Superstatistical generalizations of Wishart–Laguerre ensembles of random matrices

Abstract

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 β = 1,2, and 4. The entries of the data matrix are Gaussian random variables whose variances η fluctuate from one sample to another according to a certain probability density f (η) and a single deformation parameter γ. Three superstatistical classes for f (η) are usually considered: χ 2 -, inverse χ 2 -and log-normaldistributions. 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 χ 2 -distribution. The corresponding macroscopic spectral density is given by a γ-deformation of the semicircle and Marčenko-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 γ-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 χ 2 -and inverse χ 2 -class to empirical data from financial covariance matrices.

Joint probability densities of level spacing ratios in random matrices

“…and normalizing the answer to obey (17). It is natural to generalize these well-known arguments by considering not 2 × 2 matrices but 3 × 3 ones.…”

“…Enforcing the normalization conditions (17) leads to the following expression for the 3×3 nearest-neighbor distributions (22)- (24), red). From left to right: GOE, GUE and GSE.…”

Extreme Value Laws for Superstatistics

“…Ultimately all expectation values relevant for the complex system under consideration are averaged over this distribution g(β). Many applications have been described in the past, including modeling the statistics of classical turbulent flow [2,[13][14][15], quantum turbulence [16], space-time granularity [17], stock price changes [8], wind velocity fluctuations [18], sea level fluctuations [19], infection pathways of a virus [20], and much more [5,[21][22][23][24][25][26][27]. Superstatistical systems, when integrated over the fluctuating parameter, are effectively described by more general entropy measures than the Boltzamnn-Gibbs entropy [10,12].…”

Generalized statistical mechanics for superstatistical systems

“…The environment is represented by a suitable parameter entering the stochastic differential equation describing the mesoscopic system. The superstatistics concept can be applied in quite a general way, and a couple of interesting applications for a variety of complex systems have been pointed out recently [17][18][19][20][21][22][23][24][25][26][27][28]. Essential for this approach is the existence of sufficient time-scale separation so that the system has enough time to relax to a local equilibrium state and to stay within it for some time.…”

“…In general, the superstatistical parameter b need not be an inverse temperature but can be an effective parameter in a stochastic differential equation, a volatility in finance or just a local variance parameter extracted from some experimental time series. There are interesting applications in hydrodynamic turbulence [2,19,29,30], defect turbulence [17], cosmic rays [25] and other scattering processes in high-energy physics [31,32], solar flares [18], share price fluctuations [26,27,33,34], random matrix theory [20,21,35], random networks [36], multiplicative-noise stochastic processes [37], wind velocity fluctuations [23,24], hydroclimatic fluctuations [22], the statistics of train departure delays [38] and survival statistics of cancer patients [39]. Maximum entropy principles can be generalized in a suitable way to yield the relevant probability distributions that characterize the various important universality classes in superstatistics [5,[40][41][42][43].…”