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  - Houdebert, Pierre
A1  - Zass, Alexander
T1  - An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions
JF  - Journal of applied probability / Applied Probability Trust
N2  - We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature beta. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.
KW  - Gibbs point process
KW  - DLR equations
KW  - uniqueness
KW  - Dobrushin criterion;
KW  - cluster expansion
KW  - disagreement percolation
Y1  - 2022
U6  - https://doi.org/10.1017/jpr.2021.70
SN  - 0021-9002
SN  - 1475-6072
VL  - 59
IS  - 2
SP  - 541
EP  - 555
PB  - Cambridge Univ. Press
CY  - Cambridge
ER  -