Abstract: A crossing probability is the probability of finding a critical cluster that touches specified boundary arcs. We consider formulas for the prototypical example, the horizontal crossing probability Ph, which gives the probability of connecting the left and right sides of a rectangle containing a system at the two-dimensional…

Abstract: Recent experiments have found an universal value for the dynamic conductivity in graphene, fully confirming theoretical predictions based on a non interacting tight binding model. This however poses a problem since one could expect a many body renormalization of the non interacting value of the conductivity due to the interaction. In…

Abstract: : In contrast to the cases of single particle Schrödinger operators and ideal Bose gases the effects of random external potentials on many-body systems of interacting particles is still poorly understood. In the lecture the effects of random scatterers on the ground state of the one-dimensional Lieb-Liniger model of interacting…

Abstract: In this talk I will explain how to prove the existence of the thermodynamic limit for a many-body quantum crystal in which the electrons are quantum and the nuclei are classical point particles, with disordered charges and locations around a lattice. This is a collaboration with Xavier Blanc, based on previous work with Christian…

Abstract: We describe the first rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis…

Abstract: Nonequilibrium statistical mechanics close to equilibrium is a physically satisfactory theory centered on the linear response formula of Green-Kubo. This formula results from a formal first order perturbation calculation without rigorous justification. A rigorous derivation of Fourier's law for heat conduction from the laws of mechanics…

Abstract: I will review a method to control irrelevant terms in the renormalisation group and describe an application to self-repelling walk in four dimensions.

Abstract: Quantum fluctuations can drive phase transitions in ice. This happens when the protons tunnel between the two equivalent sites on a hydrogen bond. The corresponding dynamics is collective and dominated by strong local correlations originating from the ice rules. As a result, the state of the proton is entangled, an effect that can be…

ABSTRACT: A chaotic system under periodic forcing can develop a periodically visited strange attractor. We discuss simple models in which the phenomenon, quite easy to see in numerical simulations, can be completely studied analytically.

Abstract: I will review both old and recent work about the time dependence and steady states of isolated macroscopic systems as well as those in contact with infinite thermal reservoirs. The emphasis will be on quantum systems and will include a discussion of the micro/macro connection in isolated ones and the derivation of a master equation…

Abstract: The Mott insulator is accepted as the appropriate ground manifold for the strongly interacting Fermion Hubbard model, with solid He-3 as the simplest exemplar. It is a manifold because of the spin degrees of freedom, which order antiferromagneticallly due to atom exchange, below a critical temperature. No corresponding effect…

Abstract: Review of rigorous results on thermostats. Families of exact formal solutions of the BBGKY hierarchy for hard sphere systems with free boundary conditions at collisions and Fourier equation emergence, to first order in the temperature difference, after boundary conditions are imposed. Formal means that the solutions are given by…

Title“Hidden Symmetries at the Percolation Point in Two Dimensions

Title: "Rounding of 1st Order Quantum Phase Transitions in Low- Dimensional Systems with Quenched Disorder " Abstract: The addition of quenched disorder has a rounding effect on 1st order phase trans ition in systems of sufficiently low dimension (d=2, and up to � 4 in case of co ntinuous symmetry). The talk will focus on the recent…

Title: Critical velocites in rotating Bose gases Abstract: Some of the remarkable phenomena that emerge when a trapped, ultracold Bose gas is set in rapid rotational motion will be reviewed. In anharmonic traps, where the rotational velocity can in principle be arbitrarily large, one can distinguish three critical velocities at which the…

Title: Bethe-Ansatz for the two species totally asymmetric diffusion Model Abstract: We study the two species asymmetric diffusion model that describes two species and vacancies diffusing asymmetrically on a one-dimensional lattice. Our method is the algebraic Bethe Ansatz. We will explain this technique which we use to find the finite-size…

Title: Characterization of Lee-Yang polynomials Abstract: I shall give a characterization of the multi-affine polynomials in n variables that occur in the proof of the Lee-Yang Circle Theorem. In particular, it will be seen that the spin interactions for which the Circle Theorem holds at all temperatures must be the ferromagnetic pair…

Title: Characterization of Lee-Yang polynomials AAbstract: I shall give a characterization of the multi-affine polynomials in n variables that occur in the proof of the Lee-Yang Circle Theorem. In particular, it will be seen that the spin interactions for which the Circle Theorem holds at all temperatures must be the ferromagnetic pair…

Title: From microscopic hamiltonian dynamics to heat equation Abstract: One of the main problems in non-equilibrium statistical mechanics is to derive, by space-time rescaling, macroscopic irreversible diffusive evolution for the co nserved quantities of an (large) hamiltonian system. I will describe the mathematical setup of the problem,…

Title: "Microscopic Models of Macroscopic Transport: A Selective Overview" Abstract: I will describe various attempts to derive, heuristically or rigorously, diffusive behavior of energy (particle) transport, i.e. Fourier's law (Fick's law) from classical microscopic models (mostly deterministic). Computer simulations showing presence or…