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 $d\ge 2$, 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.