Critical configurations and tube of typical trajectories for the Potts and Ising models with zero external field

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.

arXiv comment44 pages, 15 figures. All comments are welcome!

References in corpus (16)

• Tunneling and Metastability of continuous time Markov chains
• Metastability in Glauber dynamics in the low-temperature limit: Beyond exponential asymptotics
• Metastability and Nucleation for the Blume-Capel Model. Different mechanisms of transition
• Tunneling and Metastability of continuous time Markov chains II, the nonreversible case
• Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States
• Metastability for reversible probabilistic cellular automata with self--interaction
• Metastable states, quasi-stationary distributions and soft measures
• Metastability in the two-dimensional Ising model with free boundary conditions
• Metastability for general dynamics with rare transitions: escape time and critical configurations
• Competitive nucleation in reversible Probabilistic Cellular Automata
• Sum of exit times in a series of two metastable states
• Hitting times asymptotics for hard-core interactions on grids
• Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics
• Metastability of hard-core dynamics on bipartite graphs
• Probabilistic Cellular Automata for low temperature Ising model
• Effect of energy degeneracy on the transition time for a series of metastable states: application to Probabilistic Cellular Automata

Cited by in corpus (3)

• Metastability for the degenerate Potts Model with negative external magnetic field under Glauber dynamics
• Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions. Extended version
• Metastability of Ising and Potts models without external fields in large volumes at low temperatures