We prove that the persistence diagram of the sublevel set filtration of the quadratic form f(x) = x^T M x restricted to the unit sphere S^{n-1} is analytically determined by the eigenvalues of the symmetric matrix M. By Morse theory, the diagram has exactly n-1 finite bars, with the k-th bar living in homological dimension k-1 and having length equal to the k-th eigenvalue spacing s_k = λ_{k+1} - λ_k. This identification transfers random matrix theory (RMT) universality to persistence diagram universality: for matrices drawn from the Gaussian Orthogonal Ensemble (GOE), we derive the closed-form persistence entropy PE = log(8n/π) - 1, and verify numerically that the coefficient of variation of persistence statistics decays as n^{-0.6}. Different random matrix ensembles (GOE, GUE, Wishart) produce distinct universal persistence diagrams, providing topological fingerprints of RMT universality classes. As a practical consequence, we show that persistence entropy outperforms the standard level spacing ratio \langle r \rangle for discriminating GOE from GUE matrices (AUC 0.978 vs. 0.952 at n = 100, non-overlapping bootstrap 95% CIs), and detects global spectral perturbations in the Rosenzweig-Porter model to which \langle r \rangle is blind. These results establish persistence entropy as a new spectral diagnostic that captures complementary information to existing RMT tools.
Figures & tables
Figure 1: Eigenvalue empirical CDF for 200 independent GOE (100) matrices (blue), overlaid with the Wigner semicircle CDF (black). The near-perfect collapse demonstrates eigenvalue universality, which by Theorem 2 implies persistence diagram universality.
Figure 2: Unfolded bulk spacing distributions (central 80% of eigenvalues) for GOE, GUE, and Wishart ensembles ( n=100 , 200 samples each). Histograms match the Wigner surmise: p1(s)=(π/2)sexp(−πs2/4) for β=1 (GOE, Wishart) and p2(s)=(32/π2)s2exp(−4s2/π) for β=2 (GUE). Since bar lengths equal eigenvalue spacings, these are also the universal bar length distributions of the persistence diagrams.
Statistic
n=50
n=100
n=200
TP
0.032
0.022
0.013
PE
0.009
0.006
0.004
μ (max bar/TP)
0.223
0.228
0.259
Table 1: Coefficient of variation (CV) of persistence statistics for GOE (n) , 200 samples per n . TP and PE exhibit universality (CV →0 ); μ is an extreme-value statistic with O(1) fluctuations.
n
PE (numerical)
PE (analytical)
Bias (%)
50
3.721±0.002
3.847
3.3
100
4.410±0.002
4.540
2.9
200
5.104±0.001
5.233
2.5
500
6.017±0.001
6.149
2.2
1000
6.707±0.001
6.843
2.0
Table 2: Persistence entropy: numerical mean ± SEM (200 GOE samples) vs. analytical prediction log(8n/π)−1 . The bias is systematic (edge corrections) and decreases as ∼n−0.17 .
Statistic
GOE
GUE
Wishart
TP
3.88±0.08
3.83±0.06
2.73±0.09
PE
4.41±0.03
4.47±0.02
4.25±0.05
μ
0.038±0.009
0.033±0.007
0.066±0.020
Table 3: Persistence statistics for three ensembles ( n=100 , 200 samples). GOE and GUE share the same density but differ in level repulsion ( β ); Wishart has a distinct density.
Figure 3: (a) Coefficient of variation of TP and PE vs. matrix size n , confirming universality (CV →0 ). (b) Persistence entropy: numerical values (200 GOE samples per n , error bars show ±1 std) vs. the closed-form prediction PE=log(8n/π)−1 (solid line).
Statistic
n=50
n=100
n=200
PE
0.921
0.978 [.971, .985]
0.996
⟨r⟩
0.862
0.952 [.939, .964]
0.991
Spacing variance
0.880
0.931
0.966
PE +⟨r⟩
0.922
0.978
0.998
Table 4: AUC for GOE vs. GUE discrimination (500 samples per class, bootstrap 95% CIs at n=100 ). PE captures the global shape of the spacing distribution, complementing the local-ratio information in ⟨r⟩ .
Figure 4: AUC for GOE vs. GUE discrimination as a function of matrix size n (500 samples per class). PE (circles) consistently outperforms ⟨r⟩ (squares). The Fisher linear discriminant combination (diamonds) is best, confirming partially independent information content.
Figure 5: Signal-to-noise ratio (deviation from GOE reference divided by GOE standard deviation) for three spectral diagnostics across the Rosenzweig–Porter model ( n=100 , 300 samples per λ ). ⟨r⟩ (squares) remains at its GOE value for all λ≤5 , while PE (circles) and the spacing variance (triangles) detect the global density change at 3σ by λ=0.7 and 0.5 , respectively.