Nuclei, Primes and the Random Matrix Connection

Abstract

In this article, we discuss the remarkable connection between two very different fields, number theory and nuclear physics. We describe the essential aspects of these fields, the quantities studied, and how insights in one have been fruitfully applied in the other. The exciting branch of modern mathematics, random matrix theory, provides the connection between the two fields. We assume no detailed knowledge of number theory, nuclear physics, or random matrix theory; all that is required is some familiarity with linear algebra and probability theory, as well as some results from complex analysis. Our goal is to provide the inquisitive reader with a sound overview of the subjects, placing them in their historical context in a way that is not traditionally given in the popular and technical surveys.

arXiv comment54 pages, 11 images

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• Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros
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• Identifying the Riemann zeros by periodically driving a single qubit
• Towards an 'average' version of the Birch and Swinnerton-Dyer Conjecture
• Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta functions
• Spectrum statistics in the integrable Lieb-Liniger model
• Determining optimal test functions for $2$-level densities
• Determining Optimal Test Functions for Bounding the Average Rank in Families of $L$-Functions
• An investigation of the non-trivial zeros of the Riemann zeta function
• Spectral Statistics of Non-Hermitian Random Matrix Ensembles
• A Fowler-Nordheim Integrator can Track the Density of Prime Numbers
• Constructing exchangeable pairs by diffusion on manifolds and its application
• On the theorem of Conrey and Iwaniec
• Random Matrix Ensembles with Split Limiting Behavior
• Limiting Spectral Measures for Random Matrix Ensembles with a Polynomial Link Function
• A class of 2x2 correlated random-matrix models with Brody spacing distribution