Rising GUE Eigenvalue Process from a Fixed Level

TL;DR

This paper constructs a multilevel correlation kernel for the rising GUE eigenvalue process from a fixed initial state and proves its convergence to the extended semi-discrete sine kernel, establishing universality under minimal moment conditions.

Contribution

It introduces a new approach to prove bulk universality for complex Hermitian Wigner matrices without Dyson Brownian motion relaxation, under near-optimal moment assumptions.

Findings

Kernel converges on short time scales to the extended semi-discrete sine kernel.

Establishes fixed-energy universality for Wigner matrices matching GUE covariance.

Achieves universality results with only finite 4+ε moments, avoiding Dyson Brownian motion.

Abstract

We construct the multilevel correlation kernel for the rising GUE eigenvalue process starting from a fixed initial configuration $x^{(m)}$, and show that it converges on short time scales (as quickly as $\text{polylog}(m)$) to the extended semi-discrete sine kernel. As an application, we show fixed-energy universality of bulk local statistics of complex Hermitian Wigner matrices matching the covariance structure of GUE and with a finite $4+\varepsilon$ moment for $\varepsilon>0$. This application demonstrates that it is possible to obtain universality of bulk local statistics under near-optimal moment assumptions without using a Dyson Brownian motion relaxation step, which was a key ingredient in many results on this topic.