Jacobi Ensemble, Hurwitz Numbers and Wilson Polynomials

Abstract

We express the topological expansion of the Jacobi Unitary Ensemble in terms of triple monotone Hurwitz numbers. This completes the combinatorial interpretation of the topological expansion of the classical unitary invariant matrix ensembles. We also provide effective formulæ for generating functions of multipoint correlators of the Jacobi Unitary Ensemble in terms of Wilson polynomials, generalizing the known relations between one point correlators and Wilson polynomials.

Grothendieck's Dessins d'Enfants in a Web of Dualities. III

“…We note that the method used in [45] for deriving explicit formula for the generating series of connected LUE correlators (96) is different from ours: In [45,46], isomonodromic tau-functions of certain Riemann-Hilbert problems are used.…”

“…The following is a table of the initial values 1 for various tau-functions of the Toda lattice hierarchy (cf. [33,37,45,46,78,86]) appearing in matrix models:…”

“…For JUE, to make the normalized JUE partition function [46] a tau-function for the Toda lattice hierarchy, we need to add the following correction factor (using (71) and the second line of the above table)…”

The matrix-resolvent method to tau-functions for the nonlinear Schrödinger hierarchy

“…We note that the feature of the improvement for the algorithm is the following: one is to solve the system of first-order difference equations (1.31) instead of to solve the second-order difference equation (4.5), and the system (1.31) can be recast into a sequence of first-order evolutionary PDEs, recursively, as explained in Remark 3.1 and Theorem 1.3. The improved algorithm works for the GUE [20], LUE [26] and JUE [27] solutions. Let us explain in more details the simple algorithm by means of examples, which include GUE (see Example 1) below.…”

Topological Expansion of Oscillatory BGW and HCIZ Integrals at Strong Coupling

“…Although interest in monotone Hurwitz numbers has exploded in recent years, expanding the scope of Hurwitz theory [3,4,13,22] and leading to new connections with between enumerative geometry and matrix models [11,31,47,48,87], they were originally conceived in order to address the HCIZ case of Conjecture 1.1, with [52] taking the first steps in this direction. In this paper, the program begun in [52] achieves a considerably enhanced fulfillment of its initial purpose.…”