Resolvents of R-Diagonal Operators

TL;DR

This paper derives a universal asymptotic formula for the norm of the resolvent of R-diagonal operators in a II_1-factor, using complex analysis, combinatorics, and introducing new partition structure diagrams.

Contribution

It provides the first asymptotic formula for the resolvent norm of R-diagonal operators and introduces partition structure diagrams as a new combinatorial tool in free probability.

Findings

Universal asymptotic formula for resolvent norm

Calculation of R-transform of |λ-c|^2 for circular operator c

Asymptotic formulas for negative moments of |λ-a|^2

Abstract

We consider the resolvent $(\lambda-a)^{-1}$ of any $R$-diagonal operator $a$ in a $\mathrm{II}_1$-factor. Our main theorem gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the $R$-transform of the operator $|\lambda-c|^2$ where $c$ is Voiculescu's circular operator, and give an asymptotic formula for the negative moments of $|\lambda-a|^2$ for any $R$-diagonal $a$. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce {\em partition structure diagrams}, a new combinatorial structure arising in free probability.