The Annals of Probability

Uniqueness of Gibbs measures for continuous hardcore models

David Gamarnik and Kavita Ramanan

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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 $\lambda =1$.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 1949-1981.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B820
Secondary: 82B27: Critical phenomena 68W25: Approximation algorithms

Hardcore model independent set Gibbs measures phase transition partition function linear programming polytope volume computation convex polytope computational hardness regular graphs


Gamarnik, David; Ramanan, Kavita. Uniqueness of Gibbs measures for continuous hardcore models. Ann. Probab. 47 (2019), no. 4, 1949--1981. doi:10.1214/18-AOP1298.

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