The Marchenko-Pastur Law

The Wishart (Laguerre) ensembles LOE (\(\beta=1\)), LUE (\(\beta=2\)), and LSE (\(\beta=4\)) are built from the sample covariance matrix \(W = X X^\dagger\) of an \(n \times m\) Gaussian data matrix \(X\) (\(n\) variables, \(m\) samples, \(m \geq n\)), with real, complex, or quaternionic entries respectively. The relevant control parameter is the aspect ratio \(\gamma = n/m \in (0, 1]\).

As \(n, m \to \infty\) with \(\gamma\) fixed, the empirical spectral density of \(W/m\) converges to the Marchenko-Pastur law, supported on \([\lambda_-, \lambda_+]\) with

\[\lambda_\pm = \sigma^2\left(1 \pm \sqrt{\gamma}\right)^2,\]

and density

\[p(x) = \frac{\sqrt{(\lambda_+ - x)(x - \lambda_-)}} {2\pi\,\gamma\,\sigma^2\, x}, \qquad \lambda_- < x < \lambda_+,\]

where \(\sigma^2\) is the entry variance (here \(\sigma^2=1\)). This example reproduces that law for LOE, LUE, and LSE across several values of \(\gamma\).

Reference: V. A. Marchenko, L. A. Pastur, “Distribution of eigenvalues for some sets of random matrices”, Mat. Sb. 72 (1967) 507.

Run:

python examples/paper_replications/marchenko_pastur_demo.py

Saved marchenko_pastur_replication.png

import matplotlib.pyplot as plt
import numpy as np

import physicskit.rmt as rmt

N = 600
N_SAMPLES = 30
SEED = 2026
GAMMAS = [0.1, 0.4, 0.9]

fig, axes = plt.subplots(1, len(GAMMAS), figsize=(13, 4), sharey=False)

for ax, gamma in zip(axes, GAMMAS, strict=True):
    m = int(N / gamma)
    benchmark = rmt.validation.MarchenkoPastur(gamma=gamma)
    lo, hi = rmt.stats.mp_support(gamma)
    x = np.linspace(lo, hi, 400)

    for cls, color in [
        (rmt.ensembles.LOE, "steelblue"),
        (rmt.ensembles.LUE, "indianred"),
        (rmt.ensembles.LSE, "seagreen"),
    ]:
        ensemble = cls(n=N, m=m, seed=SEED)
        spectrum = ensemble.sample(n_samples=N_SAMPLES)
        centers, counts = rmt.stats.empirical_density(spectrum, bins=60)
        ax.plot(centers, counts, color=color, alpha=0.7, lw=1.5, label=cls.__name__)

    ax.plot(x, benchmark.theoretical_pdf(x), "k--", lw=2, label="Marchenko-Pastur")
    ax.set_title(f"gamma = n/m = {gamma}")
    ax.set_xlabel("eigenvalue")
    ax.legend(fontsize=8)

axes[0].set_ylabel("density")
fig.suptitle(
    f"Marchenko-Pastur law across aspect ratios -- n={N}, {N_SAMPLES} samples per ensemble",
)
fig.tight_layout()
out_path = "marchenko_pastur_replication.png"
fig.savefig(out_path, dpi=150)
print(f"Saved {out_path}")