On the Limit of the Tridiagonal Model for $β$-Dyson Brownian Motion

Abstract

In previous work, a description of the result of applying the Householder tridiagonalization algorithm to a G$β$E random matrix is provided by Edelman and Dumitriu. The resulting tridiagonal ensemble makes sense for all $β>0$, and has spectrum given by the $β$-ensemble for all $β>0$. Moreover, the tridiagonal model has useful stochastic operator limits which was introduced and analyzed in subsequent studies. In this work, we analogously study the result of applying the Householder tridiagonalization algorithm to a G$β$E process which has eigenvalues governed by $β$-Dyson Brownian motion. We propose an explicit limit of the upper left $k \times k$ minor of the $n \times n$ tridiagonal process as $n \to \infty$ and $k$ remains fixed. We prove the result for $β=1$, and also provide numerical evidence for $β=1,2,4$. This leads us to conjecture the form of a dynamical $β$-stochastic Airy operator with smallest $k$ eigenvalues evolving according to the $n \to \infty$ limit of the largest, centered and re-scaled, $k$ eigenvalues of $β$-Dyson Brownian motion.

Figures (8)

• Figure 1: Householder Tridiagonalization (right) of a $10\times 10$ GOE process (left)
• Figure 2: Comparing empirical means and variances of $A_j(t)$ (black) and $a_j(t)$ (red) for an arbitrary choice $j=9$. One can observe that the mean is around zero and the variance is near 2 (green dashed line) for both processes at each time, as their stationary distiribution is $\mathcal{N}(0, 2)$. We used 10000 samples of $2000\times 2000$ tridiagonalized GOE processess and 10000 samples of the Ornstein-Uhlenbeck process \ref{['eq:aj']} with $n=2000$. For both samples we sampled from $t=0$ to $t=0.5$ with $dt=10^{-3}$.
• Figure 3: Sample covariances $\text{Cov}(a_j(0), a_j(t))$ and $\text{Cov}(A_j(0), A_j(t))$ with an arbitrary choice of $j=5$. The yellow line is the theoretical covariance of the process $A_j(t)$. We used the same samples used in Figure \ref{['fig:meanvara']}.
• Figure 4: Comparing empirical means and variances of $\hat{b}_j(t)$ (black) and $b_j(t)$ (red) for an arbitrary choice $j=25$. One can observe that for each $t$, the mean is around $\sqrt{2}\frac{\Gamma((n-j+1)/2)}{\Gamma((n-j)/2)} \sim \sqrt{n-j} = 44.44$, and the variance is around $n-j-2(\frac{\Gamma((n-j+1)/2)}{\Gamma((n-j)/2)})^2 \sim \frac{1}{2}$, which is represented by the green dashed lines. We used the samples from Figure \ref{['fig:meanvara']} for the tridiagonalized GOE model $b_j(t)$. We used 10000 samples of the Ornstein-Uhlenbeck process $B_j(t)\sim \mathrm{OU}(2j)$ and computed $\hat{b}_j(t)$ for each sample.
• Figure 5: The first plot shows that the sum $a_j(0)+a_j(t)$ is not a Gaussian (for $j>1$) as we observe nonzero (excess) kurtosis at each time. Note that for $j=1$ the process is indeed Gaussian. The second plot illustrates that the process $a_j(t)$ gets closer to a Gaussian process as $n$ gets larger, with an arbitrary choice of $j=3$. (Notice that the blue line represents $a_3(t)$ for $n=5$ in both bottom plots.) Recall from Theorem \ref{['Main Theorem Statement']}$a_j(t)$ converges weakly to a Ornstein-Uhlenbeck process as $n\to\infty$. We sampled 100000 tridiagonalized GOE processes for each $n=5, 40, 320$.

Theorems & Definitions (66)

• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4: Householder Reflector
• Remark 2.5
• Remark 2.7
• Remark 2.8
• Lemma 2.9
• proof
• Remark 2.10