Banded matrix models

Computing the eigenvalues of a full \(N\times N\) random matrix is \(\mathcal{O}(N^3)\), and can thus become prohibitive for large \(N\). A way to circumvent the problem is to adopt the equivalent banded models i.e. diagonalize banded matrices.

The first tridiagonal models for the Hermite Ensemble and Laguerre Ensemble were revealed by [DE02], who left the Jacobi Ensemble as an open question, addressed by [KN04]. Such tridiagonal formulations made sampling possible at cost \(\mathcal{O}(N^2)\) but also unlocked sampling for generic \(\beta>0\)!

Note that [KN04] also derived a quindiagonal model for the Circular Ensemble.

Hermite Ensemble

Take for reference measure \(\mu=\mathcal{N}(\mu, \sigma)\)

\[(x_1,\dots,x_N) \sim \left|\Delta(x_1,\dots,x_N)\right|^{\beta} \prod_{i= 1}^N e^{- \frac{(x_i-\mu)^2}{2\sigma^2}} % \indic_{\bbR}(x_i) \ d x_i.\]

Note

Recall that from the definition in (32)

\[\left|\Delta(x_1,\dots,x_N)\right| = \prod_{i<j} |x_i - x_j|.\]

The equivalent tridiagonal model reads

\[\begin{split}\begin{bmatrix} \alpha_1 & \sqrt{\beta_2}& 0 & 0 & 0 \\ \sqrt{\beta_2} & \alpha_2 & \sqrt{\beta_3}& 0 & 0 \\ 0 & \ddots & \ddots & \ddots & 0 \\ 0 & 0 & \sqrt{\beta_{N-1}} & \alpha_{N- 1} & \sqrt{\beta_{N}} \\ 0 & 0 & 0 & \sqrt{\beta_N} & \alpha_{N} \end{bmatrix},\end{split}\]

with

\[\alpha_{i} \sim \mathcal{N}(\mu, \sigma^2) \quad \text{and} \quad \beta_{i+1} \sim \Gamma\left(\frac{\beta}{2}(N - i), \sigma^2\right).\]

To recover the full matrix model for Hermite Ensemble, recall that \(\Gamma(\frac{k}{2}, 2)\equiv \chi_k^2\) and take

\[\mu = 0 \quad \text{and} \quad \sigma^2 = 2.\]

That is to say,

\[\alpha_{i} \sim \mathcal{N}(0, 2) \quad \text{and} \quad \beta_{i+1} \sim \chi_{\beta(N - i)}^2.\]
from dppy.beta_ensembles import HermiteEnsemble


hermite = HermiteEnsemble(beta=5.43)  # beta can be >=0, default beta=2
# Reference measure is N(mu, sigma^2)
hermite.sample_banded_model(loc=0.0, scale=1.0, size_N=500)
# hermite.plot(normalization=True)
hermite.hist(normalization=True)

(Source code, png, hires.png, pdf)

../_images/ex_plot_hermite_banded_matrix_model.png

Fig. 26 Tridiagonal matrix model for the Hermite ensemble

See also

Laguerre Ensemble

Take for reference measure \(\mu=\Gamma(k,\theta)\)

\[(x_1,\dots,x_N) \sim \left|\Delta(x_1,\dots,x_N)\right|^{\beta} % \prod_{i= 1}^N x_i^{k-1} e^{- \frac{x_i}{\theta}} % \indic_{\bbR}(x_i) \ d x_i.\]

Note

Recall that from the definition in (32)

\[\left|\Delta(x_1,\dots,x_N)\right| = \prod_{i<j} |x_i - x_j|.\]

The equivalent tridiagonal model reads

\[\begin{split}\begin{bmatrix} \alpha_1 & \sqrt{\beta_2}& 0 & 0 & 0 \\ \sqrt{\beta_2} & \alpha_2 & \sqrt{\beta_3}& 0 & 0 \\ 0 & \ddots & \ddots & \ddots & 0 \\ 0 & 0 & \sqrt{\beta_{N-1}} & \alpha_{N- 1} & \sqrt{\beta_{N}} \\ 0 & 0 & 0 & \sqrt{\beta_N} & \alpha_{N} \end{bmatrix} = \begin{bmatrix} \sqrt{\xi_1} & & & \\ \sqrt{\xi_2} & \sqrt{\xi_3} & & \\ & \ddots & \ddots & \\ & & \sqrt{\xi_{2N-2}} & \sqrt{\xi_{2N-1}} \end{bmatrix} \begin{bmatrix} \sqrt{\xi_1} & \sqrt{\xi_2} & & \\ & \sqrt{\xi_3} & \ddots & \\ & & \ddots & \sqrt{\xi_{2N-2}} \\ & & & \sqrt{\xi_{2N-1}} \end{bmatrix},\end{split}\]

