Metastable crossover time for the Stochastic Ising Model on the torus

Determine the precise large-N asymptotics of the average crossover time α_N for the Stochastic Ising Model with Glauber dynamics on the d-dimensional torus Λ_N = [0,N)^d ∩ Z^d (with periodic boundary conditions) in the low-temperature regime β > β_d, specifically prove that α_N = exp[κ_d(β) N^{d−1} (1+o(1))] as N→∞, where κ_d(β) is the free energy of the Wulff droplet of volume 1/2 in R^d representing the barrier between the minus and plus magnetised states.

Background

In the finite-volume Stochastic Ising Model on Λ_N, metastability arises for β > β_d: the system intermittently transitions between minus and plus magnetised states. The characteristic metastable crossover time α_N governs these transitions and is expected to grow exponentially in a boundary order of N. The Wulff droplet free energy κ_d(β) is conjectured to represent the dominant barrier for such transitions.

While heuristic and physics-based arguments suggest the form α_N = exp[κ_d(β) N{d−1}(1+o(1))], a rigorous derivation of this asymptotic remains open and is connected to deep problems in metastability and droplet nucleation for lattice systems.

References

It is expected that $$ \alpha_N = \exp\big[\kappa_d(\beta)N{d-1}(1 + o(1))\big] $$ with $\kappa_d(\beta)$ the free energy of the so-called Wulff droplet of volume $\tfrac12$ in $\mathbb{R}d$ representing the barrier between $\nu-\beta,\nu+\beta$. The proof remains a challenge.

— Interacting Particle Systems on Random Graphs (2410.17766 - Capannoli et al., 23 Oct 2024) in Lecture 1, SIM on the torus (after Theorem 1.6)