The Fyodorov-Hiary-Keating Conjecture

In 2012, Fyodorov, Hiary & Keating and Fyodorov & Keating proposed a series of conjectures describing the statistics of large values of zeta in short intervals of the critical line. In particular, they relate these statistics to the ones of log-correlated Gaussian fields and of characteristic polynomials of random matrices. In this lecture, I will present recent results that answer many aspects of these conjectures, including upper and lower bounds for the maximum that are sharp up to the order of fluctuations. The connections with log-correlated Gaussian fields, in particular branching random walks, will be emphasized. Based on joint works with P. Bourgade and M. Radziwill.
 

Arguin-2022-06-22