Kesten-Stigum Thresholds and Critical Multiplicative Chaos: A Dichotomy Analogy for the Riemann Hypothesis

Abstract

This essay mines the depth-dichotomy governing message-passing error on sparse random graphs—specifically the Kesten-Stigum threshold at κ = γ²Δ separating geometric saturation from geometric productivity—for structural patterns that might illuminate the critical behavior of the Riemann zeta function.

Overview

This essay explores a speculative bridge between two threshold phenomena: the Kesten-Stigum dichotomy in deep message passing on sparse graphs, and the critical line in the analytic theory of the Riemann zeta function ζ(s). The source paper (arXiv:2607.16676v1) proves that the classification error E(ℓ) of a depth-ℓ message-passing classifier on a sparse stochastic block model exhibits a sharp transition at the ratio κ = γ²Δ = 1. Below this threshold (κ < 1), deeper layers provide geometrically diminishing returns; above it (κ > 1), depth is geometrically productive.

We propose that this dichotomy structurally echoes the phase transition in random multiplicative functions and critical multiplicative chaos models of ζ(s). In these probabilistic models, the “depth” ℓ corresponds to the truncation level of a Dirichlet series or Euler product, and the “signal-to-noise” ratio κ corresponds to the real part σ = Re(s). The critical line σ = 1/2 emerges as the analogue of the Kesten-Stigum threshold: for σ > 1/2 (subcritical), truncations converge and saturate; for σ < 1/2 (supercritical), the L² norm diverges, mirroring the productive accumulation of error above κ = 1.

Analogy strength: Suggestive Metaphor. While the geometric scaling laws are formally similar (exponential decay versus growth in second moments), the source domain relies on a tree-structured branching process with binary labels, whereas the zeta function lives on the complex plane with a rigid arithmetic structure. The essay details these parallels honestly, marking all new claims as speculative and proposing concrete numerical tests (unexecuted) to discriminate the correspondence.

Key Structural Parallels

• Source: Poisson Galton-Watson tree with branching number Δ, edge signal γ, and depth-ℓ aggregate T_ℓ = Σ γᵏ · (evidence at distance k).
• Zeta Analog: Random Euler product ζ(s,X) = ∏_p (1 − X_p p⁻ˢ)⁻¹ truncated at prime p ≤ exp(ℓ), or multiplicative chaos measure on a tree with weight p⁻ᵒ at level corresponding to log p.
• Threshold: κ = γ²Δ = 1 separates L²-boundedness (convergence) from unboundedness; σ = 1/2 separates convergence from divergence of the random model’s second moment.
• Observable: Misclassification error E(ℓ) in the source; truncation error or partition function variance in the chaos model.

Obstructions

The analogy fails to be a formal isomorphism because the integers possess multiplicative cycles (pq = qp) that violate the tree independence assumption, and because the Riemann Hypothesis concerns exact zero locations rather than second-moment phase transitions. The essay concludes by cataloging these mismatches.

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