TY  - JOUR
A1  - Dereudre, David
A1  - Houdebert, Pierre
T1  - Sharp phase transition for the continuum Widom-Rowlinson model
JF  - Annales de l'Institut Henri Poincaré. B, Probability and statistics
N2  - The Widom-Rowlinson model (or the Area-interaction model) is a Gibbs point process in R-d with the formal Hamiltonian defined as the volume of Ux epsilon omega B1(x), where. is a locally finite configuration of points and B-1(x) denotes the unit closed ball centred at x. The model is also tuned by two other parameters: the activity z > 0 related to the intensity of the process and the inverse temperature beta >= 0 related to the strength of the interaction. In the present paper we investigate the phase transition of the model in the point of view of percolation theory and the liquid-gas transition. First, considering the graph connecting points with distance smaller than 2r > 0, we show that for any beta >= 0, there exists 0 <(similar to a)(zc) (beta, r) < +infinity such that an exponential decay of connectivity at distance n occurs in the subcritical phase (i.e. z <(similar to a)(zc) (beta, r)) and a linear lower bound of the connection at infinity holds in the supercritical case (i.e. z >(similar to a)(zc) (beta, r)). These results are in the spirit of recent works using the theory of randomised tree algorithms (Probab. Theory Related Fields 173 (2019) 479-490, Ann. of Math. 189 (2019) 75-99, Duminil-Copin, Raoufi and Tassion (2018)). Secondly we study a standard liquid-gas phase transition related to the uniqueness/non-uniqueness of Gibbs states depending on the parameters z, beta. Old results (Phys. Rev. Lett. 27 (1971) 1040-1041, J. Chem. Phys. 52 (1970) 1670-1684) claim that a non-uniqueness regime occurs for z = beta large enough and it is conjectured that the uniqueness should hold outside such an half line ( z = beta >= beta(c) > 0). We solve partially this conjecture in any dimension by showing that for beta large enough the non-uniqueness holds if and only if z = beta. We show also that this critical value z = beta corresponds to the percolation threshold (similar to a)(zc) (beta, r) = beta for beta large enough, providing a straight connection between these two notions of phase transition.
KW  - Gibbs point process
KW  - DLR equations
KW  - Boolean model
KW  - Continuum
KW  - percolation
KW  - Random cluster model
KW  - Fortuin-Kasteleyn representation
KW  - Randomised tree algorithm
KW  - OSSS inequality
Y1  - 2018
U6  - https://doi.org/10.1214/20-AIHP1082
SN  - 0246-0203
SN  - 1778-7017
VL  - 57
IS  - 1
SP  - 387
EP  - 407
PB  - Association des Publications de l'Institut Henri Poincaré
CY  - Bethesda, Md.
ER  - 
TY  - JOUR
A1  - Hofer-Temmel, Christoph
A1  - Houdebert, Pierre
T1  - Disagreement percolation for Gibbs ball models
JF  - Stochastic processes and their application
N2  - We generalise disagreement percolation to Gibbs point processes of balls with varying radii. This allows to establish the uniqueness of the Gibbs measure and exponential decay of pair correlations in the low activity regime by comparison with a sub-critical Boolean model. Applications to the Continuum Random Cluster model and the Quermass-interaction model are presented. At the core of our proof lies an explicit dependent thinning from a Poisson point process to a dominated Gibbs point process. (C) 2018 Elsevier B.V. All rights reserved.
KW  - Continuum random cluster model
KW  - Disagreement percolation
KW  - Dependent thinning
KW  - Boolean model
KW  - Stochastic domination
KW  - Phase transition
KW  - Unique Gibbs state
KW  - Exponential decay of pair correlation
Y1  - 2019
U6  - https://doi.org/10.1016/j.spa.2018.11.003
SN  - 0304-4149
SN  - 1879-209X
VL  - 129
IS  - 10
SP  - 3922
EP  - 3940
PB  - Elsevier
CY  - Amsterdam
ER  -