Diffusion in the space of complex hermitian matrices : microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial
Diffusion in the space of complex hermitian matrices : microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial
Diffusion in the space of complex hermitian matrices : microscopic properties of the averaged characteristic polynomial and the averaged inverse characteristic polynomial
author:
Blaizot Jean-Paul, Grela Jacek , Nowak Maciej , Warchoł Piotr
journal title:
Acta Physica Polonica. B
title of volume:
Random Matrix Theory : foundations and applications
We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with the Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like equations. These equations are valid for matrices of arbitrary size and for any initial condition assigned to the process. The solutions have compact integral representation that allows for a simple study of their asymptotic behavior, uncovering the Airy and Pearcey functions.
number of pulisher's sheets:
1,5
affiliation:
Wydział Fizyki, Astronomii i Informatyki Stosowanej : Instytut Fizyki im. Mariana Smoluchowskiego