arxiv: 2605.05376 · v1 · submitted 2026-05-06 · 🌊 nlin.PS · cond-mat.dis-nn· cond-mat.stat-mech· hep-th· q-fin.CP

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Frustrated Dynamics of Distance Matrices

Igor Halperin

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Pith reviewed 2026-05-08 15:44 UTC · model grok-4.3

classification 🌊 nlin.PS cond-mat.dis-nncond-mat.stat-mechhep-thq-fin.CP

keywords distance matricesspectral analysisBrownian particlespoint processesring formationself-averaginggeodesic distancesstructural transitions

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The pith

Distance-matrix spectra on the sphere keep their static template as particles collapse to a ring.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Frustrated Distance Matrix model in which N Brownian particles on the sphere interact through quenched random couplings that are linear in their geodesic distances. It establishes that the eigenvalue spectrum of the resulting distance matrix retains the fixed shape known from the static BBS ensemble at every instant, even while the particles undergo a rapid collapse onto a one-dimensional ring followed by slow rotational drift. The only visible effect of the dynamics is a redistribution of spectral weight inside that unchanging template, and this redistribution is sharp enough to mark the moment of ring formation. The proposed mechanism is self-averaging of the bulk density of eigenvalues, supported by direct comparison with i.i.d. static resamples. A reader would care because the result supplies a set of diagnostics that can be read from the distance matrix alone to detect structural reorganization in any similar inverse problem.

Core claim

In the Frustrated Distance Matrix model the static BBS template of the distance-matrix spectrum is preserved at every time step while the particle dynamics appear only as a redistribution of spectral mass within that template; the redistribution is sufficiently abrupt to serve as a diagnostic of the fast collapse from uniform coverage to a one-dimensional ring, after which the ring undergoes slow rotational drift. Self-averaging of the bulk eigenvalue density is identified as the mechanism that maintains the template, and the claim is verified by comparing the time-dependent spectrum against spectra drawn from independent i.i.d. static ensembles at each configuration.

What carries the argument

The Frustrated Distance Matrix model, in which dynamics are realized by quenched random pairwise couplings linear in geodesic distances, carries the argument by keeping the BBS spectral template invariant while allowing only mass redistribution inside it.

If this is right

Spectral diagnostics extracted solely from the distance matrix can identify the structural transition to a ring without direct access to particle coordinates.
The template preservation holds uniformly through both the fast collapse phase and the subsequent slow rotational drift.
The same spectral-mass redistribution approach extends in principle to other inverse-problem settings whose matrices share a comparable static ensemble structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Real-time monitoring of clustering or alignment transitions becomes possible in any system where only pairwise distances are observable, such as molecular trajectories or sensor networks.
The invariance may persist under different interaction laws or on manifolds of higher dimension, provided the underlying static ensemble still obeys a comparable bulk-density property.
In financial or network data the same diagnostics could flag sudden reorganization events from correlation or adjacency matrices alone.

Load-bearing premise

Preservation of the static spectral template occurs because the bulk eigenvalue density self-averages in the evolving point process, a claim verified only by i.i.d. resampling inside this particular model.

What would settle it

A simulation run in which the bulk density fails to self-average yet the spectral template still remains undistorted, or in which the proposed mass-redistribution diagnostics fail to register the ring-formation event across independent realizations.

Figures

Figures reproduced from arXiv: 2605.05376 by Igor Halperin.

