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Partial openings in a chaotic quantum map produce a smooth crossover from quasi-1D to Ginibre-like resonance statistics, controlled jointly by channel number and reflectivity.

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2026-07-10 16:09 UTC pith:JKQOEKOM

load-bearing objection Careful numerics map a smooth M-and-ρ-controlled crossover of complex spacing ratios from quasi-1D to Ginibre-like in the partially open asymmetric baker; the continuous ρ knob is the practical takeaway. the 2 major comments →

arxiv 2607.07878 v1 pith:JKQOEKOM submitted 2026-07-08 quant-ph cond-mat.stat-mechnlin.CD

Complex spacing ratio statistics in the partially open asymmetric quantum baker map

Leonardo Ermann , Pablo Sesin , Alejandro M. F. Rivas , Pablo D. Bergamasco , Gabriel G. Carlo This is my paper
classification quant-ph cond-mat.stat-mechnlin.CD
keywords complex spacing ratioopen quantum mapsasymmetric baker mappartial openingsPTCUEGinibre statisticsquantum chaosresonance spectra
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies how the complex resonances of a fully chaotic quantum map change when the system is only partly open. Using the asymmetric baker map with three different opening shapes and a tunable amplitude reflectivity, the authors track the joint statistics of the complex spacing ratio, a local measure of nearest-neighbor geometry that needs no unfolding. They show that the spectrum evolves continuously from a quasi-one-dimensional ring-like arrangement (eigenvalues hugging the unit circle, phase peaks at 0 and ±π) into a two-dimensional Ginibre-like cloud with fully developed level repulsion. Both the number of open channels and the reflectivity act as continuous control knobs; the random-matrix model of partially truncated circular unitary matrices captures the large-opening limit for all three geometries. The result matters because real optical cavities, microwave billiards and quantum dots have finite wall reflectivity, so the same continuous crossover should appear in laboratory spectra once the openings and reflectivities can be tuned.

Core claim

In the partially open asymmetric quantum baker map the joint distribution of the complex spacing ratio z undergoes a smooth, continuous crossover from a quasi-1D regime (eigenvalues clustered near the unit circle, P(θ_z) peaked at 0 and ±π, power-law exponent β near zero) to a two-dimensional Ginibre-like regime (nearly uniform phase distribution, β o 2.5). The crossover is controlled jointly by the number of open channels M and the amplitude reflectivity ρ; all three opening geometries converge to the partially truncated circular unitary ensemble at large M, and no abrupt transition is observed.

What carries the argument

The complex spacing ratio z = (λ_NN − λ)/(λ_NNN − λ), whose joint modulus-phase distribution P(|z|, θ_z) serves as an unfolding-free probe of two-dimensional spectral correlations; it is benchmarked against the partially truncated circular unitary ensemble (PTCUE).

Load-bearing premise

The hand-chosen cutoff that discards every resonance with modulus less than 0.1 leaves the spacing-ratio statistics of the remaining long-lived states unbiased and representative of the physically relevant spectrum.

What would settle it

Compute the complex-spacing-ratio distributions for the same baker map while systematically varying the modulus cutoff (e.g., 0.05, 0.1, 0.2) at fixed large M and ρ = 0; if the extracted β and P(θ_z) change qualitatively with the cutoff, the claimed crossover is an artifact of the filter.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Cavity or billiard experiments can drive the same 1D-to-2D crossover by tuning wall reflectivity even at fixed opening size.
  • Spectral diagnostics that rely on complex spacing ratios remain reliable across partial openings once M and ρ place the system past the quasi-1D regime.
  • Localized openings require larger M (or smaller ρ) than random or uniform openings to reach PTCUE statistics, matching their slower classical escape.
  • The absence of an abrupt transition implies that effective analytic interpolations between ring-like and disk-like ensembles should exist for the PTCUE family.

