via $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$; each switch toggles $d+1$ lamps, and ON lamps cost energy $e^{2\beta}$. The ground states form precisely the Ledrappier subshift.

The central result is:

• Infinite volume (Theorem 1.1): for all $d \geq 2$ there exists $\tau \approx e^{\beta^2}$0 such that, for every $\tau \approx e^{\beta^2}$1,

$\tau \approx e^{\beta^2}$2

and the Gibbs measure is unique. The lower bound confirms the Newman–Moore conjecture; the upper bound is doubly exponential in $\tau \approx e^{\beta^2}$3, inherited from strong spatial mixing results of Chleboun–Faggionato–Martinelli–Toninelli combined with standard mixing-to-relaxation-time machinery. Uniqueness of the Gibbs state rules out an equilibrium phase transition despite the glassy dynamics.

• Dyadic tori (Theorem 1.2): in dimension $\tau \approx e^{\beta^2}$4, on tori of side length $\tau \approx e^{\beta^2}$5 with $\tau \approx e^{\beta^2}$6 and $\tau \approx e^{\beta^2}$7,

$\tau \approx e^{\beta^2}$8

i.e. $\tau \approx e^{\beta^2}$9, exactly the setting of the original physics prediction.

• Non-periodic finite volumes (Theorem 1.3): quite surprisingly, on simplices $d \geq 2$0 with $d \geq 2$1,

$d \geq 2$2

so $d \geq 2$3 — polynomially larger than the torus value at comparable scales. The authors emphasize that this does not transfer to infinite volume because of the subcriticality constraint $d \geq 2$4; they expect a speed-up in larger volumes so that the $d \geq 2$5 lower bound is closer to the truth there. This coexistence of two regimes — slow relaxation in small non-periodic boxes versus faster relaxation on large tori — is the most counterintuitive claim of the paper and had not been predicted in the physics literature.

Combinatorial ingredients

Three deterministic results drive the proofs.

Logarithmic bottleneck. Any chain of lamp configurations (each step toggling one plaquette) that removes a lone ON lamp from a ball of radius $d \geq 2$6 must at some intermediate time have more than $d \geq 2$7 ON lamps in the ball (Theorem 2.4). The proof is a renormalisation argument: the map $d \geq 2$8, contracting inverted plaquettes $d \geq 2$9, sends chains to chains (Lemma 2.8), enabling induction on nested simplices. A delicate parity issue arises for odd $\mathbb{Z}^d$0 (a plaquette toggles an even number of lamps), resolved by a swap argument showing minimality is violated. The bound is sharp up to constants, witnessed by Sierpiński gasket configurations.

Entangled configurations. Admissible sets (finite symmetric differences of plaquettes) correspond under the Laurent-polynomial representation over $\mathbb{Z}^d$1 to multiples of $\mathbb{Z}^d$2. Small admissible sets are classified: any admissible set of cardinality at most $\mathbb{Z}^d$3 is a "large plaquette" $\mathbb{Z}^d$4 (Lemma 2.13). Introducing a metric $\mathbb{Z}^d$5 counting decompositions into long vectors ($\mathbb{Z}^d$6, $\mathbb{Z}^d$7), Lemma 2.15 establishes exponential decay $\mathbb{Z}^d$8 with $\mathbb{Z}^d$9 for every admissible set. Combining this with a greedy construction yields, inside every simplex $T_d = \{0, e_1, \dots, e_d\}$0, an admissible set of size $T_d = \{0, e_1, \dots, e_d\}$1 that is $T_d = \{0, e_1, \dots, e_d\}$2-$T_d = \{0, e_1, \dots, e_d\}$3-separated, and hence entangled: reducing it to empty forces creating $T_d = \{0, e_1, \dots, e_d\}$4 additional ON lamps (Theorem 2.20).

Cycle generating function. Cycles are switch configurations whose induced lamp configuration vanishes; their weighted count is encoded in $T_d = \{0, e_1, \dots, e_d\}$5. The key estimate (Theorem 2.24): if $T_d = \{0, e_1, \dots, e_d\}$6 and $T_d = \{0, e_1, \dots, e_d\}$7 with $T_d = \{0, e_1, \dots, e_d\}$8, then $T_d = \{0, e_1, \dots, e_d\}$9. The proof combines a recursive inequality $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$0 — obtained by slicing cycles into parity classes, using that restrictions of cycles are renormalised cycles via the Frobenius identity $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$1 — with crude bounds at small scales. Physically, this says that high-temperature expansion clusters are negligible once the volume exceeds roughly $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$2 with $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$3.

