Wigner Surmise-Inspired Analytical Calculation

• Wigner surmise-inspired calculations are analytical methods that adapt simple closed-form distributions from low-dimensional matrices to describe complex quantum spectra.
• The technique uses surmise formulas to compute weighted averages of level spacing ratios, providing parameter-free benchmarks for quantum ergodicity across different Dyson classes.
• Combining analytical simplicity with numerical precision via Painlevé-V ODEs, the method effectively distinguishes chaotic (ergodic) regimes from Poissonian (non-ergodic) behavior in many-body systems.

A Wigner surmise-inspired analytical calculation is a class of techniques within random matrix theory (RMT) and quantum spectral statistics where simple, closed-form distributions—originally motivated by the exact results for small-dimensional (e.g., 2×2 or 3×3) Gaussian random matrices—are adapted to estimate statistical observables in high-dimensional, physically relevant systems. These surmises, despite their mathematical simplicity, capture essential signatures of level repulsion and quantum ergodic behavior in many-body systems, with particular applications to the analysis of level spacing ratios and related probes of quantum chaos.

1. Wigner Surmise-Inspired Formulas and Level Spacing Statistics

The Wigner surmise is the exact expression for the nearest-neighbor level spacing distribution P(s) for 2×2 Gaussian random matrices of Dyson index β (β = 1, 2, 4 for GOE, GUE, GSE), generalized as

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

where A and B ensure normalization and unit mean spacing. This framework has been adapted ("Wigner surmise-inspired") to ratios of consecutive level spacings, $r_n = s_{n+1}/s_n$, where $s_n$ is the $n$th spacing. The surmise for the distribution of ratios for 3×3 matrices, for instance, is

$$P_{\mathrm{W},\beta}(r) = \frac{1}{Z} \frac{(r + r^2)^{\beta}}{(1 + r + r^2)^{1 + \frac{3}{2}\beta}},$$

with Z a normalization constant. Analytical observable averages can then be computed, such as the average of a logarithmic variable $q = \ln(1 + r)$, which is invariant under rescaling of spacings. Such closed-form, Wigner surmise-inspired analytical results provide parameter-free predictions to compare with statistics from large random matrices and many-body quantum spectra (Buijsman, 19 Aug 2025).

2. Analytical Strategies for Weighted Ratio Averages

The weighted ratio average (editor's term: ⟨q⟩ probe) is defined as

$$\langle q \rangle = \int_0^{\infty} \ln(1 + r) P(r) dr,$$

where $P(r)$ is the distribution of the ratio of consecutive level spacings. In addition to direct integration of the Wigner surmise-inspired P(r), the average can be expressed using connections between the level spacing distributions of different symmetry classes:

$$\langle q \rangle = \int_0^\infty \left[ \frac{1}{2} P_4(s/2) - P_1(s) \right] \ln(s) ds,$$

where $P_{1,4}$ denote the spacing distributions for the orthogonal and symplectic ensembles, respectively. Inserting the surmise-based P(s), this leads to analytical estimates such as ⟨q⟩ ≈ 0.8041 and, using the surmise-based ratio distribution, ⟨q⟩ ≈ 0.8069 for β = 1. This difference is less than 1%, demonstrating the efficacy of the surmise approach in producing reference values for the infinite-dimensional limit.

3. Numerical Determination via Painlevé-V Differential Equation

To obtain numerically exact statistics for infinite-dimensional random matrices, the gap probabilities $E_{\beta}(s)$ and level spacing distributions $P_{\beta}(s)$ are computed by solving a system of coupled Painlevé-V ordinary differential equations with appropriate boundary conditions:

• For β = 2:

$$E_2(s) = \exp\left\{ \int_0^{\pi s} \frac{\sigma(t)}{t} dt \right\},$$

$$P_2(s) = \frac{1}{s^2} [\pi s \sigma'(\pi s) - \sigma(\pi s) + (\sigma(\pi s))^2] E_2(s),$$

where $\sigma(t)$ is obtained from the PV ODE system with initial conditions ensuring analyticity at $t \to 0^+$. For β = 1 or 4, analogous expressions are used, involving inter-relations between $E_1$, $E_2$, and $E_4$ due to underlying mathematical structure. High-precision numerical quadrature, with carefully chosen starting points (e.g., $t_0 = 10^{-14}$), is necessary to avoid degeneracies and obtain accurate values for observables like ⟨q⟩.

