Perturbed Toeplitz operators and radial determinantal processes (Q376682)

Consider \(n\) complex random variables \(Z_1,\dots,Z_n\) with the joint distribution NEWLINE\[NEWLINE p(z_1,\dots, z_n) = C \prod_{1\leq j < k\leq n} |z_j-z_k|^2 \prod_{k=1}^n d m(z_k), NEWLINE\]NEWLINE where \(m\) is a radially symmetric measure on the complex plane. Examples include the Ginibre ensemble, the circular unitary ensemble, and the roots of certain random polynomials. The authors prove central limit theorems for the linear statistics of the form NEWLINE\[NEWLINE X_n:=\sum_{k=1}^n f(\arg Z_k), NEWLINE\]NEWLINE where \(f\) is a sufficiently smooth function. To prove their results, the authors use an operator-theoretic approach which exploits an exact formula relating the generating function of \(X_n\) to the determinant of a perturbed Toeplitz matrix.