Probabilistic arguments

Lower bounds follow the KCM playbook: the image of Glauber dynamics under $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$4 is a Markov chain on lamp configurations with product Bernoulli invariant measure of parameter $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$5. For dyadic tori in even dimension, $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$6 is bijective (via Sierpiński gasket preimages); applying the bottleneck theorem to the event that a box contains a single ON lamp yields $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$7. In infinite volume, bijectivity fails, but uniqueness of the Gibbs measure (proved independently below) shows $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$8, recovering the same bound with $H(\sigma) = -\sum_x [\sigma]_{x+T_d}$9 and giving $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$0. For finite simplices, the entangled configuration defines a cut set separating $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$1 from a low-energy state; comparing variance and Dirichlet form across this bottleneck gives $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$2.

Upper bounds. For dyadic tori, a bisection/canonical-paths induction gives $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$3, matching the lower bound's exponent when $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$4. For infinite volume, the cycle estimate implies partition functions under different boundary conditions differ by at most $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$5 at scale $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$6. This uniform closeness feeds a path-coupling contraction for a block dynamics resampling simplices of side $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$7: discrepancies contract at rate $\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$8, yielding exponential ergodicity of the block dynamics, uniqueness of the Gibbs measure, and finally

$\Phi(\sigma)_x = (1 - [\sigma]_{x+T_d})/2$9

Applications beyond statistical mechanics

Geometric group theory. Baumslag's metabelian group $d+1$0 has lamp configurations equal to admissible sets modulo the Ledrappier relation. Reading a word as a lamp walk, chains correspond to combinatorial null-homotopies in van Kampen diagrams. Consequently, the $d+1$1-filling length function satisfies $d+1$2 for words of size $d+1$3, and there exist identity words of configuration diameter $d+1$4 with filling length at least $d+1$5 where $d+1$6 is the number of $d+1$7-letters. The authors note that a tight group-theoretic reformulation would be homological rather than homotopical, since null-homotopies carry strictly more information than chains.

Ergodic theory. Arenas-Carmona, Berend and Bergelson showed the Ledrappier subshift is $d+1$8-mixing modulo special $d+1$9-gons and asked how many large plaquettes are needed to represent an admissible set of size $e^{2\beta}$0; their own bound was cubic. The paper's separated-admissible-set construction improves this to a superlinear lower bound $e^{2\beta}$1 for some $e^{2\beta}$2, refuting their conjectured linear examples $e^{2\beta}$3 as asymptotically optimal.

Limitations and open questions

Several gaps remain explicit in the paper. The gap between the infinite-volume bounds $e^{2\beta}$4 and $e^{2\beta}$5 is wide; closing it would require either improving the upper bound beyond what strong spatial mixing techniques give or refining the lower bound's dependence on volume. The true infinite-volume asymptotics are conjectured but not proved: the authors expect $e^{2\beta}$6, with the finite-volume bottleneck of Theorem 1.3 being an artifact of subcritical sizes, yet the mechanism of the anticipated speed-up in larger volumes is not established. The exact constant in the torus asymptotics is computed only for even $e^{2\beta}$7 (the map $e^{2\beta}$8 fails to be surjective for odd $e^{2\beta}$9, where the dynamics itself is not irreducible on the full state space). Finally, the classification of small admissible sets and the exponential decay lemma are stated for the simplex plaquette; extensions to other plaquette shapes, which govern whether a given plaquette model is Arrhenius or super-Arrhenius, are not addressed here.

Conclusion

This work settles a long-standing conjecture by proving $d \geq 2$0 super-Arrhenius relaxation for the triangular plaquette model in every dimension $d \geq 2$1, while establishing uniqueness of its Gibbs measure and thus the absence of an equilibrium phase transition. Methodologically, it demonstrates that renormalisation ideas native to the Ledrappier subshift — the parity contraction, the Laurent-polynomial algebra of admissible sets, and recursive counting of cycles — suffice to control both dynamical bottlenecks and boundary-condition influence. The same combinatorial toolkit yields quantitative answers in geometric group theory (filling length of Baumslag's group) and ergodic theory (representation complexity of special polygons), underscoring that the fine structure of ground-state configurations, rather than kinetic constraints alone, encodes fragile-glass dynamics.