Norm-dependent Random Matrix Ensembles in External Field and Supersymmetry

TL;DR

This paper investigates norm-dependent random matrix ensembles under external fields, deriving a supersymmetric mapping and explicit correlation functions for unitary symmetry, advancing analytical tools in random matrix theory.

Contribution

It introduces an exact supersymmetric mapping for norm-dependent ensembles and provides explicit correlation functions for the unitary case, enhancing analytical methods.

Findings

Derived a transformation formula for superspace probability densities.

Performed an exact mapping to superspace for orthogonal, unitary, and symplectic ensembles.

Obtained explicit correlation functions for the unitary symmetry case.

Abstract

The class of norm-dependent Random Matrix Ensembles is studied in the presence of an external field. The probability density in those ensembles depends on the trace of the squared random matrices, but is otherwise arbitrary. An exact mapping to superspace is performed. A transformation formula is derived which gives the probability density in superspace as a single integral over the probability density in ordinary space. This is done for orthogonal, unitary and symplectic symmetry. In the case of unitary symmetry, some explicit results for the correlation functions are derived.