Locating Zeros Of Entire Functions Using The Extended Cumulant Method

Abstract

Occurence of phase transitions at the thermodynamic limit can be predicted from the behaviour of Lee-Yang zeros of the partition function in finite systems. The cumulant method was developed to locate these Lee-Yang zeros. This method was extended to locate the zeros of Loschmidt echos, an analogue of the partition function in the context of dynamical phase transitions. In this thesis, our objective is to investigate and prove properties of the extended cumulant method.

Through numerical experiments, we observe the influence of the base point’s position and the convergence of the approximate roots to their corresponding exact roots with increasing cumulant order. Additionally, we observe an interesting property when applying the method to entire functions with real zeros. The subsequent mathematical analysis aims to prove these properties.

The proofs are constrained to entire functions of finite rank due to convergence con- straints. We prove the convergence of the approximate roots to their corresponding exact roots as the cumulant order tends to infinity for entire functions of finite rank. The remaining properties are proven under additional assumptions. These include the real line property and the spiral-like convergence path of the approximate roots. We derive an upper bound for the error that decreases exponentially with increasing cumulant order. A discussion on the optimality and utility of this bound is presented.