Analysis And Probability Day

Wednesday February 11, 2026

Location: Elizabeth Templeton Lecture Room, New College, Mound Pl., Edinburgh

Nicolas Burq (Université Paris-Saclay)

Time: 2:30 pm

Coffee break: 3:30 pm – 4:00 pm

Entropy and large deviations for pairs of random unitary matrices

Tim Austin (University of Warwick)

Time: 4:00 pm

Abstract: Consider a pair of n-by-n unitary matrices chosen independently at random.  By combining them using sums and products, we can then form many other random matrices as well.  The random operator norms or spectra of all these different combinations tell us a lot about how the original unitaries “sit together” as operators on the same space.  An active sub-field of random matrix theory studies the limiting behaviours of these norms or other quantities as n diverges.

This talk will be about a large deviations principle that governs the random fluctuations of those operator norms.  We obtain this principle via a new quantity called “almost periodic entropy”, which is a kind of analog of Lewis Bowen’s “sofic entropy” from ergodic theory.  I will sketch the origins of this new quantity and its relevance to the large-deviations question.  I will not assume any prior knowledge about sofic entropy.