Fisher-Hartwig asymptotics, Szego expansions, and applications to statistical physics

March 27 to March 31, 2017

at the

American Institute of Mathematics, San Jose, California

organized by

Hajo Leschke, Alexander V. Sobolev, and Wolfgang Spitzer

Original Announcement

This workshop will focus on the theory of Toeplitz matrices (TM) and Wiener-Hopf operators (WHO).

In the recent ten or so years, the theory of these operators (TM and WHO) attracted a great deal of renewed attention from mathematical and theoretical physics communities due to the mathematical beauty of problems involved, and due to new applications in physics. For example, novel developments in quantum information theory and statistical physics concern the asymptotics of the (quantum) entanglement entropy of thermal equilibrium states, the overlap of ground states in the Anderson orthogonality catastrophe and the emptiness formation probability in quantum spin models. These quantities can be deduced from suitable Szego limit theorems, that is, from asymptotic results for traces of certain functions of TM and WHO.

The workshop will focus on the following problems.

Double scaling limits for TM and WHO with symbols which, apart from the scaling parameter $L$, depend on another parameter, say on temperature $T$, as $L \rightarrow \infty$ and $T \downarrow 0$ in a coordinated way.
Asymptotics for block TM and WHO with Fisher-Hartwig singularities.
The study of multi-term asymptotics for TM and WHO with the aim to detect topological entropy, that is, a term depending, for instance, on the Euler characteristics of the spatial domain.
Extension to TM and WHO which are translation invariant with respect to a group different from (all) Euclidean translations. Examples include periodic potentials, magnetic fields and random operators.

The purpose of this workshop is to bring together experts from mathematics and theoretical/mathematical physics in order to address the topics outlined above, and to identify new directions in the field Szego expansions, stimulated by recent developments in both fields.