Pathwise asymptotic behavior of random determinants in the Jacobi ensemble.
Résumé
This is a companion paper of arxiv math.PR/050921. It concentrates on asymptotic properties of determinants of random matrices in the Jacobi ensemble. Let $M \in {\cal M}_{n_1 + n_2,r}(`R)$ (with $r \leq n_1 + n_2$) be a matrix whose entries are standard i.i.d. Gaussian. If $M^T = (M_1^T , M_2^T)$ with $M_1 \in {\cal M}_{n_1,r}$ and $M_2 \in {\cal M}_{n_2,r}$, then, $W_1 := M_1^T M_1$ and $W_2 := M_2^T M_2$ are independent $r\times r$ Wishart matrices with parameters $n_1$ and $n_2$ and $M^T M = W_1 + W_2$ is Wishart with parameter $n_1+ n_2$. Then ${\cal Z} := (W_1 + W_2)^{-1/2} W_1 (W_1 + W_2)^{-1/2}$ has a Beta matrix variate distribution with parameters $n_1/2, n_2/2$ (sometimes called the Jacobi distribution). We set $n_1 = \lfloor n\tau_1 \rfloor$, $n_2 = \lfloor n\tau_2 \rfloor$, $r= \lfloor nt\rfloor$ $t\in [0, \tau_1)$ and let $n \rightarrow \infty$; we define ${\cal Z}_n (t)$ as the corresponding matrix and $\Theta_n (t) := |{\cal Z}_n(t)|$ as its determinant. In the Jacobi ensemble, the Kshirsagar's theorem decomposes $\Theta_n (t)$ into a product of independent beta distributed variables. This allows us to study the process $\frac{1}{n} \left(n^{-1} \log \Theta_n (t), t \in [0,\tau_1)\right)$ and the asymptotic behavior of the sequence $\{\frac{1}{n} n^{-1}\log \Theta_n \}_n$ as $n\rightarrow \infty$ with $\tau_1$ and $\tau_2$ fixed : a.s. convergence, fluctuations, large deviations. We connect the results for marginals (fixed $t$) with those obtained by the study of the empirical spectral distribution. In the whole paper, we consider the problem of general $\beta$, where the particular cases $\beta = 1,2,4$ correspond to real, complex, and quaternionic matrices.}