Sharp phase transition for the continuum Widom–Rowlinson model

The Widom–Rowlinson model (or the Area-interaction model) is a Gibbs point process in Rd with formal Hamiltonian defined as volume of ∪x∈ωB1(x), where ω locally finite configuration points and B1(x) denotes unit closed ball centred at x. also tuned by two other parameters: activity z>0 related to intensity inverse temperature β≥0 strength interaction. In present paper we investigate phase transition view percolation theory liquid-gas transition. First, considering graph connecting distance smaller than 2r>0, show that for any β≥0, there exists 0z˜ca(β,r)). These results are spirit recent works using randomised tree algorithms (Probab. Theory Related Fields 173 (2019) 479–490, Ann. Math. 189 75–99, Duminil-Copin, Raoufi Tassion (2018)). Secondly study standard uniqueness/non-uniqueness states depending on parameters z, β. Old (Phys. Rev. Lett. 27 (1971) 1040–1041, J. Chem. Phys. 52 (1970) 1670–1684) claim non-uniqueness regime z=β large enough it conjectured uniqueness should hold outside half line (z=β≥βc>0). We solve partially this conjecture dimension showing β if only z=β. critical value corresponds threshold z˜ca(β,r)=β enough, providing straight between these notions