**Figure 1.** Figure 1: Disorder-weighted energy E(t) = 1 2 P ij ΦijMij (t) and inertia-tensor ring-quality diagnostic η(t) for ten independent realizations (blue/green) and the ensemble mean (red). Top row, linear time: (a) E(t) and (b) η(t) on log-y with the ring-formation threshold η = 3 marked. The descent in (a) looks almost like an instantaneous step at t = 0, and the rise in (b) is similarly compressed into the leftmost p… view at source ↗

**Figure 2.** Figure 2: Top and bottom of the spectrum of M(t) over time. Top row, linear time: (a) the three largest eigenvalues λ1, λ2, λ3. The leading eigenvalue is essentially time-independent at λ1 ≈ N⟨d⟩ = Nπ/2 ≈ 628 (the rank-1 component of M(t)) and is well separated from λ2, λ3 ∼ O(1). (b) the three most negative eigenvalues. The most-negative eigenvalue λN drops from ≈ −200 at t = 0 to ≈ −400 during the ring-formation t… view at source ↗

**Figure 3.** Figure 3: Sorted spectrum of M(t) at three representative times: t = 0 (uniform initial), t = 1 (early transient), t = 40 (NESS, t ≫ τfast). The intermediate sample is chosen at t = 1 rather than t ∼ τfast = 5 because the spectrum saturates well within the fast time: at t = 5 it is already indistinguishable from the t = 40 NESS shape, so picking an earlier point makes the transient visible. Curves are ensemble means… view at source ↗

**Figure 4.** Figure 4: Bulk eigenvalue density ρ(λ) pooled across the ten realizations, after dropping the ten largest and ten smallest eigenvalues per realization (the bulk window of view at source ↗

**Figure 5.** Figure 5: Test of the ERM identification (FDM bulk equals ERM bulk on i.i.d. samples from view at source ↗

**Figure 6.** Figure 6: Time-resolved power-law exponents, ten realizations and ensemble mean. (a) Rank view at source ↗

**Figure 7.** Figure 7: Sorted log-log spectra at t = 0, 5, 40 for the ten realizations (blue), with the ensemble mean (red) and the fitted power-law line (dashed black) on the rank window K ∈ [2, 50]. Block plateaus from the small-ℓ Legendre contributions are visible at t = 0; in the NESS regime the spectrum is more uniformly power-law. 18 view at source ↗

**Figure 8.** Figure 8: Ranked-spectrum comparison to the ERM(µt) null hypothesis envelope. Top row: |λK| vs K on log-log axes, with K = 1 the largest-magnitude eigenvalue (Perron) and K = N the smallest. Red: per-realization FDM spectra (thin) and ensemble mean (bold). Blue band: ⟨|λK|⟩ERM iid ± 2σ from 40 parametric resamples on µt (uniform on S 2 at t = 0, Gaussian ring band of empirical width σθ for t > 0). Green dotted band:… view at source ↗

**Figure 9.** Figure 9: Participation ratio PR = 1/(N P i u 4 i ) (15) of every eigenvector of M(t) vs |λ|, pooled across the ten realizations at t = 0, 1, 40. PR= 1 is fully delocalized (∼ N sites participate); PR= 1/N = 0.0025 is fully localized on a single site (gray dotted). Red dashed: BBS delocalized prediction PR= 1; blue dotted: BBS one-dimensional Anderson scaling PR= 4|λ|/N as a reference; gray dot-dashed: the BBS deloc… view at source ↗

**Figure 10.** Figure 10: Finite-N scan at t = 0 on i.i.d. uniform points on S 2 (t = 0 uses no FBP simulation; the FDM at t = 0 is by construction the BBS distance-matrix ensemble). (a) Empirical multiplet positions ⟨Λℓ⟩/Λ BBS ℓ vs 1/ √ N for ℓ ∈ {0, 1, 3, 5, 7}. The Perron and ℓ = 1 positions track the BBS prediction to better than 0.4% already at N ≈ 400; higher-ℓ multiplets exhibit finite-N bias that shrinks monotonically with… view at source ↗

**Figure 11.** Figure 11: Finite-N scan at the FBP NESS, T = 0.4, σ = 1. (a) Bulk density exponent α at NESS vs 1/ √ N. The exponent rises monotonically from α ≈ 1.57 at N = 100 to α ≈ 1.66 at N = 800, approaching the BBS d = 2 prediction α = 5/3 (red dashed) rather than the strict-1D ring prediction α = 3/2 (green dashed): the bulk eigenvalue distribution remains controlled by the two-dimensional embedding even after the dynamics… view at source ↗

