Year: 2026
Type: preprint
Abstract: Using the theory of von Neumann algebras, Vaughan Jones showed abstractly that there is a bridge between holomorphic cusp forms and random matrices. This bridge is supplied by a distinguished function he called a {\it tracelike vector}. He posed the following problem: construct a tracelike vector to make this bridge explicit. We carry out Jones' vision for the full modular group. We construct holomorphic tracelike vectors for every finite-covolume Fuchsian group throughout the permissible Bergman range. For the modular group at the critical parameter $s=13$, we use the resulting unit tracelike vector to assign explicit group coefficients to cusp form multiplication operators, and form positive finite random matrices from a shared Haar-distributed unitary matrix. Almost surely, their normalised traces recover all joint operator moments, first as the matrix size tends to infinity, and then as the group-word cutoff tends to infinity. We also identify the limiting mixed second moment with a series of Petersson norms that involve Maass-Shimura operators of cusp forms. The exact nature of our use of AI is detailed in the introduction.
Source: arXiv (Cornell University)
Authors: Diamantis, Nikolaos, Rolen, Larry
Institution: Vanderbilt University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7215123310