# Statistical Mechanics Seminar

## 2007/08 - Term 2

Current Statistical Mechanics Seminars

10.01.2008 | Dimitrios Tsagkarogiannis (Max Planck Institut, Leipzig) Coarse-graining schemes for stochastic lattice systems In this talk we present a method to construct higher order Monte Carlo numerical schemes for the coarse-graining of stochastic lattice systems with short and long range interactions. The main tool is the cluster expansion of the partition function of the conditioned (to the coarse-grained variables) measure. We also discuss a strategy on how to recover microscopic information from the coarse-grained one. NOTE: Special room, B3.01. |

17.01.2008 | Stefan Adams (University of Warwick) Strict convexity of the free energy for non-convex gradient models We consider a gradient interface model on the lattice with non-convex interaction potential. We show using a multi scale analysis that for sufficiently low temperature and sufficiently small tilt of the interface the free energy is a strictly convex function of the tilt. |

24.01.2008 | Gunter Schütz (Forschungszentrum Jülich) Random Matrices and Current Fluctuations in the Zero Range Process The totally asymmetric simple exclusion process (TASEP) and several related stochastic lattice gas models are shown to have determinantal transition probabilities. With this result, derived using Bethe ansatz, we obtain (i) a (fairly) simple rederivation and generalizations of Johansson's celebrated result (CMP, 2000) for the current distribution in the TASEP, (ii) some surprising properties of the current distribution in the zero-range process with open boundaries, and (iii) the exact solution of the Bernoulli sequence matching problem for finite sequence length. |

31.01.2008 | Lenka Zdeborová (Université de Paris Sud, Orsay) Phase transitions in random constraint satisfaction problems Solving constraint satisfaction problems is of immense importance in many computer science related applications. Our goal is to understand better why and when are some constraint satisfaction problems computationally hard. As an example we consider the coloring a large random graph with a given number of colors such that no adjacent vertexes have the same color. We first introduce interesting questions about the problem and then explain how some of them can be answered using ideas from statistical physics of disordered systems. In particular we show that as the average degree is increased, the space of proper colorings first decomposes into exponential number of clusters (pure states), then it condenses over a finite number of the largest clusters, and, eventually, no more solutions exist. We describe in detail the nature and properties of the clustered (glassy) phase, and highlight some connections to the algorithmic hardness. The talk will be self-contained. |

07.02.2008 | Bill Faris (University of Arizona) Tree equations and lattice gas models This talk is an elementary and self-contained account of an important relation between combinatorics and equilibrium statistical mechanics. In fact, the existence of a tree graph fixed point is the classic criterion for convergence of a cluster expansion. The talk will explore properties of this fixed point equation, pointing out some apparently open questions. |

14.02.2008 | Stefano Olla (Université Paris Dauphine) Energy diffusion in system of oscillators with conservative noise: weak coupling and kinetic limits We consider a system of coupled oscillators whose Hamiltonian dynamics is perturbed by stochastic terms that conserve energy (and eventually momentum). We study the macroscopic thermal conductivity and the diffusion of energy in the weak coupling limit and in the kinetic limit. |

21.02.2008 | Charles-Edouard Pfister (Ecole Polytechnique Fédérale, Lausanne) On the nature of isotherms at first order phase transition for classical lattice models I shall present two main results: 1) Isakov's Theorem, and its generalization by Friedli and Pfister, about the impossibility of an analytic continuation of the pressure at first order phase transition. 2) Friedli's work about the restoration of the analytic continuation of the pressure in the van der Waals limit. I shall give an historical introduction to these questions (van der Waals' theory versus Mayer's theory), which were much debated among physicists during several decades last century. |

28.02.2008 | Wolfgang König (Universität Leipzig) A variational formula for the free energy of a many-Boson system We consider $N$ Bosons in a box with volume $N/\rho$ under the presence of a mutually repellent pair potential. Denote by $H_N$ the corresponding Hamilton operator with either zero or periodic boundary condition. The symmetrised trace of $e^{-\beta H_N}$ describes the Bosons at positive temperature $1/\beta$. Our main result is a variational formula for the limiting free energy, for any fixed values of the particle density $\rho$ and the inverse temperature $\beta$, in any dimension. The main tools are a description in terms of a marked Poisson point process and a large-deviation analysis of the stationary empirical field. The resulting variational formula in particular describes the asymptotic cycle structure that is induced by the symmetrisation in the Feynman-Kac formula. We close with a short discussion of the relation to Bose-Einstein condensation. (joint work in progress with S. Adams and A. Collevecchio) |

06.03.2008 | François Germinet (Université de Cergy-Pontoise) Bernoulli decomposition and applications We recall recent Bernoulli decompositions of any given non trivial real random variable. While our main motivation is a proof of universal occurence of Anderson localization in continuum random Schrödinger operators, we review other applications like Sperner theory of antichains, anticoncentration bounds of some functions of random variables, as well as singularity of random matrices. |

13.03.2008 | Yvon Vignaud (Technische Universität Berlin) Phase transitions for a Potts model in the continuum We consider a q-Potts model in the continuum, where particles interact in a ferromagnetic Kac potential (with very large but finite range R), the chemical potential being independant of the type of particles. In the spirit of the seminal work by Lebowitz, Mazel and Presutti, we show that our model can be considered as a perturbation of a suitable mean field model. In particular we show coexistence of ordered and disordered Gibbs measures, with a jump of the total density of particles at the transition point. |

