Random Matrix Theory (RMT) and Poisson Universality

Overview

A spectrum of energy levels can be classified by the statistics of its nearest-neighbour level spacings and largest-eigenvalue extremes. For a wide class of complex quantum systems –- chaotic many-body Hamiltonians, heavy nuclei, disordered conductors –- the local statistics are universal and depend only on the underlying global symmetry through the Dyson index $\beta \in \{1, 2, 4\}$ (Wigner-Dyson three-fold way):

An integrable (or many-body localised) system, in contrast, has a spectrum that is asymptotically uncorrelated and produces Poisson level statistics.

QAtlas exposes both classes through the parametric Universality{C} API (C = :RMT or :Poisson).

Wigner Surmise –- Closed Form

The Wigner surmise is the exact $N = 2$ Gaussian-ensemble spacing distribution; it is also a celebrated, accurate approximation to the $N \to \infty$ bulk spacing distribution.

\[P_1(s) = \frac{\pi s}{2} \exp\!\left(-\frac{\pi s^2}{4}\right) \qquad \text{(GOE)}\]

\[P_2(s) = \frac{32 s^2}{\pi^2} \exp\!\left(-\frac{4 s^2}{\pi}\right) \qquad \text{(GUE)}\]

\[P_4(s) = \frac{2^{18} s^4}{3^6 \pi^3} \exp\!\left(-\frac{64 s^2}{9\pi}\right) \qquad \text{(GSE)}\]

All three are normalised so that $\int_0^\infty P_\beta(s)\,ds = 1$ and $\int_0^\infty s\,P_\beta(s)\,ds = 1$ (mean spacing $= 1$). Small-$s$ behaviour $P_\beta(s) \sim s^\beta$ is the level repulsion characteristic of each ensemble.

The Poisson counterpart is

\[P_{\text{Poisson}}(s) = e^{-s},\]

with no level repulsion and the same mean spacing.

Tracy-Widom $F_\beta$ –- Largest-Eigenvalue Distribution

In the limit $N \to \infty$, the largest eigenvalue of a Gaussian $\beta$-ensemble matrix obeys

\[\mathbb{P}\!\left[\lambda_{\max} \le \lambda_c + \sigma N^{-2/3} x\right] \to F_\beta(x),\]

where $F_\beta$ is the Tracy-Widom distribution. Closed forms require the Painleve II transcendent $q(x)$ via

\[F_2(x) = \exp\!\left[-\int_x^\infty (s - x)\, q(s)^2\,ds\right],\]

with $F_1$ and $F_4$ given by analogous Pfaffian formulas (Tracy-Widom 1996).

Phase 1 implementation

QAtlas Phase 1 evaluates $F_\beta$ from an embedded high-precision table compiled from Bornemann (Math. Comp. 79, 871, 2010), Table 1, covering $x \in [-4, 4]$ for all three $\beta$. Inside the table support the interpolant is piecewise-linear and monotone non-decreasing; outside the support the function returns the Tracy-Widom 1994/1996 tail asymptotics

\[F_\beta(x) \sim \tau_\beta \exp\!\left[-\tfrac{\beta}{24} |x|^3\right] \qquad (x \to -\infty),\]

\[1 - F_\beta(x) \sim \exp\!\left[-\tfrac{2\beta}{3} x^{3/2}\right] \qquad (x \to +\infty),\]

continuously matched at the table boundary. Reference checkpoints pinned by the standalone test:

A direct Painleve-II ODE integrator (DifferentialEquations.jl-based) is deferred to Phase 2 of issue #151.

Mean Ratio $\langle r \rangle$

For a level sequence $\{E_n\}$ with spacings $s_n = E_{n+1} - E_n$, the consecutive-spacing ratio

\[r_n = \frac{\min(s_n, s_{n+1})}{\max(s_n, s_{n+1})}\]

is dimensionless (no spectral unfolding needed). Its mean takes universal values:

QAtlas returns these literature values directly from Atas-Bogomolny-Giraud-Roux, Phys. Rev. Lett. 110, 084101 (2013).

QAtlas API

using QAtlas

# Wigner-Dyson surmise
P_GUE_at_1 = QAtlas.fetch(Universality(:RMT), WignerSurmise(); beta=2, s=1.0)

# Tracy-Widom CDF
F_GUE_at_0 = QAtlas.fetch(Universality(:RMT), TracyWidom(); beta=2, x=0.0)
# ~ 0.9694 (Bornemann 2010 Table 1)

# Mean ratio
r_GUE = QAtlas.fetch(Universality(:RMT), MeanRatio(); beta=2)   # 0.5996

# Poisson (integrable baseline)
P_int = QAtlas.fetch(Universality(:Poisson), WignerSurmise(); s=1.0)  # 1/e
r_int = QAtlas.fetch(Universality(:Poisson), MeanRatio())             # 2 log 2 - 1

(In the actual code use the Greek letter beta keyword as β.)

References

• M. L. Mehta, Random Matrices, 3rd ed., Elsevier (2004).
• E. P. Wigner, Conference on Neutron Physics by Time-of-Flight, Oak Ridge Natl. Lab. Rep. ORNL-2309, 59 (1957) –- surmise.
• F. J. Dyson, J. Math. Phys. 3, 140 (1962) –- three-fold way.
• C. A. Tracy, H. Widom, Level-spacing distributions and the Airy kernel, Commun. Math. Phys. 159, 151 (1994) –- $F_2$.
• C. A. Tracy, H. Widom, On orthogonal and symplectic matrix ensembles, Commun. Math. Phys. 177, 727 (1996) –- $F_1, F_4$.
• F. Bornemann, On the numerical evaluation of Fredholm determinants, Math. Comp. 79, 871 (2010) –- high-precision $F_\beta$ table.
• Y. Y. Atas, E. Bogomolny, O. Giraud, G. Roux, Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett. 110, 084101 (2013) –- $\langle r \rangle$.
• V. Oganesyan, D. A. Huse, Phys. Rev. B 75, 155111 (2007) –- ratio statistic in many-body localisation.