"RANDOM MATRICES AND RELATED TOPICS" |
Monthly Colloquia |
Abstract:
A basic problem in the theory of expander graphs, formulated by Lubotzky and
Weiss, is to what extent being an expander familyfor a family of Cayley graphs
is a property of the groups alone, independent of the choice of generators.
While recently Alon, Lubotzky and Wigderson constructed an example demonstrating
that expansion is not in general a group property, the problem is open for
``natural'' families of groups. In particular for SL(2, p) numerical experiments
indicate that it might be an expander family for ``generic'' choices of
generators (Independence Conjecture).
A basic conjecture in Quantum Chaos, formulated by Bohigas, Giannoni, and Shmit,
asserts that the eigenvalues of a quantized chaotic Hamiltonian behave like the
spectrum of a typical member of the appropriate ensemble of random matrices.
Both conjectures can be viewed as asserting that a deterministically constructed
spectrum ``generically" behaves like the spectrum of a large random matrix: ``in
the bulk'' (Quantum Chaos Conjecture) and at the ``edge of the spectrum''
(Independence Conjecture). After explaining this approach in the context of the
spectra of random walks on groups, we will review some recent related results
and numerical experiments.