Weighted Ratio of Level Spacings

• Weighted ratio of consecutive level spacings is an invariant statistical measure that captures local correlations and distinguishes ergodic from Poissonian distributions.
• The approach utilizes advanced numerical Painlevé-based methods alongside analytical Wigner surmise approximations to accurately determine average spectral measures.
• Its diagnostic capability aids in analyzing quantum chaos and ergodicity in diverse systems, providing precise benchmarks for complex and experimental quantum models.

The weighted ratio of consecutive level spacings is a statistical measure central to modern spectral analysis in quantum chaos, random matrix theory, and complex quantum systems. Unlike traditional level spacing statistics, which can be hampered by the requirement to unfold the spectrum (i.e., rescale to account for nonuniform mean level density), weighted ratios—such as the logarithmically weighted ratio qₙ = ln(1 + rₙ) = ln(sₙ + sₙ₊₁) – ln(sₙ), with rₙ = sₙ₊₁/sₙ—are invariant under monotonic transformations of the spectrum and directly probe the local correlations and ergodicity of the underlying system (Buijsman, 19 Aug 2025). The robust nature of weighted ratios makes them indispensable as quantitative diagnostics of quantum ergodicity, especially in the thermodynamic or bulk limit of large random matrices.

1. Definition and Theoretical Motivation

Let {Eₙ} be an ordered set of energy levels, and define the nearest-neighbor spacings as sₙ = Eₙ₊₁ – Eₙ. The simplest unweighted ratio is rₙ = sₙ₊₁ / sₙ. In the "weighted" ratio statistic considered here, the primary observable is

$qₙ = \ln(1 + rₙ) = \ln(sₙ + sₙ₊₁) - \ln(sₙ)$

This transformation is advantageous because qₙ is insensitive to rescalings of the spectrum and provides a one-to-one mapping between the joint distribution of two consecutive spacings and the ratio, while strongly suppressing the impact of fluctuations in the global mean spacing. The average ⟨q⟩ over the spectrum thus constitutes a fingerprint for distinguishing ergodic (Wigner–Dyson–like) statistics from Poissonian (uncorrelated) statistics.

2. Painlevé-Based Numerical Determination for Infinite-Dimensional Environments

For ensembles with time-reversal symmetry (Gaussian Orthogonal Ensemble, GOE), the infinite-dimensional limit is approached using integrable systems methods—specifically, via the solution of coupled first-order ordinary differential equations of Painlevé type for the gap probabilities. The system takes the schematic form

$$\frac{d}{d t} \begin{pmatrix} σ(t) \ σ'(t) \end{pmatrix} = \begin{pmatrix} σ'(t) \ - \frac{2}{t} \sqrt{[σ(t) - t σ'(t)] [t σ'(t) - σ(t) + (σ'(t))^2]} \end{pmatrix}$$

with initial conditions set in the small-t regime, such as

$$\lim_{t \to 0^+} σ(t) = -\frac{t}{\pi} - \left( \frac{t}{\pi} \right)^2$$

The gap probability E₁(s) for the orthogonal ensemble is built from σ(t), and the level spacing distribution P₁(s) is computed as the second derivative of E₁(s). This leads ultimately to the determination of ⟨q⟩ via either:

$⟨q⟩ = \int_0^\infty \ln(1 + r) P_1(r) dr$

or, equivalently,

$$⟨q⟩ = \int_0^\infty \Big[ \frac{1}{2} P_4(s/2) - P_1(s) \Big] \ln s \, ds$$

where P₄(s) is the spacing distribution for the symplectic ensemble (related by symmetry transformations to P₁(s)). This approach yields the highly accurate reference value

$⟨q⟩_{\mathrm{GOE}} \approx 0.8100699350$

for infinite-dimensional orthogonal random matrices (Buijsman, 19 Aug 2025).

3. Analytical (Wigner Surmise) Approximations

Inspired by the classical Wigner surmise, one can estimate ⟨q⟩ analytically for each universality class by using the closed-form expressions for the nearest-neighbor spacing distributions for 2×2 matrices,

$P_{W,\beta}(s) = A s^\beta \exp(-B s^2)$

with β = 1 (GOE), 2 (GUE), or 4 (GSE). Substituting into the q integral, two routes are available:

• Via the spacing distribution:

$$⟨q⟩_{s} = \frac{4}{3} + \ln\left(\frac{3\pi}{16}\right) \approx 0.8041$$

• Via the joint spacing distribution and explicit r-integration:

$⟨q⟩_{r} = \frac{3}{2} - \ln(2) \approx 0.8069$

Both values are within 1% of the numerically exact result, confirming the qualitative predictive power of the surmise even in the infinite-matrix limit.

4. Contrast with Poisson and Diagnostic Capability

For Poissonian (uncorrelated) statistics, the distribution of spacings is exponential, and a straightforward calculation yields the mean weighted ratio

$⟨q⟩_{\mathrm{Poisson}} = 1$

The substantial difference between ⟨q⟩ in the ergodic (GOE/WD) case and the Poisson case forms the quantitative basis for using ⟨q⟩ as a probe of quantum ergodicity. In chaotic systems, correlations and level repulsion reduce ⟨q⟩ below unity, whereas in the uncorrelated regime it remains at its maximum value.

5. Applications, Implications, and Universality

This construction provides a robust, unfolding-invariant measure of spectral correlations. It is directly applicable to the spectral analysis of quantum many-body systems, complex nuclei, condensed matter models, and experimental platforms seeking to probe the onset of chaos or many-body localization. Notably:

• ⟨q⟩ serves as a precise quantitative benchmark for the ergodic regime in quantum chaos.
• The methodology is extendable to other symmetry classes (unitary, symplectic) and to spectral edges, offering the potential to reveal universal and non-universal behavior.

The stark difference between the ergodic and Poissonian values of the average weighted ratio is a sensitive diagnostic, especially suited to finite data sets and real-world scenarios where unfolding is ambiguous or impractical.

6. Future Directions and Research Outlook

Open avenues for further research include:

• Exact calculation of ⟨q⟩ for the unitary and symplectic universality classes in the infinite-matrix limit.
• Fredholm determinant methods for even higher-precision benchmarks.
• Analysis of edge effects and their influence on weighted ratios, relevant for exploring non-standard universality in systems with unusual symmetries or boundary conditions.
• Applications to experimental quantum systems with tunable disorder or interactions, where fine distinctions between ergodic and nonergodic behavior are of central interest (Buijsman, 19 Aug 2025).

Table: Reference Values for ⟨q⟩

Ensemble	Average Weighted Ratio ⟨q⟩	Description
GOE (infinite-N)	≈ 0.81007	Ergodic (orthogonal)
GUE (Wigner surmise)	≈ 0.8347	Ergodic (unitary, approximate)
Poisson	1.0	Uncorrelated (integrable)

Summary

The logarithmically weighted ratio q = ln(1 + r) provides a universal, unfolding-free probe for quantum ergodicity, with precise reference values computable in the infinite-dimensional random matrix limit by advanced numerical (Painlevé) and analytical (Wigner surmise) techniques. The marked discrepancy between these values for ergodic (Wigner–Dyson) versus Poissonian statistics makes ⟨q⟩ an essential tool in diagnosing complex spectral correlations and the onset of quantum chaos (Buijsman, 19 Aug 2025).