Open Access
July 2019 Uniqueness of Gibbs measures for continuous hardcore models
David Gamarnik, Kavita Ramanan
Ann. Probab. 47(4): 1949-1981 (July 2019). DOI: 10.1214/18-AOP1298


We formulate a continuous version of the well-known discrete hardcore (or independent set) model on a locally finite graph, parameterized by the so-called activity parameter $\lambda >0$. In this version the state or “spin value” $x_{u}$ of any node $u$ of the graph lies in the interval $[0,1]$, the hardcore constraint $x_{u}+x_{v}\leq 1$ is satisfied for every edge $(u,v)$ of the graph, and the space of feasible configurations is given by a convex polytope. When the graph is a regular tree, we show that there is a unique Gibbs measure associated to each activity parameter $\lambda >0$. Our result shows that, in contrast to the standard discrete hardcore model, the continuous hardcore model does not exhibit a phase transition on the infinite regular tree. We also consider a family of continuous models that interpolate between the discrete and continuous hardcore models on a regular tree when $\lambda =1$ and show that each member of the family has a unique Gibbs measure, even when the discrete model does not. In each case the proof entails the analysis of an associated Hamiltonian dynamical system that describes a certain limit of the marginal distribution at a node. Furthermore, given any sequence of regular graphs with fixed degree and girth diverging to infinity, we apply our results to compute the asymptotic limit of suitably normalized volumes of the corresponding sequence of convex polytopes of feasible configurations. In particular this yields an approximation for the partition function of the continuous hard core model on a regular graph with large girth in the case $\lambda =1$.


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David Gamarnik. Kavita Ramanan. "Uniqueness of Gibbs measures for continuous hardcore models." Ann. Probab. 47 (4) 1949 - 1981, July 2019.


Received: 1 August 2017; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114708
MathSciNet: MR3980912
Digital Object Identifier: 10.1214/18-AOP1298

Primary: 60K35 , 82B820
Secondary: 68W25 , 82B27

Keywords: computational hardness , convex polytope , Gibbs measures , hardcore model , Independent set , linear programming polytope , Partition function , phase transition , regular graphs , volume computation

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 4 • July 2019
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