Where Pith is reading between the lines

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

  • The same continuous crossover should appear in any fully chaotic open map once reflection symmetry and time-reversal are broken, independent of the particular baker dynamics.
  • Finite-N corrections to the fractal Weyl law may be extractable from the rate at which β saturates as a function of M/N and ρ.
  • If an experimental microcavity can independently control both aperture size and wall reflectivity, a single device could map the entire quasi-1D-to-Ginibre path by recording successive resonance clouds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies complex eigenvalue statistics of the asymmetric quantum baker map (discontinuity at q=2/3) subject to partial projective openings of three geometries (localized, random, uniform), controlled by the number of open channels M and a continuous amplitude reflectivity ho ∈ [0,1]. Using the complex spacing ratio z (nearest-to-next-nearest neighbor ratio in the complex plane) and its joint distribution P(|z|, heta_z), and benchmarking against the partially truncated circular unitary ensemble (PTCUE), the authors report a smooth crossover from a quasi-1D regime (eigenvalues near the unit circle, P( heta_z) peaked at 0 and ±π, power-law exponent eta near 0) to a 2D Ginibre-like regime (nearly uniform P( heta_z), eta o ~2.5). Both M and ho jointly control the crossover; all three geometries converge to PTCUE at large M, with largest deviations for the localized opening at small M; no abrupt transition is observed.

Significance. If the reported smooth (M, ho)-controlled crossover is robust, the work supplies a clean, symmetry-free numerical laboratory for resonance statistics of partially open chaotic systems and strengthens the case that complex spacing ratios diagnose the quasi-1D to Ginibre transition without unfolding. The continuous ho knob is experimentally relevant for microcavities and microwave billiards with finite reflectivity, and the systematic comparison of three opening geometries against PTCUE is a useful addition to the open-quantum-chaos literature. Strengths include the use of a fully chaotic asymmetric map free of reflection degeneracies, averages over many realizations, and the joint 2D z-distribution (rather than marginals alone). The results are primarily numerical; no analytic formula for the PTCUE z-distribution is derived, but the falsifiable claim of smooth universal crossover is clearly stated and illustrated.

major comments (2)
  1. [Sec. II.B] Sec. II.B and the filtering statement after Eq. (9): the ad-hoc cut |λ| ≥ 0.1 is used for all subsequent spacing-ratio statistics. For ho = 0 and large M the cut removes most eigenvalues (explicitly noted as producing noise in Fig. 3), so the reported eta(M) and P( heta_z) in that corner of parameter space rest on a severely depleted sample. A short sensitivity check (e.g., thresholds 0.05 and 0.2, or a soft radial weight) should be added to confirm that the smooth crossover shape itself is not an artifact of the hard cut; without it the claim that the filter leaves the long-lived statistics unbiased remains an untested assumption.
  2. [Sec. IV.B / Fig. 4] Fig. 4 and the accompanying text: the power-law exponent eta is extracted by linear regression of P(|z|) ~ |z|^eta only on |z| ∈ [0, 0.75]. The upper edge of the window is arbitrary and the GinUE analytic form already deviates from pure power-law there; reporting the fit range dependence (or using a maximum-likelihood estimator against the known GinUE density) would make the claimed saturation at eta ≈ 2.5 more convincing, especially for the high- ho curves that remain far from saturation until M ~ 1000.
minor comments (5)
  1. [Abstract / Sec. VI] Abstract and Sec. VI: the phrase "This crossover which suggests a universal behavior" is missing a comma and is slightly overstated; the data show convergence among three openings and PTCUE for this map, not a proof of universality across all open chaotic systems.
  2. [Fig. 5] Fig. 5 caption: the heat-maps are normalized to the maximum of each row rather than to unit integral; this choice should be stated more prominently in the main text so that readers do not misinterpret absolute densities.
  3. [Sec. II.A.1] Eq. (5) and surrounding text: the Kantz–Grassberger estimate uses h_KS of the closed map as a proxy for the Lyapunov exponent on the repeller; a brief remark on the expected error for large openings (M/N ~ 0.2) would help.
  4. [Sec. I / Sec. VI] References: the recent Signor et al. work on the leaky standard map (arXiv:2512.09038) is cited; a short explicit comparison of the AI† versus CUE symmetry classes would clarify why the baker map is free of the short-range anomalies reported there.
  5. [Sec. VI] Typographical: "conjeture" → "conjecture" (last paragraph of Sec. VI); "St a tistics" spacing artifacts in section headings; occasional missing spaces after periods.

