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Cutoff for the Ising model on the lattice. (English) Zbl 1273.82014
The authors prove that the Glauber dynamics for the Ising model on the discrete torus exhibits a total-variation cutoff, an abrupt drop of the total-variation distance of the Markov chain from equilibrium from near 1 to near 0. They also compute the location of cutoff in terms of the spectral gap of the dynamics on the infinite-volume lattice and get bounds on the cutoff window.
All the results hold in any dimension and for any values of the temperature and the external field for which the corresponding static Gibbs distribution satisfies the so-called strong spatial mixing. In particular, the results hold in the one dimensional case without external field and for any temperature, in the two dimensional case with a non-zero external field and any temperature, and in the two dimensional case without external field and for all temperatures above critical.
The techniques used in the paper are rather general and apply to many other monotone and anti-monotone spin-systems as well as boundary conditions different from periodic. It is the first instance when cutoff is shown for a Markov chain for which its stationary distribution is not completely understood.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82D40 Statistical mechanical studies of magnetic materials
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