@article{DereudreHoudebert2018,
  author    = {Dereudre, David and Houdebert, Pierre},
  title     = {Sharp phase transition for the continuum Widom-Rowlinson model},
  series = {Annales de l'Institut Henri Poincar{\´e}. B, Probability and statistics},
  volume    = {57},
  journal   = {Annales de l'Institut Henri Poincar{\´e}. B, Probability and statistics},
  number    = {1},
  publisher = {Association des Publications de l'Institut Henri Poincar{\´e}},
  address   = {Bethesda, Md.},
  issn      = {0246-0203},
  doi       = {10.1214/20-AIHP1082},
  pages     = {387 -- 407},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}
@article{DereudreMazzonettoRoelly2017,
  author    = {Dereudre, David and Mazzonetto, Sara and Roelly, Sylvie},
  title     = {Exact simulation of Brownian diffusions with drift admitting jumps},
  series = {SIAM journal on scientific computing},
  volume    = {39},
  journal   = {SIAM journal on scientific computing},
  number    = {3},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1064-8275},
  doi       = {10.1137/16M107699X},
  pages     = {A711 -- A740},
  year      = {2017},
  abstract  = {In this paper, using an algorithm based on the retrospective rejection sampling scheme introduced in [A. Beskos, O. Papaspiliopoulos, and G. O. Roberts,Methodol. Comput. Appl. Probab., 10 (2008), pp. 85-104] and [P. Etore and M. Martinez, ESAIM Probab.Stat., 18 (2014), pp. 686-702], we propose an exact simulation of a Brownian di ff usion whose drift admits several jumps. We treat explicitly and extensively the case of two jumps, providing numerical simulations. Our main contribution is to manage the technical di ffi culty due to the presence of t w o jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift.},
  language  = {en}
}
@article{DereudreRoelly2017,
  author    = {Dereudre, David and Roelly, Sylvie},
  title     = {Path-dependent infinite-dimensional SDE with non-regular drift},
  series = {Annales de l'Institut Henri Poincar{\´e} : B, Probability and statistics},
  volume    = {53},
  journal   = {Annales de l'Institut Henri Poincar{\´e} : B, Probability and statistics},
  number    = {2},
  publisher = {Inst. of Mathematical Statistics},
  address   = {Bethesda},
  issn      = {0246-0203},
  doi       = {10.1214/15-AIHP728},
  pages     = {641 -- 657},
  year      = {2017},
  abstract  = {We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither bounded or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy and a finite second moment. The originality of our method leads in the use of the specific entropy as a tightness tool and in the description of such infinite-dimensional stochastic process as solution of a variational problem on the path space. Our result clearly improves previous ones obtained for free dynamics with bounded drift.},
  language  = {en}
}