Circularity Check

0 steps flagged

No significant circularity: the quasi-1D to Ginibre-like crossover is measured directly from eigenvalues of the partially open baker map and compared to independently generated PTCUE matrices.

full rationale

The paper constructs the asymmetric baker propagator U, multiplies selected columns by the free parameter ρ to obtain U^(ρ), extracts its complex eigenvalues (after an explicit |λ|≥0.1 filter), and computes the complex spacing ratio z from those eigenvalues. The same construction is applied to Haar-random CUE matrices to produce the PTCUE benchmark. The reported observables—β of P(|z|), P(θ_z), and Bhattacharyya overlaps—are therefore direct numerical measurements, not quantities forced by a fitted ansatz or by a self-citation that already contains the result. Self-citations supply the classical map, its quantization, and earlier short-orbit studies of partial openings; none of them pre-supplies the joint (M,ρ) dependence of the z-statistics that constitutes the central claim. The |λ|≥0.1 filter is an ad-hoc choice that reduces statistics for ρ=0 and large M, but it is not used to define or predict the crossover itself. Consequently the derivation chain is self-contained against an external RMT benchmark and exhibits no circular reduction.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 1 invented entities

The work is a numerical exploration of a well-defined quantum map against an RMT benchmark. Free parameters are the usual numerical cut-offs and fit windows; axioms are standard quantum-chaos and RMT statements plus the modeling choice that partial column scaling by ρ correctly represents partial reflectivity. No new physical entities are postulated beyond the PTCUE construction, which is a direct, transparent extension of the known TCUE.

free parameters (5)
  • eigenvalue modulus filter threshold = 0.1
    Only eigenvalues with |λ| ≥ 0.1 are retained for spacing-ratio statistics (Sec. II.B); the value is chosen by hand and becomes severe for fully open large-M cases.
  • power-law fit window for β = [0, 0.75]
    β is extracted by linear regression of P(|z|) ~ |z|^β on the interval |z| ∈ [0, 0.75] (Fig. 4 caption); the upper cut-off is arbitrary.
  • Hilbert-space dimension N = 5001
    N = 5001 = 3 imes 1667 is fixed for all production runs to guarantee integer block sizes; robustness is only asserted for N = 10002.
  • number of opening realizations = 20 / 50
    Averages over 20 (baker) / 50 (PTCUE) independent channel configurations; the numbers are conventional but affect reported smoothness.
  • Bhattacharyya histogram grid = 80 imes80
    2-D distributions are discretized on an 80 imes 80 grid before computing the overlap S (Sec. V).
axioms (5)
  • domain assumption Fully chaotic open quantum maps obey Ginibre (or truncated-CUE) spectral statistics in the appropriate limit (GHS conjecture).
    Invoked throughout as the target 2-D regime; cited from Grobe-Haake-Sommers and subsequent open-map literature.
  • domain assumption The closed asymmetric baker map has CUE eigenvalue statistics.
    Used to justify PTCUE as the natural partially-open counterpart (Sec. III.A); rests on earlier numerical and semiclassical work on the map.
  • domain assumption Multiplying selected columns of the unitary propagator by a real factor ρ ∈ [0,1] correctly models partial amplitude reflectivity.
    Central modeling step (Eqs. 8–9); taken from earlier partially-open tribaker studies.
  • domain assumption The complex spacing ratio z performs a local unfolding that renders P(z) independent of the global density of states.
    Justifies the use of the joint (|z|, heta_z) distribution without further unfolding (Sec. III.B); taken from Sá et al. 2020.
  • domain assumption Kantz–Grassberger relation d_I = 2 - γ / h_KS estimates the fractal dimension of the classical repeller for moderate openings.
    Used to interpret classical survival curves (Eq. 5, Fig. 1); standard approximation for small-to-moderate openings.
invented entities (1)
  • partially truncated circular unitary ensemble (PTCUE) independent evidence
    purpose: Random-matrix benchmark obtained by scaling M columns of a CUE matrix by ρ; used as the universal target for all three opening geometries.
    Direct, transparent extension of the known TCUE; no new physical degree of freedom is introduced, only a convenient name for the ensemble already implied by the map construction.

