Attractive regular stochastic chains: perfect simulation and phase transition

Author(s)

Gallo, Sandro ; Takahashi, Daniel Y.

• Document type: Journal article
• Source: Ergodic Theory and Dynamical Systems; v. 34, p. 20-pg., 2014-10-01.

Abstract

We prove that uniqueness of the stationary chain, or equivalently, of the g-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an independent and identically distributed (i.i.d.) process with countable alphabet; (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson-Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order. (AU)

• FAPESP's process: 09/09809-1 - Stochastic processes with variable length memory: Monge-Kantorovich problem, bootstrap and particle systems
• Grantee: Alexsandro Giacomo Grimbert Gallo
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• FAPESP's process: 08/08171-0 - Modeling populations of neurons with multicomponent systems with variable range interactions
• Grantee: Daniel Yasumasa Takahashi
• Support Opportunities: Scholarships in Brazil - Post-Doctoral