On the asymptotics of occurrence times of rare events for stochastic spin systems

We consider translation-invariant attractive spin systems. LetTΛ,xv be the first time that the average spin inside the hypercube Λ reaches the valuex when the process is started from an invariant measure ν with density smaller thanx. We obtain sufficient conditions for (1) ¦Λ¦−1 logTΛ,xv →ϕ(x) in distribution as ¦Λ¦ → ∞, and ¦Λ¦−1 logTΛ,xv →ϕ(x) as ¦Λ¦ → ∞, where ϕ(x):= −limΛ¦Λ¦−1 log ν{(average spin inside Λ) ⩾ x. And (2)TΛ,xv/ETΛ,xv converges to a unit mean exponential random variable as ¦Λ¦ → ∞. Both (1) and (2) are proven under some type of rapid convergence to equilibrium. (1) is also proven without extra conditions for Ising models with ferromagnetic pair interactions evolving according to an attractive reversible dynamics; in this case ϕ is a thermodynamic function. We discuss also the case of finite systems with boundary conditions and what can be said about the state of the system at the timeTΛ,xv.

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