A universality theorem for ratios of random characteristic polynomials (Q435181)

The authors consider orthogonal polynomial ensembles of \(n\) particles on the real line described by a positive Borel measure \(\mu\) with finite moments, and the associated distribution function for the particles \(\{x_1,x_2,\ldots,x_n\}\) of the form NEWLINE\[NEWLINE dP_{\mu,n}(x)=\frac1{Z_n}\,\prod_{1\leq j<i\leq n}(x_i-x_j)^2\,\prod_{i=1}^n d\mu(x_i), NEWLINE\]NEWLINE where \(Z_n\) is a normalization constant. For symmetric functions \(f(x_1,x_2,\ldots,x_n)\) the average of \(f\) is NEWLINE\[NEWLINE \langle f(x)\rangle_\mu:=\int f(x) dP_{\mu,n}(x), \quad x=(x_1,x_2,\ldots,x_n). NEWLINE\]NEWLINE The goal of the paper is to study the large \(n\) asymptotics of the averages \(\langle f_n(x)\rangle_\mu\) where NEWLINE\[NEWLINE f_n(x)=\frac{D_n(\alpha_1)\ldots D_n(\alpha_k)}{D_n(\beta_1)\ldots D_n(\beta_k)}\,, \quad D_n(z)=\prod_{i=1}^n (z-x_i), NEWLINE\]NEWLINE and \(\alpha_1,\ldots,\beta_k\) are complex numbers. The main result says that for \(\mu\) locally absolutely continuous with a bounded Radon--Nikodym derivative, universality for the reproducing kernels implies universality for \(f_n(x)\). It should be emphasized that aside for the existence of moments, the authors don't make global assumptions on \(\mu\). In particular, the global absolute continuity or compactness of the support of \(\mu\) is not assumed.