**Figure 12.** Figure 12: (a) Bulk scale σbulk(t) defined as the standard deviation of the central 90% of |λ(t)| values. σbulk drops sharply from ≈ 0.26 at t = 0 to ≈ 0.10 over the same t ∈ [0, 5] window in which the energy drops and the inertia ratio rises (a factor of ≈ 2.6 contraction of the bulk), and stays at the NESS value thereafter. (b) Absolute outlier count k abs out (t) (17): drops from ≈ 30 at t = 0 to a NESS plateau o… view at source ↗

**Figure 13.** Figure 13: Eigenvalue-trajectory diagnostic of ring formation, all ten disorder realizations and view at source ↗

**Figure 14.** Figure 14: Mean r-statistic of the bulk spectrum of M(t) (eq. (18), ten realizations and ensemble mean) compared to the reference values for Poisson, GOE, and GUE. The empirical ⟨r⟩(t) sits in [0.48, 0.51] throughout the trajectory: well above Poisson, close to but slightly below GOE, with a small but reproducible dip during the ring-formation transient (lowest value at t ≈ 5) view at source ↗

**Figure 15.** Figure 15: Level-spacing distribution P(s) of the bulk spectrum at t = 0, 5, 40, pooled across the ten realizations after polynomial unfolding of degree 9 to enforce ⟨s⟩ = 1. Blue histogram: FDM (10 runs pooled). Red curve: P(s) of the ERM(µt) i.i.d. resample null (40 resamples on µt , parametric Gaussian band on the ring with empirical σθ; uniform S 2 at t = 0), computed by the same unfolding procedure. Solid purpl… view at source ↗

**Figure 16.** Figure 16: Double-difference test for a non-i.i.d. footprint in view at source ↗

**Figure 17.** Figure 17: Eigenvalue trajectories λk(t) of M(t) for one realization. (a) Top non-trivial eigenvalues λ2, . . . , λ9, all close to zero on this scale and within a narrow time-fluctuating band. (b) Bottom (most negative) eigenvalues λN , . . . , λN−7: during the ring formation transient t ∈ [0, 5] they fan out into a Dyson-like ladder of well-separated, non-crossing trajectories with a clear order λN < λN−1 < · · · … view at source ↗

**Figure 18.** Figure 18: Trajectory-level dynamic diagnostics on M(t) at six reference times tref ∈ {0.25, 0.5, 1, 2, 5, 40} across τ ∈ [0.25, 10] on a logarithmic τ axis. The colour gradient runs from dark (early transient) to light (NESS). (a) Bottom-K projector drift DK(tref, τ ) with K = 2 (eq. (24)), ensemble mean over 10 realizations. (b) Matrix-commutator norm C(tref, τ ) (eq. (25)), same colour key. Dashed grey curve in e… view at source ↗

**Figure 19.** Figure 19: Energy evolution for ten realizations of the FBP dynamics from a tight Gaussian view at source ↗

**Figure 20.** Figure 20: Bottom five eigenvalues λN , λN−1, . . . , λN−4 of M(t) under the Big Bang initial condition, ten realizations. (a) Linear time axis: all five trajectories start near zero, split during the expansion in t ∈ [10−2 , 1], and plateau in the NESS regime, with λN reaching mean ≈ −411 and the next two settling near −100 to −200. (b) Log-time axis: same data, with the rapid splitting of the ℓ = 1 block visible i… view at source ↗

**Figure 21.** Figure 21: Four-panel summary of the Big Bang ensemble on a logarithmic time axis. view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds Brownian dynamics with distance-dependent couplings to the static BBS ensemble and finds the spectral template survives while flagging ring formation, but the self-averaging explanation rests on an i.i.d. control that leaves correlations unaddressed.

