Hyperpfaffian point processes when beta is a square integer?
Monday, June 18, 2012 - 3:30pm - 4:45pm
The prototypical joint density of eigenvalues of a random matrix contains as a factor a power of the absolute value of the Vandermonde determinant in the variables of integration. The most nuanced statistical information can be derived when this power, denoted beta, takes the value 1, 2 or 4. Of particular importance in this talk, is the fact that, when beta is 1 or 4, the marginal densities (vis correlation functions) can be expressed as Pfaffians of antisymmetric matrices formed from a (matrix) kernel. That is, in this situation, the eigenvalues form a Pfaffian point process. In this talk, I will explain one path to the derivation of the Pfaffian point process from the relatively basic fact that the partition functions for these ensembles are themselves expressed as Pfaffians. After introducing the notion of the hyperpfaffian, and showing that, when beta is a square integer, the partition function is a hyperpfaffian, I will ask the question: Does there a hyperpfaffian point process for the eigenvalues when beta is a square integer. I will give evidence for and against such a thing, and sketch out a possible method of proof, assuming the question has an affirmative answer.