Logarithmic Flux Corrections and the De Bruijn–Newman Constant: A Scaling Analogy

Abstract

This essay explores a formal analogy between the finite-$N$ corrections to the particle flux in the one-dimensional Fleming--Viot process and the logarithmic scale of the Riemann zero spacing.

Overview

This essay mines the recent preprint by Brunet and Derrida on the Fleming--Viot process for structural patterns that might resonate with the analytic theory of the Riemann zeta function. The Fleming--Viot process describes $N$ particles diffusing on a semi-infinite line with an absorbing boundary; in the large $N$ limit, the particle density evolves deterministically and exhibits a "pulled" front with a logarithmic cut-off scale $L\sim\ln N$. The flux of absorbed particles approaches a critical value $\kappa_c$ with a leading correction of order $(\ln N)^{-2}$.

We propose a speculative bridge to the de Bruijn--Newman theory of the Riemann $\xi$-function. The de Bruijn--Newman constant $\Lambda$ governs the heat-flow deformation of $\xi$: for times $t\ge\Lambda$, the deformed function has only real zeros. We observe that the $(\ln N)^{-2}$ scaling in the Fleming--Viot process matches the asymptotic order of the squared mean spacing between Riemann zeros at height $T$, which is $\sim(\ln T)^{-2}$. This suggests a formal analogy where the particle number $N$ corresponds to the spectral height $T$, and the cut-off scale $L$ corresponds to the logarithmic density of zeros.

The correspondence is rated as a formal analogy: the asymptotic shapes match, but the underlying axioms differ fundamentally. The Fleming--Viot correction arises from a non-linear boundary-value problem with a discrete cut-off, whereas the de Bruijn--Newman flow is a linear convolution with a Gaussian. The essay details what would have to be true for the dictionary to hold—specifically, that a truncated or regularized version of the $\xi$-function must exhibit a spectral "flux" converging to its limit with the same logarithmic scaling.

We conclude by proposing numerical experiments on truncated heat-flow integrals to test the scaling, and by analyzing the failure modes: the presence of a boundary condition at the origin in the Fleming--Viot process, the non-linearity of the mean-field equation, and the lack of an obvious Euler product or functional equation analogue on the particle side.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.