pith-pipeline@v1.1.0-grok45 · 19766 in / 4016 out tokens · 46199 ms · 2026-07-10T16:09:12.346695+00:00 · methodology

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read the original abstract

We study the complex eigenvalue statistics of the asymmetric quantum baker map with partial projective openings. The classical asymmetric baker map, with its discontinuity at $q=2/3$, is fully chaotic, has no reflection symmetry, and provides a clean setting with tunable escape rate and fractal repeller dimension. We consider three distinct opening geometries in position space: localized (contiguous channels), random, and uniform (equispaced channels), all controlled by a tunable amplitude reflectivity parameter $\rho$ that interpolates between the fully open ($\rho=0$) and the closed ($\rho=1$) limits. We use the partially truncated circular unitary ensemble (PTCUE) as the random matrix theory benchmark. The main focus is on the joint distribution of the complex spacing ratio $z$, defined as the ratio of the distances from an eigenvalue to its nearest and next-nearest neighbors in the complex plane. We find a smooth crossover from a quasi-1D spectral regime, where eigenvalues cluster near the unit circle and the phase distribution of $z$ is peaked, to a two-dimensional Ginibre-like regime, where the distribution becomes nearly uniform and level repulsion is fully developed. Both the number of open channels $M$ and the reflectivity $\rho$ modulate this crossover, and $\rho$ provides an additional continuous control even at fixed opening size. All three opening models converge to PTCUE statistics at large $M$, while differences are most pronounced for the localized model at small $M$. No evidence of an abrupt transition is found. This crossover which suggests a universal behavior, has deep consequences for open quantum and wave-chaotic experiments.

Figures

Figures reproduced from arXiv: 2607.07878 by Alejandro M. F. Rivas, Gabriel G. Carlo, Leonardo Ermann, Pablo D. Bergamasco, Pablo Sesin.

Figure 1
Figure 1. Figure 1: FIG. 1. Representation of the classical transformation, quan [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Spectral properties and complex spacing ratio distributions for [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Statistical properties of eigenvalues with [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the power-law exponent β, extracted from the relation P(|z|) ∼ |z| β , as a function of the num￾ber of open channels M on a logarithmic scale. The re￾sults are presented for reflectivities ρ = 0, 0.5, and 0.9 (panels a, b, and c, respectively), comparing the three baker map models with the PTCUE. For the fully open case (ρ = 0, panel a), the exponent β at the smallest [PITH_FULL_IMAGE:figures/full_f… view at source ↗
Figure 5
Figure 5. Figure 5: illustrates the evolution of the phase distri￾bution P(θz) as a function of the number of open chan￾nels M. Panels (a) through (d) display P(θz) (horizon￾tal axis) versus M (vertical axis, logarithmic scale) as heatmaps. Panels (a) and (b) show the distributions for the random baker map model and the PTCUE, respec￾tively, at ρ = 0. The other two deterministic models exhibit behavior very similar to the ran… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Statistical similarity and convergence of complex [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Statistical similarity and convergence of complex [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗

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