Disagreement percolation for the hard-sphere model

Abstract

Disagreement percolation connects a Gibbs lattice gas and iid site percolation such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds improve upon known bounds obtained by cluster expansion techniques. The proof uses a novel dependent thinning from a Poisson point process to the hard-sphere model, with the thinning probability related to a derivative of the free energy.Keywords: hard-sphere model, disagreement percolation, unique Gibbs measure, stochastic domination, Boolean model, absence of phase transition, dependent thinning MSC 2010: 82B21 (60E15 60K35 60G55 82B43 60D05) * math@temmel.me † The author acknowledges the support of the VIDI project "Phase transitions, Euclidean fields and random fractals", NWO 639.032.916. arXiv:1507.02521v6 [math.PR] 7 Sep 2017Hofer-Temmel Hard-sphere disagreement percolation two hard-sphere realisations with high probability on a small domain inside a larger domain. Taking a limit along an exhaustive sequence of bounded domains implies the uniqueness of the Gibbs measure of the hard-sphere model.The disagreement coupling connects the activity of the hard-sphere models and the intensity of the Poisson point process. Hence, lower bounds on the critical intensity of the Boolean model imply lower bounds on the critical activity of the hard-sphere model. In one dimension, the results replicate Tonk's classic result of the complete absence of phase transitions [26]. In two dimensions, the new bounds improve upon the best known cluster expansion bounds [23,9] by at least a factor of two. They even exceed the best theoretical largest activities achievable by cluster expansion techniques. In high dimensions, extrapolation of known upper bounds on the activities achievable in the discrete case to the continuum suggests that the disagreement percolation bounds always go beyond the region attainable by cluster expansion techniques.This work exclusively treats the hard-sphere model. One reason is its central importance in statistical mechanics and its easy and emblematic definition. Another reason is the comparison with the cluster expansion bounds. More important though, the bounds in this paper stem from a twisted disagreement coupling optimised for the hard-sphere model. While a generalisation of the disagreement approach to simple finite-range Gibbs point processes with bounded interaction range seems possible, the twisted coupling depends critically on the hard-sphere constraint. The twisted approach takes inspiration from a disagreement percolation tailored to the hard-core model [31].The twisted disagreement coupling is defined in a recursive fashion a...

Analyticity for Classical Gasses via Recursion

“…Hofer-Temmel [18] and Dereudre [7] gave a new bound for uniqueness and exponential decay of correlations for the hard sphere model based on disagreement percolation and the critical activity for Poisson-Boolean percolation. In high dimensions this gives uniqueness for λ < 1 + o d (1).…”

“…Therefore to move beyond the limits of cluster expansion convergence one must use properties of positive activities or utilize regions of the complex plane that are not symmetric around 0. Two previous approaches in this direction are probabilistic: disagreement percolation [4,7,18] and Markov chain mixing [17]. In the case of the hard sphere model these techniques surpass the bounds for uniqueness given by the cluster expansion.…”