
TL;DR
This paper introduces the Frustrated Distance Matrix model, a dynamic spectral analysis tool for understanding structural changes in point processes, demonstrated through particle collapse and rotation on a sphere.
Contribution
It extends static distance matrix theory to a dynamic setting, providing spectral diagnostics for structural changes in evolving point configurations.
Findings
Spectral diagnostics can detect ring formation in particle dynamics.
Static spectral templates are preserved during dynamic evolution.
Self-averaging explains the spectral template preservation.
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…
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