Exponential Localization in Sparse Games and Subconvexity for Zeta: A Formal Analogy

Abstract

This essay proposes a speculative formal analogy between the exponential decay of influence in sparse stochastic differential network games (Cirant and Redaelli) and the problem of bounding the growth of the Riemann zeta function on the critical line.

Overview

This essay explores a connection between the analysis of large population stochastic differential games on sparse graphs and the analytic theory of the Riemann zeta function. The source paper establishes that in sparse network games, the optimal strategy of a given player can be approximated by considering only a local neighborhood of nearby players, with an error that decays exponentially in the graph distance. This "unimportance of distant players" is a strong form of spatial localization.

We speculate that this structure might formally analogize to the truncation principles used in the study of the Riemann zeta function ζ(s) on the critical line Re(s) = 1/2. Specifically, the approximate functional equation approximates ζ(1/2 + it) by a finite Dirichlet polynomial, and the Lindelöf Hypothesis posits that the growth of these partial sums is uniformly controlled. The analogy maps "graph distance" to "frequency distance" and the "exponential decay of influence" to the decay of the tail in the approximate functional equation.

The strength of this correspondence is rated as a Formal Analogy: both domains involve bounding a global quantity by a local truncation, but the mechanisms and decay rates differ fundamentally. The essay concludes by identifying the primary failure mode: the Riemann zeta function lacks the sparse, locally interacting graph structure that is essential to the exponential localization proven in the source paper.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.