Quantative universality for cokernels of matrices with symmetries

TL;DR

This paper establishes universality results for cokernels of random symmetric and alternating matrices, providing explicit error bounds and a novel proof approach that improves on classical methods.

Contribution

It introduces a new quantitative method for proving universality of cokernels in symmetric and alternating matrices, with explicit error bounds and improved convergence rates.

Findings

Proves universality for cokernels of symmetric matrices.

Reproves Hodges' universality theorem with explicit bounds.

Addresses effective convergence rates in cokernel universality.

Abstract

We prove universality for cokernels of random integral matrices with symmetries via an approach different from the classical surjection moment method introduced by Wood (arXiv:1402.5149). In the symmetric case, we reprove Hodges' universality theorem (arXiv:2311.07078), i.e. the version incorporating the canonical pairing from Wood's setting, and in the alternating case we reprove the local universality theorem of Nguyen-Wood (arXiv:2210.08526). A key advantage of our method is that it is quantitative: we obtain explicit error bounds, which are exponentially small in most regimes, thereby addressing Wood's question on effective convergence rates. Our argument is inspired by Maples' exposure-process and coupling viewpoint (arXiv:1301.1239) and uses a generalized form of Fourier-analytic estimates in the exponentially sharp style of Ferber-Jain-Sah-Sawhney (arXiv:2106.04049).