Probability and Mathematical Physics Seminar

Quantum Ergodicity for Periodic Graphs

Speaker: Theo McKenzie, Stanford University

Location: Warren Weaver Hall 1302

Date: Friday, September 22, 2023, 11:10 a.m.


Quantum ergodicity (QE) is a notion of eigenvector delocalization, that large eigenvector entries are “well spread” throughout the entire graph. Such a property is true of ``chaotic’’ manifolds and graphs, such as random regular graphs and Riemannian manifolds with ergodic geodesic flow. Focusing on graphs, outside of very specific examples, QE was previously only known to hold for families of graphs with a tree local limit. In this talk we show how QE is in satisfied for many families of operators on periodic graphs, including Schrodinger operators with periodic potential on the discrete torus and on the honeycomb lattice.

In order to do this, we use new ideas coming from analyzing Bloch varieties, and some methods coming from proofs in the continuous setting.

Based on joint work with Mostafa Sabri.