File:Spectral density of Wishart-Laguerre ensemble (8, 15).png
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Spectral density of Wishart-Laguerre ensemble (8, 15).png
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DescriptionSpectral density of Wishart-Laguerre ensemble (8, 15).png |
English: A reconstruction of Figure 1 of Moments of Wishart-Laguerre and Jacobi ensembles of random matrices (Livan 2011)
https://arxiv.org/pdf/1103.2638.pdf ```python import numpy as np import matplotlib.pyplot as plt
betas = 1, 2, 4 NMs = [(8, 15)]
Nmatr = 100000 Es = {} for n, m in NMs: for beta in betas: if beta == 1: # Wishart Orthogonal Ensemble X = np.random.randn(Nmatr, n, m) M = np.einsum('ijk,ilk->ijl', X, X) E = np.linalg.eigvals(M.reshape(Nmatr, n, n)).flatten() elif beta == 2: # Wishart Unitary Ensemble X_real = np.random.randn(Nmatr, n, m) X_imag = np.random.randn(Nmatr, n, m) X = X_real + 1j * X_imag M = np.einsum('ijk,ilk->ijl', X, X.conjugate()) E = np.linalg.eigvals(M.reshape(Nmatr, n, n)).flatten() elif beta == 4: # Wishart Symplectic Ensemble A = np.random.randn(Nmatr, n,m) + 1j * np.random.randn(Nmatr, n,m) B = np.random.randn(Nmatr, n,m) + 1j * np.random.randn(Nmatr, n,m) X = np.block([[A, B],[-np.conj(B), np.conj(A)]]) M = np.einsum('ijk,ilk->ijl', X, X.conjugate()) E = np.linalg.eigvals(M.reshape(Nmatr, 2 * n, 2 * n)).flatten() Es[(n, m, beta)] = E for n, m in NMs: plt.figure(figsize=(16, 8)) legends = {1: "LOE", 2:"LUE", 4:"LSE"} colors={1:"blue", 2:"red", 4:"green"} for beta in betas: color=colors[beta] E = Es[(n, m, beta)] xs = np.real(E) / np.sqrt(beta) bin_heights, bin_borders, _ = plt.hist(xs, bins=500, density=True, color=color, alpha=0.1) bin_centers = bin_borders[:-1] + np.diff(bin_borders) / 2 # Compute sliding window average window_size = 5 window = np.ones(window_size) / window_size smoothed_heights = np.convolve(bin_heights, window, mode='same') # Plot sliding window average plt.plot(bin_centers, smoothed_heights, label=legends[beta], color=color) # Add plot labels and title plt.xlabel('x', fontsize=14) plt.ylabel('ρ(x)', fontsize=14) plt.title(r'Eigenvalues $/\sqrtTemplate:\beta$, with (N, M) = {}'.format((n, m)), fontsize=18) plt.grid(True) plt.legend() plt.show()``` |
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Author | Cosmia Nebula |
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