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>0, we show that, in any dimension d≥ 2, the infinite volume relaxation time satisfies eβ2/C/C ≤ Trel≤ CeeCβ for some C>0. Our result entails that the Gibbs measure is unique. The eβ2 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 2k, when β→∞ and k/β→0, we have Trel=e2β k(1+o(1)). Quite surprisingly, however, we also prove that, on non-periodic finite domains of size n≤ eβ/C for large C>0, we have the much larger asymptotics ln Trel=β nΘ(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.
Related
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- Stretched exponential asymptotics for bases of triangular bootstrap percolation
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- The discontinuous Galerkin method for solving the two-dimensional transport equation with distributed-order memory kernel on triangular meshes
BibTeX
@misc{bartholdi2026superarrhenius,
title = {{Super-Arrhenius relaxation of the triangular plaquette model in any dimension}},
author = {Laurent Bartholdi and Ivailo Hartarsky and Ivan Mitrofanov},
howpublished = {HAL},
year = {2026},
month = jun,
eprint = {2606.16259},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05764430v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Super-Arrhenius relaxation of the triangular plaquette model in any dimension
AU - Laurent Bartholdi
AU - Ivailo Hartarsky
AU - Ivan Mitrofanov
PY - 2026
DA - 2026/06/15
PB - HAL
UR - https://hal.science/hal-05764430v1
AB - 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≥ 2, the infinite volume relaxation time satisfies e^β^2/C/C ≤ T_rel≤ 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→∞ and k/β→0, we have T_rel=e^2β k(1+o(1)). Quite surprisingly, however, we also prove that, on non-periodic finite domains of size n≤ e^β/C for large C>0, we have the much larger asymptotics ln T_rel=β n^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.
ER -