A computational study of Random Matrix Theory (RMT), exploring eigenvalue densities and level spacing statistics across five classical random matrix ensembles.
Random Matrix Theory was introduced by Eugene Wigner in the 1950s to model the energy spectra of heavy atomic nuclei whose Hamiltonians were too complex to write down explicitly. The key insight is that the statistical properties of eigenvalues depend only on the global symmetries of the matrix ensemble, not on the precise form of the entries — a form of universality with broad applications in quantum chaos, condensed matter physics, number theory, and wireless communications.
This project studies five ensembles:
| Ensemble | Entries | Structure | Physical interpretation |
|---|---|---|---|
| GUE | Gaussian | Complex Hermitian | Systems without time-reversal symmetry |
| GOE | Gaussian | Real symmetric | Systems with time-reversal symmetry |
| CUE | Cauchy | Complex Hermitian | Heavy-tailed Gaussian analogue |
| COE | Cauchy | Real symmetric | Heavy-tailed GOE analogue |
| UUE | Uniform [-1,1] | Complex Hermitian | CLT and trace statistics study |
The acceptance-rejection sampler uses a double-exponential proposal P_α(x) = (α/2) exp(−α|x|). The optimal α is found analytically by maximising the acceptance probability √(π/2) · α · exp(−α²/2).
The proposal distribution matches the analytic curve exactly, confirming the inversion sampler is correct.
The acceptance-rejection output agrees with N(0,1) across the full range, validating the generator.
The measured efficiency peaks at α = 1 (~76%), matching the analytic optimum. This is the parameter used throughout.
Both constructors fill the upper triangle with the Gaussian generator and reflect to enforce Hermitian (GUE) or real-symmetric (GOE) symmetry.
The custom density estimator reproduces the analytic N(0,1) curve accurately, confirming it is suitable for spectral density estimation.
For large N, the spectral density of GUE converges to:
ρ(ε) = (2/πR²) √(R² − ε²), R = √(8N)
Already at N = 32 the empirical density is indistinguishable from the semicircle. By N = 128 convergence is complete, confirming Wigner's law with radius R = √(8N).
ϱ_GUE(s) = (32s²/π²) exp(−4s²/π)
Even at N = 4, the quadratic vanishing at s = 0 (level repulsion, β = 2) and the Gaussian tail are clearly observed.
At N = 32 the empirical distribution overlaps the Wigner surmise almost perfectly.
N = 64: the agreement is essentially exact across the full range of s.
N = 128: the Wigner surmise is confirmed to high precision. No finite-N correction is visible.
On a log scale the tail decays as exp(−4s²/π) — faster than exponential. This Gaussian tail is a hallmark of Gaussian ensembles and contrasts sharply with Cauchy power-law tails.
The GOE follows the same semicircle law but with radius R = √(4N) — half the GUE variance, because only real entries contribute. Convergence quality with N is identical to GUE.
ϱ_GOE(s) = (π s/2) exp(−π s²/4)
The repulsion is linear (~s, β = 1), weaker than GUE's quadratic repulsion. The peak of the distribution is shifted to larger s compared to GUE.
At N = 64 the empirical curve converges to the GOE surmise with the same quality as GUE. Tails remain Gaussian.
Exact inversion: x = γ · tan(π u − π/2), u ~ Uniform(0,1). No rejection needed.
The generator reproduces the Cauchy density exactly over four decades of magnitude. The heavy tails extending to |x| ~ 50 are visible even at γ = 0.1, confirming the distribution has no finite variance.
Same structure as GUE/GOE but with Cauchy entries (γ = 0.1).
Cauchy entries have divergent variance, so the semicircle law does not apply.
Even at N = 64 the spectral densities remain broad and heavy-tailed with no sign of convergence to a bounded support. The Wigner semicircle law breaks down entirely.
The tail follows ρ(ε) ~ ε⁻² over two decades — an algebraic decay, fundamentally different from the semicircle's sharp cutoff at R.
ϱ(0) > 0: level repulsion is absent. Eigenvalues can be arbitrarily close, unlike in GOE where ϱ(0) = 0.
The finite ϱ(0) is robust with N, confirming it is a genuine property of Cauchy ensembles, not a finite-size effect.
CUE shows the same absence of level repulsion as COE. The heavy tails of the entry distribution wash out the repulsion mechanism.
N = 64: the distribution is broader than GUE and the flat behaviour near s = 0 persists.
The spacing tail decays as ~s⁻⁴ — a power law. This is a direct consequence of Cauchy entries having infinite variance, and contrasts sharply with the Gaussian tails of GUE/GOE.
Each diagonal entry is Uniform(−1, 1) with variance 1/3. The trace sums N independent entries, so by the CLT: Tr[M] → N(0, N/3).
The trace distributions for N = 1, 2, 4, 8, 16 progressively converge to the Gaussian limit. At N = 16 the CLT is already very well satisfied.
The two eigenvalues are symmetric and repel each other — the two distributions are mirror images with equal variance.
The edge eigenvalues (ε₀, ε₃) have significantly larger variance than the central ones (ε₁, ε₂). The eigenvalues are not identically distributed.
The gradient in variance from edge to centre grows with N. Interior eigenvalues are increasingly concentrated near zero.
At N = 16 the contrast is clear: the edge eigenvalues span a wide range while the central ones cluster tightly, reflecting bulk spectrum rigidity.
The off-diagonal entry is negative: ε₀ and ε₁ anti-correlate. When one shifts up, the other shifts down to conserve the trace.
Anti-correlations exist at all pairs. The sum of all entries v_4 ≈ 1.33 ≈ 4/3, confirming the analytic prediction v_N = N/3.
The strongest anti-correlations are between adjacent eigenvalues; correlations decay with distance in the spectrum.
At N = 16 the structure is rich, but v_16 ≈ 16/3 is still satisfied — the total variance is determined by the entry distribution alone, not the correlations.
The distributions of T_N are identical to those of Tr[M], confirming T_N = Tr[M] exactly — basis-independence of the trace holds numerically.
Mean is zero for all N. Variance grows exactly as N/3, in perfect agreement with the analytic result. The eigenvalue anti-correlations do not affect the total variance.
All four curves collapse onto N(0, 1/3). The CLT holds for the sum of eigenvalues despite their pairwise anti-correlations — the correlations are not strong enough to violate the CLT.
The Cauchy distribution has no finite variance. It is stable under addition with scaling 1/N instead of 1/√N.
Under the T_N/N rescaling, all curves collapse onto the Cauchy density ρ_{0.1}. The ordinary CLT fails: there is no Gaussian limit. Instead the Generalised CLT applies, with the Cauchy distribution as the stable attractor.
- Wigner semicircle law holds for GUE and GOE with radii
√(8N)and√(4N), with visible convergence already at N = 32. - Level repulsion is quadratic (β = 2) for GUE and linear (β = 1) for GOE; the Wigner surmise is accurate already at N = 4.
- Cauchy ensembles break the semicircle: spectral tails
~ε⁻², spacing tails~s⁻⁴, and no level repulsion (ϱ(0) > 0). - UUE eigenvalues are pairwise anti-correlated, but
Var(T_N) = N/3and the CLT collapseT_N/√N → N(0, 1/3)hold exactly. - CUE trace obeys the Generalised CLT:
T_N/N → Cauchy(0.1)— Cauchy stability replaces Gaussian universality when entries have infinite variance.