Why Self-Averaging of Replica Overlaps Resists a Multiplicative Chaos Bridge to the Riemann Hypothesis: An Honest Assessment

Abstract

This essay examines whether the self-averaging of replica overlaps proven in the random-field Edwards-Anderson model (Itoi and Sakamoto, 2026) offers a structural pattern applicable to the Riemann Hypothesis via the lens of critical multiplicative chaos and random multiplicative functions.

Overview

This paper assesses a speculative bridge between two probabilistic domains: the statistical mechanics of short-range spin glasses and the analytic theory of the Riemann zeta function. The source work (Itoi and Sakamoto, arXiv:2606.18752v2) rigorously establishes that in the random-field Edwards-Anderson model, the variance of the replica overlap vanishes in the thermodynamic limit—an extreme concentration property dubbed self-averaging. The suggested angle for this exploration was probabilistic models and multiplicative chaos, where similar concentration and "freezing" transitions are studied.

The essay identifies the replica overlap—a normalized correlation between two independent copies of the spin system—as the key object. In the source paper, this overlap’s variance is bounded by the squared Edwards-Anderson order parameter via Tasaki’s correlation inequality, forcing the variance to zero. The formal abstraction is a concentration inequality for quenched disorder averages in a system with short-range Gaussian interactions.

The candidate analogy maps the vanishing overlap variance to the purported vanishing of variance for the "mass" of a critical multiplicative chaos measure or for logarithmic moments of random Euler products modeling zeta. However, this analogy is rated a failed structural correspondence. Unlike the spin-glass overlap, the logarithm of the zeta function and its random models exhibit inherent, non-vanishing variance (Selberg’s central limit theorem). The disanalogy stems from the difference between short-range lattice systems (where free energy convexity suppresses fluctuations) and log-correlated systems (where long-range dependencies preserve fluctuations).

The paper includes proposed diagnostic code to test the variance behavior in truncated random multiplicative functions, confirming the disanalogy. The conclusion is that the Edwards-Anderson self-averaging result, while profound in statistical mechanics, does not suggest a viable path toward the Riemann Hypothesis.

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