Mathematical Physics Seminar

MATHEMATICAL PHYSICS SEMINAR

SPRING 2009 Lectures

Regular meeting time: Tuesdays 4:30--5:30
Place: Jadwin 343

Date	Speaker	Title
Please note special date, time, location Wed., Feb. 18 2:00 Jadwin A08	Alex Sodin, Tel Aviv Univ	Universality at the spectrum edge for random matrices with independent entries: Soshnikov's theorems and some extensions We shall discuss the distribution of extreme eigenvalues for several classes of random matrices with independent entries. In particular, we shall discuss the results of Soshnikov and some of their recent extensions, and the combinatorial questions that appear in the proofs. (Based on joint work with Ohad Feldheim).
Feb. 24	Simone Warzel, Tech. Univ. Munich	Localization bounds for multiparticle systems We discuss the spectral and dynamical properties of quantum systems of N particles on the lattice of arbitrary dimension, with a Hamiltonian which in addition to the kinetic term includes a random potential with parameters of the model are the strength of the disorder and the strength of the interparticle interaction. We present a proof that for all N there are regimes of high disorder, and/or weak enough interactions, for which the system exhibits spectral and dynamical localization. The localization bounds are expressed in terms of exponential decay in the Hausdorff distance in the configuration space. The results are derived through the analysis of fractional moments of the N-particle Green function, and related bounds on the eigenfunction correlators. (Joint work with Michael Aizenman).
Mar. 3	Yuval Peres, UC Berkeley	Mixing time of the Ising model at critical temperatures It is widely believed that on nice graphs, the relaxation time (inverse gap) of Glauber dynamics for the Ising model (with free boundary and no external field) is bounded at high temperature, satisfies a power law at criticality and is exponential at low temperature. But exponential in what? "The surface area" is a correct answer in a square, but not in a rectangle. A more general answer appears to be the "cut width". The critical behavior is still not understood even in two dimensions. In recent work, we have established the widely believed picture on the complete graph (where the cut width is the volume and the critical power law is 3/2) and on the regular tree (where the cutwidth is the height and the exact critical power law is still not known). Talk based on joint works with E. Mossel, C. Kenyon, D. Levin, M. Luczak, J. Ding and E. Lubetzky.
Mar. 10	Vincent Vargas, Univ. Paris-Dauphine	Stochastic Scale Invariance and the KPZ formula In this talk, we will prove the KPZ equation (initially introduced in the framework of quantum gravity in 2 dimensions) for the limit lognormal measures introduced by Mandelbrot. More specifically, for a given set K, we will relate it's Hausdorff dimension under the Euclidian metric to it's Hausdorff dimension under the random metric induced by the limit lognormal measure. We will see how the notion of stochastic scale invariance is crucial in the derivation of the aforementioned relation. (Joint work with R. Rhodes)
Mar. 24	Cedric Villani, *ENS Lyon*	Landau damping Sixty years ago, Landau discovered a paradoxical collisionless relaxation effect in plasmas. The Landau damping is now one of the cornerstones of classical plasma physics. From the mathematical point of view, it has remained elusive so far, since the best available results prove the existence of some damped solutions, without saying anything about their genericity. I shall report on new advances, and a whole new mathematical theory, for this problem. I will discuss the physical implications of these results. This is joint work with Clement Mouhot. Comment: This is the first of a series of related talks. In the Colloquium I shall place the Landau damping in perspective with advances in the kinetic theory of plasmas which occurred during the past ten years. In the Analysis seminar I shall go more in depth in the analysis of the Landau damping. Each of the talks will be self-contained and can be attended independently of the others.
Mar. 31	Detlev Buchholz, Univ of Goettingen	Warped Convolutions: A novel tool in the construction of quantum field theories Recently, Grosse and Lechner introduced a deformation procedure for non-interacting quantum field theories, giving rise to interesting examples of theories with non-trivial scattering matrix in any number of spacetime dimensions. In this talk we outline an extension of this procedure to the general framework of quantum field theory by introducing the concept of "warped" convolutions of operator functions. These convolutions have some intriguing properties which permit the deformation of arbitrary nets of algebras based on wedge-shaped regions of Minkowski space to nets which still satisfy Einstein's principles of relativistic covariance and causality. The deformed nets still admit a scattering theory and give rise to a deformed scattering matrix.
Apr. 7	Christian Hainzl, *University of Alabama at Birmingham*	Coupling Einstein's equations to Dirac spinors can prevent the big bang/crunch singularity in the Friedmann model We consider a spatially homogeneous and isotropic system of Dirac particles coupled to classical gravity. We recover, on the one hand, the dust and radiation dominated closed Friedmann-Robertson-Walker space-times. On the other hand, we find particular solutions where the oscillations of the Dirac spinors prevent the formation of the big bang or big crunch singularity. This is joint work with F. Finster.
Apr. 14	Paul Federbush, Univ. of Michigan	An Asymptotic Expansion for the Dimer Lambda_d The dimer problem is to count the number of ways a d-dimensional "chessboard" can be completely covered by non-overlapping dimers (dominoes), each dimer covering two nearest neighbor boxes. The number is ~exp(Lambda_d*V) as the volume V goes to infinity. It has been long known Lambda_d ~ (1/2)ln(d) +(1/2)(ln(2)-1) We derive an asymptotic expansion whose first few terms are Lambda_d ~ (1/2)ln(d) +(1/2)(ln(2)-1) +(1/8)(1/d) + (5/96)(1/d2) + (5/64)(1/d3) The last term here was calculated by computer, and we conjecture the next term will never be explicitly computed ( just by reason of required computer time ). The expansion is not yet rigorously established.
Apr. 28	Mihai Stoiciu, Williams College	Eigenvalue Statistics for Random CMV Matrices CMV matrices are the unitary analogues of one dimensional discrete Schrodinger operators. We consider CMV matrices with random coefficients and we study the statistical distribution of their eigenvalues. For slowly decreasing random coefficients, we show that the eigenvalues are distributed according to a Poisson process. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (clock distribution). For a certain critical rate of decay we obtain the circular beta distribution. This is a joint work with Rowan Killip.
Please note special date and time Fri., May 1 2:00 p.m.	Alessandro Giuliani, Rome Univ. III	The 2D Hubbard model on the honeycomb lattice We consider the 2D Hubbard model on the honeycomb lattice, as a model for graphene in the presence of screened Coulomb interactions. At half filling and weak coupling, we compute the free and ground state energies, and we construct the correlation functions up to zero temperature in terms of convergent series; analiticity is proved by making use of constructive renormalization group methods. We show that the interaction produces a modification of the Fermi velocity and of the wave function renormalization without changing the asymptotic decay of correlations; this rules out the possibility of superconducting or magnetic instabilities in the ground state. We also prove that the correlations verify a Ward Identity similar to the one for massless Dirac fermions, up to asymptotically negligible corrections and an asymmetric renormalization of the charge velocity. The talk is based on joint work with V. Mastropietro.