MATHEMATICAL PHYSICS SEMINAR
Department of Mathematics
Princeton University
Princeton University Department of Mathematics Seminars Fall 2005 Schedule
Fall 2005 Lectures
Regular meeting time:
Tuesdays 4:30--5:30
Place: Jadwin 343
| Date | Speaker | Title |
| Oct. 4 | Jeffrey Schenker, IAS |
Dynamical localization for an ensemble of fermions with Hartree-Fock interactions at positive temperature The talk will address the localization effect of random potential for interacting fermions, within the framework of a two band Hubbard-type model with Hartree-Fock ("mean field") interactions. A proof of localization for this model can be accomplished in two steps: 1) solving a temperature dependent non-linear fixed point equation for an effective correlated random potential and 2) establishing spectral and dynamical localization for the resulting effective one particle Hamiltonian. We have carried out this program at large disorder and positive temperature by showing that the solution to the fixed point problem satisfies the requirements of the Aizenman-Molchanov fractional moment technique. (Joint work S. Chiesa.) |
| Oct. 11 | Laszlo Erdos, University of Munich |
Quantum diffusion of the random Schrodinger evolution Einstein's kinetic theory of the Brownian motion, based upon light water molecules continuously bombarding the heavy pollen, provided an explanation of diffusion from Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from the Schrodinger equation. In this talk I will report on a mathematically rigorous derivation of a diffusion equation as a long time scaling limit of a random Schrodinger equation in a weak, uncorrelated disorder potential. This is a joint work with M. Salmhofer and H.T. Yau. |
| Oct. 18 | Gregory Berkolaiko, Texas A&M University |
Correlations within the spectrum of a large quantum graph We will begin with the description of the notion of quantum graphs, their spectra, the random matrix conjecture, and the trace formula which formsone of the main tools of the analysis of correlations in the spectrum. We will then discuss the combinatorial ideas behind the expansion of the related form factor, which is used to measure the correlations within the spectrum. The ideas originate from a similar work done on quantum billiards. Transplanting the theory to quantum graphs puts the derivation on a more solid mathematical footing and helps to identify the problems that still prevent us from calling it a "proof". |
| Oct. 25 | Scott Sheffield, Courant Inst. |
Conformal loop ensembles A simple conformal loop ensemble (CLE) in a planar domain D is a random collection of pairwise disjoint simple closed loops in D with a certain Markov property. We prove that there is only a one parameter family of CLEs and describe their laws explicitly in multiple ways (using Gaussian free fields, branching SLE variants, and loop soups). This talk is based on joint work with Wendelin Werner. |
| Nov. 8 | Patrik Ferrari, TU Munich |
GUE eigenvalue distribution and the space-time correlation function for the asymmetric exclusion process (TASEP) We consider the stationary totally asymmetric simple exclusion process (TASEP), with a traslation-invariant measure as initial condition, and analyze the two-point function using a non-intersecting line representation. We determine the two-point function in the large time limit, and show that it is given in term of a generalization of the GUE Tracy-Widom distribution of random matrices. |
| Nov. 15 | Thomas Chen, Princeton Univ. |
Infrared representations, number bounds and renormalization
in QED We discuss some recent work related to the infrared problem in non-relativistic Quantum Electrodynamics (QED). It is explained how some fundamental results which have long been established for Nelson's model (infrared representations, aspects of scattering theory) can now also be proved for QED. Key to the analysis is a bound on the infrared renormalized electron mass in the case where the interaction Hamiltonian has critical scaling (a problem of endpoint type). This estimate is derived by use of an isospectral renormalization group method designed for the spectral analysis of Hamiltonians in quantum field theory. This is in part based on joint work with V. Bach, J. Fr\"ohlich, and I.M. Sigal. |
| Nov. 22 | Julien Dubedat, Courant Institute |
Commutation of SLEs Schramm-Loewner Evolutions (SLEs) have proved a powerful tool for describing, in the scaling limit, a conformally invariant simple curve. In several instances, such as: percolation, Potts model clusters, and the uniform spanning tree, the curves are initially defined in a discrete setting. We will discuss questions pertaining to the joint law of these curves in the scaling limit. |
| Nov. 29 | Rafael Benguria, Universidad Catolica Santiago, Chile |
Isoperimetric inequalities for eigenvalues of the Laplacian in the hyperbolic space H_n I will review several isoperimetric inequalities for eigenvalues of the laplacian on bounded, smooth domains of $H_n$ with Dirichlet boundary conditions. In particular, I will present our recent proof of an isoperimetric inequality for the second Dirichlet eigenvalue. This is joint work with Helmut Linde. |
Please note special date and time |
Bertrand Duplantier, Theoretical Physics, Saclay |
Random Paths, SLE and Quantum Gravity The talk will focus on a unified geometrical point of view on conformally invariant random paths in the plane, such as: the paths of Brownian motion, percolation hulls, and the traces of the Stochastic Loewner Evolution (SLE). The conjectured Knizhnik, Polyakov, and Zamolodchikov (KPZ) relation permits to relate their characteristic exponents to similar ones on a random surface with fluctuating metric. There, the stochasticity of 2D quantum gravity leads to a decoupling of interacting paths and an additive structure of the exponents. The implications can be illustrated in several instances: the duality, relating SLE_k (k=kappa) with SLE_k*, at k*=16/k, mapping hulls of SLEs to their external boundaries; the ``transmutation'' of Brownian paths into multiple SLEs; the shadow exponents describing the probability that some paths screen some others from the exterior; and the multifractal harmonic and rotational spectra of the SLE curves. The validity of this KPZ tool, coming from conformal field theory, is a challenge for mathematical physics. (*) Please note: this week's Tuesday Math-Phys seminar is shifted to Friday. The speaker will also give a Physics Colloquium on Thursday Dec. 8. |
| Dec. 13 | **POSTPONED** Charles Newman. Courant Institute |
Scaling limit of two-dimensional critical percolation We discuss the continuum nonsimple loop process that represents the scaling limit of 2D critical percolation (this is joint work with Federico Camia -- see math.PR/0504036) and then, if time permits, discuss some ideas and open problems associated with its extension to scaling limits of "near-critical" percolation and related lattice models (this is joint work with Federico Camia and L. Renato Fontes -- see cond-mat/0510740). |
For more information about this seminar, contact Princeton University Department of Mathematics Seminar