Superintegrability of matrix Student's distribution

TL;DR

This paper explores how superintegrability persists in certain long-tail matrix models, even when high moments diverge, challenging the notion that divergence implies loss of integrability.

Contribution

It demonstrates that superintegrability can survive analytical continuation in long-tail matrix models despite divergences in high moments.

Findings

Superintegrability persists after analytical continuation.

Long-tail models can maintain exact factorized averages.

Divergences do not necessarily imply loss of integrability.

Abstract

For ordinary matrix models, the eigenvalue probability density decays rapidly as one goes to infinity, in other words, has "short tails". This ensures that all the multiple trace correlators (multipoint moments) are convergent and well-defined. Still, many critical phenomena are associated with an enhanced probability of seemingly rare effects, and one expects that they are better described by the "long tail" models. In absence of the exponential fall-off, the integrals for high moments diverge, and this could imply a loss of (super)integrability properties pertinent to matrix and eigenvalue models and, presumably, to the non-perturbative (exact) treatment of more general quantum systems. In this paper, we explain that this danger to modern understanding could be exaggerated. We consider a simple family of long-tail matrix models, which preserve the crucial feature of superintegrability: exact factorized expressions for a full set of basic averages. It turns out that superintegrability can survive after an appropriate (natural and obvious) analytical continuation even in the presence of divergencies, which opens new perspectives for the study of the long-tail matrix models.