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 - INPR A1 - Dereudre, David A1 - Roelly, Sylvie T1 - Propagation of Gibbsianness for infinite-dimensional gradient Brownian diffusions N2 - We study the (strong-)Gibbsian character on RZd of the law at time t of an infinitedimensional gradient Brownian diffusion , when the initial distribution is Gibbsian. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 06 KW - infinite-dimensional Brownian diffusion KW - Gibbs measure KW - cluster expansion Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51535 ER - TY - GEN A1 - Roelly, Sylvie A1 - Dereudre, David T1 - Propagation of Gibbsiannes for infinite-dimensional gradient Brownian diffusions N2 - We study the (strong-)Gibbsian character on R Z d of the law at time t of an infinitedimensional gradient Brownian diffusion , when the initial distribution is Gibbsian. KW - infinite-dimensional Brownian diffusion KW - Gibbs measure KW - cluster expansion Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6918 ER - TY - INPR A1 - Dereudre, David A1 - Roelly, Sylvie T1 - Path-dependent infinite-dimensional SDE with non-regular drift : an existence result N2 - 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 small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)11 KW - Infinite-dimensional SDE KW - non-Markov drift KW - non-regular drift KW - variational principle KW - specific entropy Y1 - 2014 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72084 SN - 2193-6943 VL - 3 IS - 11 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Dereudre, David A1 - Roelly, Sylvie T1 - Path-dependent infinite-dimensional SDE with non-regular drift BT - an existence result JF - Annales de l'Institut Henri Poincaré : B, Probability and statistics N2 - 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. N2 - Nous établissons, dans cet article, l’existence de solutions faibles pour un système infini-dimensionnel de diffusions browniennes. Le terme de dérive est véritablement général, au sens où il est supposé n’être ni borné, ni continu, ni Markovien. Nous supposons cependant que la loi initiale admet une entropie spécifique finie. L’originalité de notre méthode consiste en l’utilisation de la bornitude de l’entropie spécifique comme critère de tension et en l’identification des solutions du système comme solutions d’un problème variationnel sur l’espace des trajectoires. Notre résultat améliore clairement ceux préexistants concernant des dynamiques libres perturbées par des dérives bornées. KW - Infinite-dimensional SDE KW - Non-Markov drift KW - Non-regular drift KW - Variational principle KW - Specific entropy Y1 - 2017 U6 - https://doi.org/10.1214/15-AIHP728 SN - 0246-0203 VL - 53 IS - 2 SP - 641 EP - 657 PB - Inst. of Mathematical Statistics CY - Bethesda ER - TY - BOOK A1 - Dereudre, David A1 - Roelly, Sylvie T1 - On Gibbsianness of infinite-dimensional diffussions T3 - Preprint / Universität Potsdam, Institut für Mathematik, Mathematische Statistik un Y1 - 2004 SN - 1613-3307 PB - Univ. CY - Potsdam ER - TY - GEN A1 - Roelly, Sylvie A1 - Dereudre, David T1 - On Gibbsianness of infinite-dimensional diffusions N2 - The authors analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the d-dimensional lattice. In the first part of the paper, these processes are characterized as Gibbs states on path spaces. In the second part of the paper, they study the Gibbsian character on R^{Z^d} of the law at time t of the infinite-dimensional diffusion X(t), when the initial law is Gibbsian. AMS Classifications: 60G15 , 60G60 , 60H10 , 60J60 KW - infinite-dimensional Brownian diffusion KW - Gibbs field KW - cluster expansion Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6692 ER - TY - BOOK A1 - Dereudre, David A1 - Roelly, Sylvie T1 - On Gibbsianness of infinite-dimensional diffusions N2 - We analyse different Gibbsian properties of interactive Brownian diffusions X indexed by the lattice $Z^{d} : X = (X_{i}(t), i ∈ Z^{d}, t ∈ [0, T], 0 < T < +∞)$. In a first part, these processes are characterized as Gibbs states on path spaces of the form $C([0, T],R)Z^{d}$. In a second part, we study the Gibbsian character on $R^{Z}^{d}$ of $v^{t}$, the law at time t of the infinite-dimensional diffusion X(t), when the initial law $v = v^{0}$ is Gibbsian. T3 - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2004, 01 KW - infinite-dimensional Brownian diffusion KW - Gibbs field KW - cluster expansion Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-52630 ER - TY - JOUR A1 - Dereudre, David A1 - Mazzonetto, Sara A1 - Roelly, Sylvie T1 - Exact simulation of Brownian diffusions with drift admitting jumps JF - SIAM journal on scientific computing N2 - 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. KW - exact simulation methods KW - skew Brownian motion KW - skew diffusions KW - Brownian motion with discontinuous drift Y1 - 2017 U6 - https://doi.org/10.1137/16M107699X SN - 1064-8275 SN - 1095-7197 VL - 39 IS - 3 SP - A711 EP - A740 PB - Society for Industrial and Applied Mathematics CY - Philadelphia ER - TY - INPR A1 - Dereudre, David A1 - Mazzonetto, Sara A1 - Roelly, Sylvie T1 - Exact simulation of Brownian diffusions with drift admitting jumps N2 - Using an algorithm based on a retrospective rejection sampling scheme, we propose an exact simulation of a Brownian diffusion 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 difficulty due to the presence of two jumps thanks to a new explicit expression of the transition density of the skew Brownian motion with two semipermeable barriers and a constant drift. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 7 KW - exact simulation method KW - skew Brownian motion KW - skew diffusion KW - Brownian motion with discontinuous drift Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-91049 SN - 2193-6943 VL - 5 IS - 7 PB - Universitätsverlag Potsdam CY - Potsdam ER -