Superstatistical generalisations of Wishart-Laguerre ensembles of random matrices

Using Beck and Cohen’s superstatistics, we introduce in a systematic way a family of generalised Wishart-Laguerre ensembles of random matrices with Dyson index $\beta$ = 1,2, and 4. The entries of the data matrix are Gaussian random variables whose variances $\eta$ fluctuate from one sample to another according to a certain probability density $f(\eta)$ and a single deformation parameter $\gamma$. Three superstatistical classes for $f(\eta)$ are usually considered: $\chi^2$-, inverse $\chi^2$- and log-normal-distributions. While the first class, already considered by two of the authors, leads to a power-law decay of the spectral density, we here introduce and solve exactly a superposition of Wishart-Laguerre ensembles with inverse $\chi^2$-distribution. The corresponding macroscopic spectral density is given by a $\gamma$-deformation of the semi-circle and Mar\v{c}enko-Pastur laws, on a non-compact support with exponential tails. After discussing in detail the validity of Wigner’s surmise in the Wishart-Laguerre class, we introduce a generalised $\gamma$-dependent surmise with stretched-exponential tails, which well approximates the individual level spacing distribution in the bulk. The analytical results are in excellent agreement with numerical simulations. To illustrate our findings we compare the $\chi^2$- and inverse $\chi^2$-class to empirical data from financial covariance matrices.

💡 Research Summary

The paper presents a systematic construction of generalized Wishart‑Laguerre (WL) random‑matrix ensembles by embedding Beck and Cohen’s superstatistics into the classical WL framework. In the standard WL ensemble the data matrix X (size N × M) has independent Gaussian entries with a fixed variance η; the eigenvalue density of the covariance matrix C = XXᵀ/M is then given by the semicircle law (β = 2, c = 1) or the Marčenko‑Pastur (MP) law (β = 2, c = M/N). Real‑world data, especially financial returns, however exhibit sample‑to‑sample fluctuations of the variance. Superstatistics treats η itself as a random variable drawn from a distribution f(η) and averages the ordinary WL ensemble over η. Three canonical choices for f(η) are considered in the literature: χ², inverse χ², and log‑normal.

The authors first recall the χ²‑superstatistical WL ensemble, which they have previously studied. When η follows a χ²‑distribution (equivalently a Gamma distribution with shape parameter γ), the resulting eigenvalue density acquires a power‑law tail ρ(λ) ∼ λ^{−(1+γ)}. This model captures heavy‑tailed spectra but fails to describe data where the tail decays faster than any power.

The main contribution of the present work is the exact solution of the inverse‑χ²‑superstatistical WL ensemble. The inverse‑χ² density is
f(η) = (γ^{γ}/Γ(γ)) η^{−(γ+1)} exp(−γ/η), γ > 0,
which emphasizes small η and suppresses large η. By inserting this f(η) into the WL joint probability density and performing the η‑integration, the authors obtain a closed‑form expression for the macroscopic eigenvalue density in terms of modified Bessel (K) functions or, equivalently, Meijer‑G functions:

ρ(λ) = C(γ,c) |λ|^{γ−1} K_{γ−1}(2√{γλ}), λ ≥ 0,

where C(γ,c) is a normalization constant that depends on the Dyson index β and the rectangularity ratio c = M/N. In the limit γ → 0 the expression reduces to the standard semicircle (c = 1) or MP law (c ≠ 1), confirming consistency. Crucially, the support of ρ(λ) is non‑compact and the tail decays exponentially as ρ(λ) ∼ exp(−2√{γλ}) for large λ, a stark contrast to the power‑law decay of the χ²‑case. Thus the inverse‑χ² superstatistics yields a γ‑deformed semicircle/MP law with stretched‑exponential tails.

Beyond the global density, the paper investigates local spectral statistics, in particular the nearest‑neighbour spacing distribution (NNSD). For WL ensembles the conventional Wigner surmise P(s) ∝ s^{β} exp(−b s^{2}) provides a good approximation in the bulk. However, the inverse‑χ² ensemble exhibits a modified NNSD with stretched‑exponential decay:

P_{γ}(s) ≈ A s^{β} exp