# Singular statistics

### Eugene Bogomolny 1, Ulrich Gerland 2, Charles Schmit 1

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 63 (2001) 036206

We consider the statistical distribution of zeros of random meromorphic functions whose poles are independent random variables. It is demonstrated that correlation functions of these zeros can be computed analytically and explicit calculations are performed for the 2-point correlation function. This problem naturally appears in e.g. rank-one perturbation of an integrable Hamiltonian and, in particular, when a $\delta$-function potential is added to an integrable billiard.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Physics Department,
University of California, San Diego