论文标题
从Hua-Pickrell分布中绘制的随机统一矩阵的截断
Truncations of random unitary matrices drawn from Hua-Pickrell distribution
论文作者
论文摘要
令$ u $为从Hua-pickrell分布中绘制的随机统一矩阵$μ_ {\ mathrm {u}（n+m）}^{（δ）} $ thy unital $ \ mathrm {u}（u}（n+m）$。我们表明，被截断的统一矩阵$ [u_ {i，j}] _ {1 \ leq i，j \ leq n} $形式的确定点过程$ \ mathscr {x} _n^（m，，m，Δ） $Δ\ in \ mathbb {c} $满足$ \ mathrm {re} \，δ> -1/2 $。 我们还证明，确定点过程的$ n \ to \ infty $所采用的限制点过程$ \ mathscr {x} _n^{（m，δ）} $始终是$ \ mathscr {x}}^{[m] [m] [m] [m]} $，独立于$δ$。这里$ \ mathscr {x}^{[m]} $是$ \ mathbb {d} $上的确定点过程 \ begin {split} k^{[m]}（z，w）= \ frac {1} {（1-z \ overline w）^{m+1}}} 相对于参考度量$dμ^{[m]}（z）= \ frac {m}π（1- | z |）^{m-1}dσ（z）$，其中$dσ（z）$是$ \ mathbb {d d} $。
Let $U$ be a random unitary matrix drawn from the Hua-Pickrell distribution $μ_{\mathrm{U}(n+m)}^{(δ)}$ on the unitary group $\mathrm{U}(n+m)$. We show that the eigenvalues of the truncated unitary matrix $[U_{i,j}]_{1\leq i,j\leq n}$ form a determinantal point process $\mathscr{X}_n^{(m,δ)}$ on the unit disc $\mathbb{D}$ for any $δ\in\mathbb{C}$ satisfying $\mathrm{Re}\,δ>-1/2$. We also prove that the limiting point process taken by $n\to\infty$ of the determinantal point process $\mathscr{X}_n^{(m,δ)}$ is always $\mathscr{X}^{[m]}$, independent of $δ$. Here $\mathscr{X}^{[m]}$ is the determinantal point process on $\mathbb{D}$ with weighted Bergman kernel \begin{equation*} \begin{split} K^{[m]}(z,w)=\frac{1}{(1-z\overline w)^{m+1}} \end{split} \end{equation*} with respect to the reference measure $dμ^{[m]}(z)=\frac{m}π(1-|z|)^{m-1}dσ(z)$, where $dσ(z)$ is the Lebesgue measure on $\mathbb{D}$.
