# Seminars & Events for Mathematical Physics Seminar

##### 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).

##### 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).

##### 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.

##### 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.

##### 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 approximately $exp^{\Lambda_d*V}$ 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.

##### Eigenvalue Statistics for Random CMV Matrices

CMV matrices are the unitary analogues of one dimensional discrete Schrödinger 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.