Superintegrability of matrix Student's distribution

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.