Abstract
We examine the swapping mechanism for interacting diffusions developed by Díaz-Pérez and Mulet for potential structural correspondences with the Riemann Hypothesis.
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Overview
This essay mines the probabilistic framework of swapping interacting diffusions—where particles exchange positions at random times—for patterns that might illuminate the statistics of the Riemann zeta zeros. The source paper introduces a rigorous continuous-time model, proves a symmetrisation identity relating swapping to regime-switching via a sum over the symmetric group SN, and establishes that swapping enhances the antisymmetric spectral gap by an amount proportional to the swap rate.
We explore whether this spectral gap enhancement serves as a stochastic analogue of the level repulsion observed in the Gaussian Unitary Ensemble (GUE) conjecture. The GUE hypothesis posits that the normalized spacings between consecutive non-trivial zeros of the Riemann zeta function exhibit the same statistics as eigenvalues of large random Hermitian matrices, characterized by a "hard gap" where the probability of small spacings vanishes quadratically.
The proposed dictionary maps the swap rate to the mean zero spacing and the antisymmetric spectral gap to the repulsion coefficient. However, we rate this correspondence as a suggestive metaphor only. The analogy founders on a critical disanalogy: the source symmetrisation sums positive densities (a permanent-like structure), whereas GUE correlation functions are determinants involving sign cancellations. Furthermore, the spectral gap is an asymptotic decay rate in time, while level repulsion is a static spatial statistic. We conclude that the structural similarity, while visually compelling, does not survive technical scrutiny.
This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.