4. Quantum Ergodicity, Symmetry Classes, and Universal ⟨q⟩ Values

The weighted ratio average ⟨q⟩ serves as a quantitative measure for quantum ergodicity. In chaotic (ergodic) quantum systems, spectral statistics for large matrices are universal and depend only on Dyson symmetry class β:

• For β = 1 (orthogonal): ⟨q⟩ ≈ 0.81007 (numerical, infinite-dim limit),
• For β = 2 (unitary): ⟨q⟩ ≈ 0.7624 (surmise-inspired),
• For β = 4 (symplectic): ⟨q⟩ ≈ 0.7314 (surmise-inspired). These class-dependent reference values reflect the universal nature of level repulsion and are robust against microscopic details of the Hamiltonian, thus providing reliable benchmarks for quantum chaos and ergodicity in a variety of settings (Buijsman, 19 Aug 2025).

5. Contrast with Poissonian (Non-Ergodic) Level Statistics

In the absence of spectral correlations (integrable or non-ergodic systems), the level spacings are Poisson-distributed, $P(s) = e^{-s}$, leading to a ratio distribution $P_P(r) = 1/(1 + r)^2$. The weighted average is

$\langle q \rangle_P = 1,$

which is markedly different from the Wigner–Dyson classes. This separation (⟨q⟩_Poisson = 1 versus ⟨q⟩_GOE ≈ 0.81) provides a sensitive and practical diagnostic to distinguish between Poissonian (integrable/non-ergodic) and Wigner–Dyson (ergodic/chaotic) regimes, especially in large interacting or disordered many-body systems.

6. Reference Values and Practical Applications

The computation of ⟨q⟩ in the infinite-dimensional limit using both Wigner surmise-inspired forms and PV ODE numerics establishes universal reference values: | Symmetry Class (β) | Analytical/Surmise ⟨q⟩ | Numerical ⟨q⟩ | |--------------------|------------------------|--------------------| | 1 (orthogonal) | ≈ 0.8069 | ≈ 0.81007 | | 2 (unitary) | ≈ 0.7624 | (cf. previous work)| | 4 (symplectic) | ≈ 0.7314 | (cf. previous work)| | ∞ (Poisson) | 1.0 | 1.0 |

These values can be used as benchmarks for evaluating ergodicity and assessing proximity to chaotic or non-chaotic regimes in numerical or experimental studies. Since the calculation does not require spectral unfolding, the method bypasses common numerical pitfalls and is suitable for direct application to raw quantum spectra data. Furthermore, the surmise-based approach provides qualitatively accurate and interpretable formulas, while the Painlevé-V based numerics supply precision benchmarks.

7. Broader Impact and Extensions

The use of Wigner surmise-inspired analytical calculations, complemented by high-precision Painlevé-V ODE solutions, demonstrates a powerful method for probing universality in random matrices and quantum spectra. The methodology is widely applicable for benchmarking quantum ergodicity, diagnosing the emergence of chaos, and exploring crossovers between integrable and chaotic regimes. It is also well positioned for extension to more exotic spectral statistics, including edge distributions and generalized ensembles (e.g., Lévy matrices), and could inform the theoretical understanding of spectral statistics in disordered, many-body, or open quantum systems.

In summary, Wigner surmise-inspired analytical calculations, as exemplified in the computation of the average weighted ratio of consecutive level spacings, provide both intuitive and precise tools for quantifying quantum ergodicity and benchmarking universal spectral statistics in complex quantum systems (Buijsman, 19 Aug 2025).