Hardy-Szeg\H{o} Point Processes: Large Deviations and Strong Szeg\H{o} Asymptotics

Abstract

We study exponential-scale fluctuations of the Hardy-Szeg\H{o} zero process, investigated by Peres and Vir\'ag in the disk setting, in its upper-half-plane realization, which reveals a different probabilistic geometry. This conformally invariant determinantal zero process is equivalent to its disk realization, but the upper-half-plane coordinates make real-translation invariance explicit and single out long horizontal windows as natural observables. For the vertical window from height one to height $a>1$, let $N_a(L)$ denote the number of zeros in the corresponding horizontal window of length $L$. We identify the limiting scaled log-moment generating function explicitly. As a consequence, we prove a large deviations principle for $N_a(L)/L$, with rate function given by the Legendre transform of this limit. We also prove a strong Szeg\H{o} expansion for the log-moment generating function, including an explicit order-one correction, locally uniformly in the natural complex strip. The proof uses the planar determinantal structure before projection, reduces the problem to a one-dimensional Fredholm determinant, and combines fixed-power trace asymptotics with a Wiener-Hopf comparison.