Universality limits for random matrices and de Branges spaces of entire functions
In the theory of orthogonal polynomials the so-called ``universality law'' has been studied from different viewpoints and with measures satisfying less and less stringent conditions. If \(\{p_k(x)\}\) are the orthonormal polynomials with respect to a finite positive Borel measure \(\mu\) with compact support \(J\) on the real line, the simplest case of this law is given in terms of the reproducing kernel \[ K_n(x,y)=\sum_{k=0}^n\,p_k(x)p_k(y),\;\tilde{K}_n(x,y)=w(x)^{1/2}w(y)^{1/2}K_n(x,y). \] Let \(w={\text{d}\mu\over\text{d}x}\) be the Radon-Nikodým derivative of \(\mu\). The paper starts (Section 1) with some notation and states a typical result in the field: Theorem 1.1. Let \(\mu\) be a finite positive Borel measure on the real line with compact support. Let \(J\subset\text{supp}(\mu)\) be compact and such that \(\mu\) is absolutely continuous at each point of \(J\). The following are equivalent: A. uniformly for \(\xi\in J\) and \(a\) in compact subsets of the real line, \[ \lim_{n\rightarrow\infty}\,{K_n\left(\xi+{a\over\tilde{K}_n(\xi,\xi)}, \xi+{a\over\tilde{K}_n(\xi,\xi)}\right)\over K_n(\xi,\xi)}=1 \] B. uniformly for \(\xi\in J\) and \(a,b\) in compact subsets of the complex plane, \[ \lim_{n\rightarrow\infty}\,{\tilde{K}_n\left(\xi+{a\over\tilde{K}_n(\xi,\xi)}, \xi+{b\over\tilde{K}_n(\xi,\xi)}\right)\over K_n(\xi,\xi)}={\sin{\pi (a-b)}\over\pi (a-b)}. \] In this paper, the author explores the possible limits of subsequences of the sequence \(\{f_n\}\), where \[ f_n(a,b)={K_n\left(\xi_n+{a\over\tilde{K}_n(\xi_n,\xi_n)}, \xi_n+{b\over\tilde{K}_n(\xi_n,\xi_n)}\right)\over K_n(\xi,\xi)}, \] and \(\{\xi_n\}\) is a sequence of real numbers. Since the \(\{K_n\}\) are reproducing kernels for polynomials, it is no surprise that the limits of the subsequences turn out to be reproducing kernels for suitable entire functions; here, the setting of de Branges spaces, Hermite-Biehler class and Paley-Wiener spaces turn up. Let \(E\) be an entire function without zeros in the upper half-plane \({\mathbb C}^{+}=\{z\mid \text{Im}\,z>0\}\) for which \(|E(z)|\geq |E(\overline{z})|\) on \({\mathbb C}^{+}\) (an element of the \textit{Hermite-Biehler class}). Recall that the \textit{Hardy space} \(H^2({\mathbb C}^{+})\) is the set of all functions \(g\) analytic in the upper half-plane for which \[ \sup_{y>0}\,\int_{-\infty}^{\infty}\,|g(x+iy)|^2dx<\infty. \] Introduce \(g^{\ast}\) by \[ g^{\ast}(z)=\overline{g(\overline{z})}. \] The \textit{de Branges space} \({\mathcal H}(E)\) corresponding to an entire function \(E\) as above is the set of all entire functions \(g\) such that both \(g/E\) and \(g^{\ast}/E\) belong to \(H^2({\mathbb C}^{+})\) and \[ \|g\|_E =\left(\int_{-\infty}^{\infty}\,\left|{g\over E}\right|^2\right)^{1/2}<\infty. \] This is a Hilbert space with respect to the inner product \[ (g,h)=\int_{-\infty}^{\infty}\,{g\overline{h}\over |E|^2}. \] Finally, we need the Paley-Wiener space \(PW_{\sigma}\), consisting of entire functions of type \(\leq\sigma\) that are square integrable along the real axis (actually, a classical de Branges space with \(E(z)=\exp{(-i\sigma z)}\)) and the norm is \[ \|g\|_{L_2({\mathbb R})} =\left(\int_{-\infty}^{\infty}\,|g|^2\right)^{1/2}. \] The main result of the paper is now that, under mild conditions, \textit{universality limits in the bulk are reproducing kernels of de Branges spaces that equal classical Paley-Wiener spaces}. These results are stated in Theorems 1.3--1.7 in the paper, and to get the feeling one of these will be stated here. Theorem 1.3. Let \(\mu\) be a measure with compact support and let \(J\) be a compact set such that \(\mu\) is absolutely continuous in an open set \(O\) containing \(J\), and for some \(C>1\), \[ C^{-1}\leq\mu'\leq C\text{ in }O. \] Choose \(\{\xi_n\}\subset J\) and define \(\{f_n\}\) as above. {\parindent7mm \begin{itemize}\item[(a)] \(\{f_n(\cdot,\cdot)\}\) is a normal family in compact subsets of \({\mathbb C}^2\). \item[(b)] Let \(f(\cdot,\cdot)\) be the limit of some subsequence \(\{f_n(\cdot,\cdot)\}_{n\in S}\). Then \(f\) is an entire function of two variables that is real-valued in \({\mathbb R}^2\) and has \(f(0,0)=1\). Moreover, for some \(\sigma>0\), \(f(\cdot,\cdot)\) is entire of exponential type \(\sigma\) in each variable. \item[(c)] Define \[ L(u,v)=(u-v)f(u,v),\;u,v\in{\mathbb C}. \] Let \(a\in{\mathbb C}\), Im\,\(a>0\) and \[ E_a(z)=\sqrt{2\pi}\,{L(a,\overline{z})\over |L(a,\overline{a})|^{1/2}}. \] Then \(f\) is a reproducing kernel for \({\mathcal H}(E)\). In particular, for all \(z,\zeta\), \[ f(z,\overline{\zeta})={i\over 2\pi}\,{E_a(z)\overline{E_a(\zeta)}-E_a^{\ast}(z)\overline{E_a^{\ast}(\zeta)} \over z-\overline{\zeta}}. \] \item[(d)] Moreover, \[ {\mathcal H}(E_a)=PW_{\sigma} \] and the norms \(\|\cdot\|_{E_a}\) of \({\mathcal H}(E)\) and \(\|\cdot\|_{L_2({\mathbb R})}\) of \(PW_{\sigma}\) are equivalent. \end{itemize}} The layout of this well written paper is as follows: Section 1 (Introduction and results), 2 (notation and background), 3 (background on entire functions), 4 (de Branges spaces of polynomials), 5 (de Branges spaces of entire functions), 6 (proofs of Theorems 1.3--1.5), 7 (proofs of Theorems 1.6 and 1.7), References (38 items).
scientific article; zbMATH DE number 5565772
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