with

\[\xi_{2i-1} \sim \Gamma\left(\frac{\beta}{2}(N - i) + k, \theta \right) \quad \text{and} \quad \xi_{2i} \sim \Gamma\left(\frac{\beta}{2}(N - i), \theta \right).\]

To recover the full matrix model for Laguerre Ensemble, recall that \(\Gamma(\frac{k}{2}, 2)\equiv \chi_k^2\) and take

\[k = \frac{\beta}{2}(M-N+1) \quad \text{and} \quad \theta = 2.\]

That is to say,

\[\xi_{2i-1} \sim \chi_{\beta(M - i + 1)}^2 \quad \text{and} \quad \xi_{2i} \sim \chi_{\beta(N - i)}^2.\]
from dppy.beta_ensembles import LaguerreEnsemble


laguerre = LaguerreEnsemble(beta=2.98)  # beta can be >=0, default beta=2
# Reference measure is Gamma(k, theta)
laguerre.sample_banded_model(shape=600, scale=2.0, size_N=400)
# laguerre.plot(normalization=True)
laguerre.hist(normalization=True)

(Source code, png, hires.png, pdf)

../_images/ex_plot_laguerre_banded_matrix_model.png

Fig. 28 Tridiagonal matrix model for the Laguerre ensemble

See also

Jacobi Ensemble

Take for reference measure \(\mu=\operatorname{Beta}(a,b)\)

\[(x_1,\dots,x_N) \sim \left|\Delta(x_1,\dots,x_N)\right|^{\beta} % \prod_{i= 1}^N x_i^{a-1} (1-x_i)^{b-1} % \indic_{\bbR}(x_i) \ d x_i.\]

Note

Recall that from the definition in (32)

\[\left|\Delta(x_1,\dots,x_N)\right| = \prod_{i<j} |x_i - x_j|.\]

The equivalent tridiagonal model reads

\[\begin{split}\begin{bmatrix} \alpha_1 & \sqrt{\beta_2}& 0 & 0 & 0 \\ \sqrt{\beta_2} & \alpha_2 & \sqrt{\beta_3}& 0 & 0 \\ 0 & \ddots & \ddots & \ddots & 0 \\ 0 & 0 & \sqrt{\beta_{N-1}} & \alpha_{N- 1} & \sqrt{\beta_{N}} \\ 0 & 0 & 0 & \sqrt{\beta_N} & \alpha_{N} \end{bmatrix}.\end{split}\]
\[ \begin{align}\begin{aligned}\begin{split}\begin{aligned} \alpha_1 &= \xi_1 \quad & \\ \alpha_k &= \xi_{2k-2}+\xi_{2k-1} \quad &\beta_{k+1} &= \xi_{2k-1}\xi_{2k} \end{aligned}\end{split}\\\begin{split}\begin{aligned} \xi_1 &= c_1 \quad &\gamma_1 &= 1-c_1 \\ \xi_k &= (1-c_{k-1})c_k \quad &\gamma_k &= c_{k-1}(1-c_k) \end{aligned},\end{split}\end{aligned}\end{align} \]

with

\[c_{2i-1} \sim \operatorname{Beta} \left( \frac{\beta}{2}(N-i) + a, \frac{\beta}{2}(N-i) + b \right) \quad \text{and} \quad c_{2i} \sim \operatorname{Beta} \left( \frac{\beta}{2} (N-i), \frac{\beta}{2} (N-i-1) + a + b \right).\]

To recover the full matrix model for Laguerre Ensemble, recall that \(\Gamma(\frac{k}{2}, 2)\equiv \chi_k^2\) and take

\[a = \frac{\beta}{2}(M_1-N+1) \quad \text{and} \quad b = \frac{\beta}{2}(M_2-N+1).\]