## 2007/08 - Term 3

24.04.2008 | Jozsef Lorinczi (Loughborough University) Exponential integrability of some rough functionals Driven by applications to quantum field theory which I will explain briefly, I address the problem of constructing Gibbs measures on Brownian paths with respect to densities dependent on double stochastic integrals. In order to have a pathwise control of boundary conditions I will describe the framework of stochastic currents combining techniques of rough paths analysis and cluster expansion. |

01.05.2008 | James Martin (Oxford University) Crossing probabilities in the asymmetric exclusion process In the one-dimensional symmetric simple exclusion process (ASEP), particles jump to the right at rate p>1/2 and left at rate 1-p, interacting by exclusion. Consider an initial state with first-class particles to the left of the origin, empty spaces to the right of the origin and a "second-class particle" at the origin. (The second-class particle can jump into an empty space just as a particle does, but can also be displaced itself by first-class particles). The second-class particle has an asymptotic speed with probability one, but this speed is random. The second-class particle can be seen as a discrepancy between two ASEP systems which evolve together under a natural coupling. I'll discuss some extensions to systems with several second-class particles, and describe how they relate to questions about coupling and to questions about interfaces in models of random growth. |

08.05.2008 | Roberto Fernández (Université de Rouen) Cut-off and exit trajectories: Two sides of the same coin We present a general framework linking cut-off and exit excursions for birth-and-death processes on a countable alphabet. Under suitable hypothesis, we prove that cut-off convergence towards a (local) equilibrium is accompanied by exponentially distributed out-of-equilibrium excursions. Furthermore, atypical trajectories leading to these excursions and final cut-off trajectories are related by time inversion; in particular their time lengths have identical laws. |

15.05.2008 | Fabio Martinelli (Università Roma Tre) Dynamical relaxation of a 1D pinning model We consider paths of a one–dimensional simple random walk conditioned to come back to the origin after L steps, In the pinning model each path has a weight exp(c N(η)), where N is the number of zeros in the path and c, the pinning strength can be positive or negative. When the paths are constrained to be non–negative, the polymer is said to satisfy a hard–wall constraint. Such models are well known to undergo a localization/delocalization transition as the pinning strength is varied. In this paper we study a natural "spin flip" dynamics for these models, derive several estimates on its spectral gap and mixing time and discuss a possible signature of the phase transition on the dynamics. |

22.05.2008 | Mathieu Lewin (Université de Cergy-Pontoise) The thermodynamic limit of quantum Coulomb systems I will present a new approach for proving the existence of the thermodynamic limit for quantum systems composed of electrons and nuclei interacting via the Coulomb potential, like in ordinary matter. In particular I will provide a very general setting allowing to study many different quantum systems. This is a joint work with Christian Hainzl and Jan Philip Solovej. |

29.05.2008 | Hans-Otto Georgii (LMU Munich) Continuum models with Delaunay interaction: Gibbs measures, thermodynamic formalism, and large deviations We present some work in progress on two-dimensional point processes with an interaction that depends on the Delaunay triangulation of the points (the dual of the Voronoi tesselation). We study the associated free energy, provide a Gibbs characterisation of its minimisers, and derive a large deviation principle on process level. |

05.06.2008 | Nadia Sidorova (University College London) Phase transitions for dilute particle systems with potentials of the Lennard-Jones type We consider a dilute stationary system of N particles interacting pairwise according to a compactly supported potential, which is repellent at short distances and attractive at average distances. We are interested in the large-N behaviour of the system, in particular, in the trace of the Boltzmann factor exp(- 1/T H(N)), where H(N) is the Hamiltonian and T is the temperature. We show that at a certain scale there are phase transitions in the temperature and compute the trace explicitly in terms of a variational problem. |

12.06.2008 | Sabine Jansen (Max Planck Institute for Mathematics in the Sciences (Leipzig)) Thermodynamic limit for jellium on a cylinder We consider the classical statistical mechanics of charged particles moving in a neutralizing background on a cylinder. For even-integer values of the so-called plasma parameter and sufficiently small cylinder radius, one can show that not only the free energy but also the correlation functions have a limit when the number of particles goes to infinity, at fixed cylinder radius and fixed background density. The limiting state is periodic with respect to translations along the cylinder axis. The proof makes crucial use of a discrete renewal equality, which is reminiscent of earlier apparitions of renewal (in)equalities in the context of Coulomb systems. The talk is based on joint work with E.H. Lieb and R. Seiler. |

19.06.2008 | Michael Allman (University of Warwick) Breaking the chain We consider the motion of a Brownian particle in R, moving between a particle fixed at the origin and another moving deterministically away at slow speed epsilon > 0. The middle particle interacts with its neighbours via a potential of finite range b > 0, with a unique minimum at a > b/2. We say that the chain of particles breaks on the left- or right-hand side when the middle particle is greater than a distance b from its left or right neighbour, respectively. We study the asymptotic location of the first break of the chain in the limit of small noise, in the case where epsilon = epsilon(sigma) and sigma > 0 is the noise intensity. SPECIAL TIME: 10am. |

26.06.2008 | Daniel Ueltschi (University of Warwick) Spatial random permutations and infinite cycles We consider random permutations of points in the space, where permutations are weighed according to the length of the jumps. The main question deals with the possible occurrence of infinitely long cycles, eventhough all jumps are finite. Rigorous results can be proposed in the case of "one-body" interactions; namely, infinite cycles occur above a critical density, that is given by an explicit expression. I will also discuss the relation between spatial permutations and the Bose-Einstein condensation. This is a collaboration with Volker Betz. SPECIAL TIME: 4pm. |