Abstract
We consider the ferromagnetic q-state Potts model with zero external field in a finite volume evolving according to Glauber-type dynamics described by the Metropolis algorithm in the low temperature asymptotic limit. Our analysis concerns the multi-spin system that has q stable equilibria. Focusing on grid graphs with periodic boundary conditions, we study the tunneling between two stable states and from one stable state to the set of all other stable states. In both cases we identify the set of gates for the transition and prove that this set has to be crossed with high probability during the transition. Moreover, we identify the tube of typical paths and prove that the probability to deviate from it during the transition is exponentially small.
Highlights
Metastability is a phenomenon that occurs when a physical system is close to a first order phase transition
Since metastability occurs in several physical situations, such as supercooled liquids and supersaturated gases, many models for metastable behavior have been formulated throughout the years
The second issue is the study of the so-called set of critical configurations, i.e., the set of those configurations that are crossed by the process during the transition from the metastable state to the stable state
Summary
Number of spins s in configuration σ Set of configurations with all spins either r or s. {η ∈ X | ∃ω ∈ Ωσv,tjA : η ∈ ω}, tube of typical paths from σ to A {C ∈ M(CA+(σ )\A)|∃(C1, . N} : C j = C}, tube of typical paths from σ to A. {η ∈ X | ∃ω ∈ Ωσv,tjσ s.t. ω ∩ X s \{σ, σ } = ∅ and η ∈ ω}, restricted-tube of typical paths from σ to σ. N} : C j = C}, restricted-tube of typical paths from σ to σ. Set of configurations with all spins r , except those, which are s, in a rectangle a×b. Set of configurations with all spins s, except those, which are r , in a rectangle a×b a ×b with a bar 1×l adjacent to one of the sides of length b, with 1 ≤ l ≤ b−1.
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