That is to say,

\[c_{2i-1} \sim \operatorname{Beta} \left( \frac{\beta}{2}(M_1-i+1), \frac{\beta}{2}(M_2-i+1) \right) \quad \text{and} \quad c_{2i} \sim \operatorname{Beta} \left( \frac{\beta}{2} (N-i), \frac{\beta}{2} (M_1+M_2-N-i+1) \right).\]
from dppy.beta_ensembles import JacobiEnsemble


jacobi = JacobiEnsemble(beta=3.14)  # beta can be >=0, default beta=2
# Reference measure is Beta(a,b)
jacobi.sample_banded_model(a=500, b=300, size_N=400)
# jacobi.plot(normalization=True)
jacobi.hist(normalization=True)

(Source code, png, hires.png, pdf)

../_images/ex_plot_jacobi_banded_matrix_model.png

Fig. 30 Tridiagonal matrix model for the Jacobi ensemble

See also

Circular Ensemble

\[\left|\Delta(e^{i \theta_1},\dots, e^{i \theta_N})\right|^{\beta} \prod_{j = 1}^N \frac{1}{2\pi} \mathbf{1}_{[0,2\pi]} (\theta_j) d\theta_j.\]

Note

Recall that from the definition in (32)

\[\left|\Delta(x_1,\dots,x_N)\right| = \prod_{i<j} |x_i - x_j|.\]

Important

Consider the distribution \(\Theta_{\nu}\) that for integers \(\nu\geq2\) is defined as follows:

Draw \(v\) uniformly at random from the unit sphere \(\mathbb{S}^{\nu} \in \mathbb{R}^{\nu+1}\), then \(v_1 + iv_2\sim \Theta_{\nu}\)

Now, given \(\beta\in \mathbb{N}^*\), let

  • \(\alpha_k\sim \Theta_{\beta(N-k-1)+1}\) independent variables

  • for \(0\leq k\leq N-1\) set \(\rho_k = \sqrt{1-|\alpha_k|^2}\).

Then, the equivalent quindiagonal model corresponds to the eigenvalues of either \(LM\) or \(ML\) with

\[L = \operatorname{diag}[\Xi_0,\Xi_2,\dots] \quad \text{and} \quad M = \operatorname{diag}[\Xi_{-1},\Xi_1,\Xi_3\dots],\]

and where

\[\begin{split}\Xi_k = \begin{bmatrix} \overline{\alpha}_k & \rho_k\\ \rho_k & -\alpha_k \end{bmatrix} , \quad 0\leq k\leq N-2 , \quad \text{with} \quad \Xi_{-1} = [1] \quad \text{and} \quad \Xi_{N-1} = [\overline{\alpha}_{N-1}].\end{split}\]

Hint

The effect of increasing the \(\beta\) parameter can be nicely visualized on this Circular Ensemble. Viewing \(\beta\) as the inverse temperature, the configuration of the eigenvalues crystallizes with \(\beta\), see the figure below.

from dppy.beta_ensembles import CircularEnsemble

circular = CircularEnsemble(beta=2)  # beta must be >=0 integer, default beta=2

# See the cristallization of the configuration as beta increases
for b in [0, 1, 5, 10]:

    circular.beta = b
    circular.sample_banded_model(size_N=30)
    circular.plot()

circular.beta = 2
circular.sample_banded_model(size_N=1000)
circular.hist()

(Source code)

../_images/ex_plot_circular_banded_matrix_model_00.png

Fig. 32 (png, hires.png, pdf) Quindiagonal matrix model for the Circular ensemble

../_images/ex_plot_circular_banded_matrix_model_01.png

Fig. 33 (png, hires.png, pdf) Quindiagonal matrix model for the Circular ensemble

../_images/ex_plot_circular_banded_matrix_model_02.png

Fig. 34 (png, hires.png, pdf) Quindiagonal matrix model for the Circular ensemble

../_images/ex_plot_circular_banded_matrix_model_03.png

Fig. 35 (png, hires.png, pdf) Quindiagonal matrix model for the Circular ensemble

../_images/ex_plot_circular_banded_matrix_model_04.png

Fig. 36 (png, hires.png, pdf) Quindiagonal matrix model for the Circular ensemble

See also