Large Deviations for Non-Crossing Partitions

TL;DR

This paper establishes a large deviations principle for non-crossing partitions and applies it to derive a variational formula for the support maximum of certain probability measures using free cumulants, aiding free probability analysis.

Contribution

It introduces a large deviations framework for the empirical law of block sizes in non-crossing partitions and connects it to free probability via a variational formula.

Findings

Large deviations principle for non-crossing partitions

Variational formula for support maximum using free cumulants

Applicable when free cumulants are non-negative

Abstract

We prove a large deviations principle for the empirical law of the block sizes of a uniformly distributed non-crossing partition. As an application we obtain a variational formula for the maximum of the support of a compactly supported probability measure in terms of its free cumulants, provided these are all non-negative. This is useful in free probability theory, where sometimes the R-transform is known but cannot be inverted explicitly to yield the density.