Many-body localization is a phenomenon in quantum many-body systems that prevents the system from reaching thermal equilibrium and chaos. This has been observed in various systems and has been a major focus of research. A widely used measure of quantum chaoticity in these systems is the “gap ratio distribution,” which describes the distribution of the ratio of consecutive energy level spacings of the quantum system. Each physical system has its own density of states (DoS), but the gap ratio is insensitive to the DoS and is best suited to access an essential property (chaotic or localized) without bothering system-specific details.
Since its introduction in the field of quantum physics, the random matrix theory of the gap ratio distribution has been employed in almost every study in this field. However, while this theory presents approximate formulas for the gap ratio distribution, an analytical expression is still missing.
Addressing this gap, a study in Progress of Theoretical and Experimental Physics explains exact analytical expressions for the joint probability distribution of two consecutive eigenvalue spacings and the distribution of their ratio for the Gaussian Unitary Ensemble (or GUE) in random matrix theory.
Building on the author’s previous work on determining a conditional probability called the Jánossy density via the Tracy–Widom (TW) system of partial differential equations, the exact, analytical expressions were derived for the first time as the solution to a system of ordinary differential equations.
This expression was showcased by applying it to the zeros of the Riemann zeta function, which are hypothesized to govern the distribution of prime numbers, known as the Riemann hypothesis, a well-known problem in mathematics for many years. These zeros have been found to mirror the eigenvalues of the GUE, suggesting a link between prime numbers and quantum chaotic systems.
The analytical expression proposed in this study demonstrates a systematic convergence of the distribution of the Reiman zeta function zeros to that of GUE, further supporting this argument.
This study thus advances our understanding of quantum chaotic systems and their links to number theory.

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