Loop Equation Analysis of the Circular $ \beta $ Ensembles

TL;DR

This paper develops a hierarchy of loop equations for circular beta ensembles, enabling systematic large N expansions and detailed analysis of resolvent functions and moments, with comparisons to known special cases.

Contribution

It introduces a general framework of loop equations for circular beta ensembles applicable to various potentials and inverse temperatures, advancing analytical understanding.

Findings

Computed the second resolvent function to ten orders in the large N expansion.

Derived explicit formulas for moments and their Fourier coefficients.

Conjectured exact forms for low k moments based on the expansion.

Abstract

We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures $ {\rm Re}\,\beta>0 $ and number of eigenvalues $ N $. Using matching arguments for the resolvent functions of linear statistics $ f(\zeta)=(\zeta+z)/(\zeta-z) $ in a particular asymptotic regime, the global regime, we systematically develop the corresponding large $ N $ expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment $ m_{k} $ to an equivalent length. The leading large $ N $, large $ k $, $ k/N $ fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low $ k $ moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of $ \beta = 1,2,4 $, general $ N $ and even, positive $ \beta $, $ N=2,3 $.