Frustrated Dynamics of Distance Matrices

Paper: arXiv 2605.05376

Authors: Igor Halperin

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper leverages advanced concepts from Random Matrix Theory, spectral geometry, and non-equilibrium statistical mechanics to analyze the dynamics of distance matrices. It matches high formal complexity with clear numerical simulations of the FDM model, establishing a bridge between abstract spectral diagnostics and practical inverse-problem applications.

Research Flowchart

  flowchart TD
    A[Research Goal: Relate spectral dynamics to underlying structural changes] --> B[Inputs: N Brownian particles on S^2 & pairwise geodesic distances]
    B --> C[Computational Process: Construct Frustrated Distance Matrix FDM]
    C --> D[Method: Track spectral mass redistribution & static BBS template preservation]
    D --> E[Key Finding: Self-averaging of bulk density flags ring formation]
    E --> F[Outcome: Spectral diagnostics for structural change in inverse problems]

Cited References (34)

1. Frustrated Fields: Statistical Field Theory for Frustrated Brownian Particles on 2D Manifolds (2026) — link
2. Order Out of Noise and Disorder: Fate of the Frustrated Manifold (2026) — link
3. The Dean–Kawasaki equation and stochastic density functional theory (2024) — link
4. A First Course in Random Matrix Theory for Physicists, Engineers and Data Scientists (2023) — link
5. Classical glasses, black holes, and strange quantum liquids (2019) — link
6. Cleaning large correlation matrices: tools from random matrix theory (2016) — link
7. powerlaw: A Python Package for Analysis of Heavy-Tailed Distributions (2013) — link
8. Euclidean random matrices and their applications in physics (2013) — link

0 of 34 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.