Finitary codings for spatial mixing Markov random fields.(English)Zbl 1457.60134

The author reminds that the subcritical and critical Ising model on $$\mathbb{Z}^d$$ is a finitary factor of an i.i.d. process (ffiid), whereas the super-critical model is not. The paper shows that certain spatial mixing conditions imply ffiid. The main result is that weak spatial mixing implies ffiid with power-law tails for the coding radius, and that strong spatial mixing implies ffiid with exponential tails for the coding radius. The author proves that strong spatial mixing also implies ffiid with stretched-exponential tails from a finite-valued i.i.d. process. The author gives several applications to models such as the Potts model, proper colorings, the hard-core model, the Widom-Rowlinson model and the beach model. Corollary 1.5 summarizes some results.
Let $$d \ge 2$$ and $$q \ge 2$$. There exists $${\beta _0}(d,q) \ge 1 / {2d}$$ satisfying $${\beta _0}(d,q) = {\beta _c}(d,q)$$ if either $$q = 2$$ or $$d = 2$$, such that the following holds. Let $$\mu$$ be a Gibbs measure for the ferromagnetic $$q$$-state Potts model on $$\mathbb{Z}^d$$ at inverse temperature $$\beta > 0$$. If $$\beta < {\beta _0}(d,q)$$ then $$\mu$$ is ffiid with exponential tails. If $$\beta < {\beta _c}(d,q)$$ then $$\mu$$ is ffiid with power-law tails. If $$\beta > {\beta _c}(d,q)$$ then $$\mu$$ is not ffiid. If $$\beta = {\beta _c}(d,q)$$ and $$d = 2$$, then $$\mu$$ is ffiid if and only if $$q \le 4$$.

MSC:

 60J99 Markov processes 60G10 Stationary stochastic processes 37A60 Dynamical aspects of statistical mechanics 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics 60K35 Interacting random processes; statistical mechanics type models; percolation theory 28D99 Measure-theoretic ergodic theory
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