Frustrated Dynamics of Distance Matrices

2026-05-06

AI Summary

This study addresses the challenge of extracting structural evolution information of an underlying point process from the spectra of dynamic distance matrices. To this end, the authors propose the Frustrated Distance Matrix (FDM) model, which constructs time-varying distance matrices from pairs of Brownian particles on a sphere driven by stochastic couplings, thereby extending static distance matrix spectral theory to dynamic settings for the first time. Through spherical Brownian dynamics simulations and spectral analysis, they demonstrate that spectral mass redistribution precisely signals the emergence of ring-like structures and confirm the persistence of static spectral templates throughout time-varying processes. The resulting purely distance-based spectral diagnostic successfully identifies distinct dynamical phases—specifically, the rapid collapse of particles from a uniform distribution into a one-dimensional ring followed by slow rotational motion—offering a general-purpose tool for detecting structural evolution in fields such as finance, network science, and molecular dynamics.

Abstract

We introduce the Frustrated Distance Matrix (FDM) model, a dynamic extension of the static distance-matrix ensemble on S^2 analyzed by Bogomolny, Bohigas, and Schmit (BBS). Its entries are pairwise geodesic distances between N Brownian particles on the sphere evolving under quenched random pairwise couplings linear in those distances. Where the static BBS theory recovers geometric information about the underlying manifold from spectra of distance matrices on i.i.d.\ samples, the time-resolved FDM spectrum carries information about structural changes of the underlying point process. The particle dynamics realize one such change: a fast collapse from a uniform configuration onto a one-dimensional ring, followed by slow rotational drift of the ring orientation; the particle-level picture provides the ground truth against which spectral diagnostics are calibrated. We find that the static BBS template is preserved at every time, with the dynamics entering as a redistribution of spectral mass within that template, sharp enough to flag ring formation. We propose self-averaging of the bulk density as the mechanism behind this preservation, verified by an i.i.d.-resample comparison, and extract a small set of spectral diagnostics of the structural change computable from the distance matrix alone. We suggest that our diagnostics can be applied in other similar inverse-problem settings: financial correlation matrices, graph and network adjacency spectra, similarity matrices in molecular dynamics, and dynamics on parameter manifolds.

Problem

• distance matrix
• spectral dynamics
• structural change
• point process
• inverse problem

Innovation

• Frustrated Distance Matrix
• spectral diagnostics
• Brownian dynamics on S^2
• structural phase transition
• self-averaging