The North Pole Problem and Recent Developments in Infinitesimal Free Probability

In this thesis, we survey the state-of-the-art of the combinatorial theory of infinitesimal free probability, one of the principal extensions of free probability theory. We place particular focus on the recently developed theory of real infinitesimal free probability by Cébron and Mingo. We also briefly discuss (complex) infinitesimal free probability to highlight comparisons with the real case. We describe in detail computations of the joint real and complex infinitesimal cumulants of certain unitarily invariant matrix ensembles and their transposes in the large N limit, leading to the observation of a common phenomenon --- the vanishing of the mixed real infinitesimal cumulants --- that hold for the GUE and the complex Wishart ensemble and their transposed ensembles. This phenomenon leads to simplifications of infinitesimal cumulant calculations due to a relationship between the complex and real infinitesimal cumulants. We also explore applications of real infinitesimal freeness and other tools commonly used in free probability such as the orthogonal Weingarten function. Using these tools, we investigate the so called north pole problem, leading to free probabilistic arguments for concentration phenomena in the large N limit involving random rotations of the north pole and random projections associated with Haar orthogonal matrices.

Department dc:contributor.department

Mathematics and Statistics

Year dc:date.issued

2025

Author dc:creator

• Luo, Qixia

Advisor dc:contributor.supervisor

• Mingo, James

• Free Probability
• Random Matrix Theory

Statement dc:rights

• Attribution-NonCommercial-NoDerivatives 4.0 International

Licence dc:rights.uri

• http://creativecommons.org/licenses/by-nc-nd/4.0/

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/1974/34728

OAI identifier oai:identifier

oai:queensu.scholaris.ca:1974/34728