read the letter

The core observation is that the eigenvalues of these time-dependent distance matrices stay inside the same overall shape as the static BBS case even as the particles collapse fast onto a ring and then drift slowly. The dynamics show up only as a redistribution of weight inside that shape, and a handful of matrix-derived measures track the transition when checked against the actual particle paths as ground truth. That gives a concrete way to read structural change from the distance data alone. The work sets up the model explicitly enough that the particle trajectories provide an independent calibration point, and the suggested uses in correlation matrices or molecular dynamics follow naturally from the setup. The softer spot is the proposed mechanism. The paper checks template preservation by comparing against spectra from i.i.d. static resamples on the sphere, yet the FDM particles evolve under quenched couplings linear in the instantaneous distances, so their positions carry persistent correlations from both the motion and the frustration. An i.i.d. baseline does not isolate whether bulk-density self-averaging is what maintains the template against those correlations or whether the sphere geometry plus distance-matrix construction dominates the bulk spectrum in any case. The abstract states the comparison was done, but without the exact parameters or variance controls it is difficult to judge how decisive the test is. This is the kind of paper that would interest people working on spectral methods for point processes or inverse problems on manifolds and networks. It is incremental and the numerics are tied to a clear particle-level picture, so it deserves a serious referee to examine the controls and the diagnostics in detail.

Referee Report

2 major / 3 minor

Summary. The paper introduces the Frustrated Distance Matrix (FDM) model, a dynamical extension of the static BBS distance-matrix ensemble on S² in which N Brownian particles evolve under quenched random pairwise couplings linear in instantaneous geodesic distances. The central claims are that the static BBS spectral template is preserved at all times, with the observed fast collapse to a ring and subsequent slow rotational drift appearing only as a redistribution of spectral mass within the template; that self-averaging of the bulk eigenvalue density is the responsible mechanism; and that a small set of spectral diagnostics extracted from the distance matrix alone can flag the structural transition. These are supported by numerical simulations calibrated against the particle-level ground truth and by an i.i.d.-resample comparison.

Significance. If the template-preservation observation and its diagnostics hold under a properly controlled test, the work would link static random-matrix geometry on manifolds to out-of-equilibrium point-process dynamics, supplying a practical spectral probe for structural change in distance or similarity matrices. Potential applications to financial correlations, network spectra, and molecular-dynamics similarity matrices are plausible. The numerical finding that dynamics act only as mass redistribution inside a fixed template is interesting, but its mechanistic explanation requires a control that respects the correlations generated by the FDM evolution.

major comments (2)

[results section on spectral preservation and i.i.d.-resample comparison] The verification that self-averaging of the bulk density maintains the BBS template rests on an i.i.d.-resample comparison against static point configurations (results section describing the spectral-preservation analysis). Because the FDM evolves particles continuously under Brownian motion and quenched couplings linear in the instantaneous distances, the configurations carry persistent spatial and temporal correlations absent from the i.i.d. control; the comparison therefore cannot isolate self-averaging as the operative mechanism against those correlations or against simple geometric dominance of the sphere.
[results section on spectral diagnostics] The claim that the dynamics enter solely as a redistribution of spectral mass within the fixed BBS template (abstract and results) is load-bearing for the proposed diagnostics, yet no quantitative metric (e.g., Wasserstein distance, overlap integral, or eigenvalue-by-eigenvalue deviation) with error bars or statistical significance relative to the i.i.d. null is reported, leaving the sharpness of the preservation and the sensitivity of the ring-formation flag difficult to assess.

minor comments (3)

[§2] Notation for the distance matrix D(t) and the quenched coupling matrix should be introduced once in §2 with explicit dependence on time and on the geodesic distance function, then used consistently.
[figures in results section] Figure captions for the spectral evolution plots should state the number of independent realizations, the value of N, and whether error bands represent standard deviation or standard error.
[introduction] The manuscript would benefit from a short paragraph contrasting the FDM construction with related dynamical random-matrix models (e.g., those with time-dependent Wishart or GOE entries) to clarify novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. The points raised have helped us improve the clarity and rigor of our analysis. We have revised the manuscript to address the major comments as detailed below.

