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R-positivity and the existence of zero-temperature limits of Gibbs measures on nearest-neighbor matrices

Part of: Special processes Markov processes

Published online by Cambridge University Press:  25 September 2023

Jorge Littin Curinao Open the ORCID record for Jorge Littin Curinao [Opens in a new window]  and
Gerardo Corredor Rincón
Jorge Littin Curinao*
Affiliation:
Universidad Católica del Norte
Gerardo Corredor Rincón*
Affiliation:
Universidad Católica del Norte
*
*Postal address: Angamos 0610, Departamento de Matemáticas, Antofagasta-Chile.
*Postal address: Angamos 0610, Departamento de Matemáticas, Antofagasta-Chile.
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Abstract

We study the $R_\beta$-positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures $(\mu_{\beta}(\!\cdot\!))_{\beta \geq 0}$ at inverse temperature $\beta$ associated to a family of nearest-neighbor matrices $(Q_{\beta})_{\beta \geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of $\beta$, and (ii) the existence of weak limits as $\beta \to \infty$. Some application examples are revised to put in context the main results of this work.

Keywords

R-positivityzero-temperature limitsbirth–death chains

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60K40: Other physical applications of random processes
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 20
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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