## Electronic Journal of Probability

### Decay of correlations for the hardcore model on the $d$-regular random graph

#### Abstract

A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on random regular graphs upto a density for the largest independent set that is bounded by and goes asymptotically to the condensation threshold. We show that the hardcore measure converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence we show that the reconstruction threshold on the random graph, indicative of the onset of point to set spatial correlations, is equal to the reconstruction threshold on the $d$-regular tree for which we determine precise asymptotics. We expect that our methods will generalize to a wide range of spin systems for which the second moment method holds.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 9, 42 pp.

Dates
Accepted: 30 December 2015
First available in Project Euclid: 5 February 2016

https://projecteuclid.org/euclid.ejp/1454682890

Digital Object Identifier
doi:10.1214/16-EJP3552

Mathematical Reviews number (MathSciNet)
MR3485351

Zentralblatt MATH identifier
1342.60160

#### Citation

Bhatnagar, Nayantara; Sly, Allan; Tetali, Prasad. Decay of correlations for the hardcore model on the $d$-regular random graph. Electron. J. Probab. 21 (2016), paper no. 9, 42 pp. doi:10.1214/16-EJP3552. https://projecteuclid.org/euclid.ejp/1454682890

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