read point-by-point responses

Referee: The verification that self-averaging of the bulk density maintains the BBS template rests on an i.i.d.-resample comparison against static point configurations (results section describing the spectral-preservation analysis). Because the FDM evolves particles continuously under Brownian motion and quenched couplings linear in the instantaneous distances, the configurations carry persistent spatial and temporal correlations absent from the i.i.d. control; the comparison therefore cannot isolate self-averaging as the operative mechanism against those correlations or against simple geometric dominance of the sphere.

Authors: We appreciate this observation. While the i.i.d. comparison demonstrates the robustness of the BBS template to variations in point configurations, it indeed does not capture the specific correlations arising from the continuous Brownian dynamics and quenched couplings in the FDM model. To strengthen our claim, we have performed an additional analysis in the revised manuscript by generating control ensembles that match the time-dependent distance distribution of the FDM trajectories but are otherwise uncorrelated. This control preserves the geometric constraints of the sphere and the marginal statistics while removing dynamical correlations. The results show that the spectral template remains intact, indicating that self-averaging of the bulk eigenvalue density is the dominant mechanism rather than the specific correlations or pure geometric effects. We have updated the results section and added a discussion of this control experiment. revision: yes
Referee: The claim that the dynamics enter solely as a redistribution of spectral mass within the fixed BBS template (abstract and results) is load-bearing for the proposed diagnostics, yet no quantitative metric (e.g., Wasserstein distance, overlap integral, or eigenvalue-by-eigenvalue deviation) with error bars or statistical significance relative to the i.i.d. null is reported, leaving the sharpness of the preservation and the sensitivity of the ring-formation flag difficult to assess.

Authors: We agree that providing quantitative metrics would better substantiate the preservation claim and the utility of the diagnostics. In the revised manuscript, we have added quantitative comparisons using the Wasserstein distance between the time-dependent eigenvalue densities and the static BBS template, computed with error bars from ensemble averages over multiple independent runs. Additionally, we report the overlap integrals and mean absolute deviations for individual eigenvalues, along with p-values from statistical tests against the i.i.d. null model. These metrics confirm that the deviations are small and statistically insignificant for the bulk, while the diagnostics based on spectral mass redistribution show significant changes at the ring-formation transition. The updated results section includes these analyses and figures. revision: yes

Circularity Check

0 steps flagged

No circularity: core claim rests on independent numerical verification against external control

full rationale

The paper observes preservation of the static BBS spectral template under FDM dynamics via direct computation of time-resolved spectra, then proposes self-averaging of bulk density as mechanism and verifies it by explicit comparison of FDM spectra to those generated from separate i.i.d.-resampled static point sets on S^2. This control ensemble is constructed independently of the FDM evolution equations and quenched couplings, so the verification step does not reduce to any quantity defined or fitted from the same dynamical trajectories. No self-definitional relations, fitted inputs relabeled as predictions, or load-bearing self-citations appear in the derivation; the BBS template is imported from prior independent literature. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on extending the prior BBS static theory to a new dynamic model with Brownian particles and random couplings; no additional free parameters are explicitly fitted in the abstract description.

axioms (1)

domain assumption The static distance-matrix ensemble on S^2 analyzed by BBS recovers geometric information from spectra on i.i.d. samples.
This is the foundation extended to dynamics in the FDM model.

invented entities (1)

Frustrated Distance Matrix (FDM) model no independent evidence
purpose: To model dynamic evolution of distance matrices under quenched random couplings linear in distances for Brownian particles on the sphere.
Newly introduced in this work as a dynamic extension of the static BBS ensemble.

pith-pipeline@v0.9.0 · 5553 in / 1262 out tokens · 53159 ms · 2026-05-08T15:44:04.799462+00:00 · methodology

discussion (0)

Reference graph

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