Universality for beta ensembles

Thursday, June 21, 2012 - 3:30pm - 4:45pm
Lind 305
Paul Bourgade (Harvard University)
Wigner stated the general hypothesis that the distribution of eigenvalue
spacings of large complicated quantum systems is universal in the sense that
it depends only on the symmetry class of the physical system but not on
other detailed structures. The simplest case for this hypothesis concerns
large but finite dimensional matrices. Spectacular progress was done in the
past two decades to prove universality of random matrices presenting an
orthogonal, unitary or symplectic invariance. These models correspond to
log-gases with respective inverse temperature 1, 2 or 4. I will report on a
joint work with L. Erdős and H.-T. Yau, which yields universality for the
log-gases at arbitrary temperature at the
microscopic scale. A main step consists in the optimal localization of the
particles, and the involved techniques include a multiscale analysis and a
local logarithmic Sobolev inequality.