# 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.

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.