Semicircular limits on the free Poisson chaos: counterexamples to a transfer principle

TL;DR

This paper identifies conditions under which free Poisson integrals converge to semicircular limits and constructs counterexamples showing the transfer principle from classical to free Brownian motion does not apply to Poisson measures.

Contribution

It introduces new sufficient conditions for convergence and provides explicit counterexamples, challenging the transfer principle in the free probability framework.

Findings

Established conditions for free Poisson integrals to converge to semicircular limits

Constructed explicit counterexamples to the transfer principle between classical and free Brownian motions

Developed new formulas and bounds for free Poisson multiple integrals

Abstract

We establish a class of sufficient conditions, ensuring that a sequence of multiple integrals with respect to a free Poisson measure converges to a semicircular limit. We use this result to construct a set of explicit counterexamples, showing that the transfer principle between classical and free Brownian motions (recently proved by Kemp, Nourdin, Peccati and Speicher (2012)) does not extend to the framework of Poisson measures. Our counterexamples implicitly use kernels appearing in the classical theory of random geometric graphs. Several new results of independent interest are obtained as necessary steps in our analysis, in particular: (i) a multiplication formula for free Poisson multiple integrals, (ii) diagram formulae and spectral bounds for these objects, and (iii) a counterexample to the general universality of the Gaussian Wiener chaos in a classical setting.