From Simplicial Walks to Dynamical Zeta Functions: A Structural Analogy and Its Breakdown

Abstract

This essay investigates whether the graded stochastic operator introduced by Febbe, Fanelli, and Carletti for random walks across dimensions of a simplicial complex can serve as a finite-dimensional model for the transfer operators used in dynamical approaches to the Riemann Hypothesis.

The Source Structure

The paper Random Walks Across Dimensions: Exploring Simplicial Complexes introduces a stochastic matrix $M$ governing transitions between simplices of different dimensions. Walkers move from nodes to edges to triangles and back, with transition probabilities determined by unsigned incidence matrices $A_k$ and generalized degrees. The operator $M$ has a block-tridiagonal structure and preserves total probability, possessing a leading eigenvalue $1$ with a stationary distribution proportional to the generalized degree sequence.

The Proposed Bridge

We explore a mapping to the transfer operator framework pioneered by Ruelle, wherein the spectrum of a linear operator encodes the zeros of a dynamical zeta function. The graded structure of the simplicial complex (dimensions $0$ through $D$) suggests a symbolic dynamics with $D+1$ symbols, while the stochastic matrix $M$ appears analogous to a discretized Ruelle-Perron-Frobenius operator.

Assessment of Strength

The correspondence is rated a suggestive metaphor. While both objects are positive operators with leading eigenvalues governing asymptotic behavior, the source operator acts on a finite-dimensional space (yielding a rational spectral determinant), whereas Riemann-zeta transfer operators act on infinite-dimensional nuclear spaces. The finite state space imposes a polynomial structure incompatible with the Euler product and functional equation of the Riemann zeta function.

Key Obstructions

• Finite vs. Infinite: The simplicial walk has finitely many states; the Riemann zeta function requires an infinite-dimensional dynamical system.
• Spectrum Location: Eigenvalues of $M$ lie in the unit disk; the Riemann Hypothesis concerns zeros on a critical line $\Re(s)=1/2$.
• Analytic Class: The spectral determinant $\det(I - zM)$ is a polynomial; $\zeta(s)$ has a pole at $s=1$ and essential singularities.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.