Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals
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Annales Henri Poincar´ e
Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals Massimo Gisonni, Tamara Grava
and Giulio Ruzza
Abstract. We consider the Laguerre partition function and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven in Cunden et al. (Ann. Inst. Henri Poincar´e D, to appear) that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers that can be explicitly computed from our formulae. As a second result, we identify the Laguerre partition function with only positive couplings and a special value of the parameter α = −1/2 with the modified GUE partition function, which has recently been introduced in Dubrovin et al. (Hodge-GUE correspondence and the discrete KdV equation. arXiv:1612.02333) as a generating function for Hodge integrals. This identification provides a direct and new link between monotone Hurwitz numbers and Hodge integrals.
1. Introduction and Results 1.1. Laguerre Unitary Ensemble (LUE) and Formulae for Correlators The LUE is the statistical model on the cone H+ N of positive definite Hermitian matrices of size N endowed with the probability measure 1 detα X exp tr (−X)dX, ZN (α; 0)
(1.1)
dX being the restriction to H+ N of the Lebesgue measure on the space HN N2 R of Hermitian matrices X = X † of size N ; dX := dXii dRe Xij dIm Xij . (1.2) 1≤i≤N
1≤i −1. Writing α = M − N , a random matrix X distributed according to the measure (1.1) is called complex Wishart matrix with parameter M ; in particular, when M is an integer, there is the equality in law X = N1 W W † , where W is an N ×M random matrix with independent identically distributed Gaussian entries [36]. Our first main result, Theorem 1.1, concerns explicit and effective formulae for correlators of the LUE 1 tr X k1 · · · tr X kr detα X exp tr (−X)dX tr X k1 · · · tr X kr := ZN (α; 0) H+ N for arbitrary nonzero integers k1 , . . . , kr ∈ Z\{0}. Theorem 1.1 is best formulated in terms of connected correlators k1 kr |P|−1 ki (−1) (|P| − 1)! tr X , tr X · · · tr X c := P partition of {1,...,r}
I∈P
i∈I
(1.4) e.g., tr X k1 c := tr X k1 , tr X k1 tr X k2 c := tr X k1 tr X k2 − tr X k1 tr X k2 . The generating function for connected correlators
1 1 1 tr tr · · · tr x1 − X x2 − X xr − X c
(1.5)
can be expanded near xj = ∞ and/or xj = 0, yielding the following generating functions up to some irrelevant terms; for r = 1 1 tr X k , C0,1 (x) := − (1.6) xk−1 tr X −k , C1,0 (x) := k+1 x k≥1
k≥1
for r = 2 C2,0 (x1 , x2 ) :=
tr X k1 tr X k2 c x1k1 +1 x2k2 +1
k1 ,k2 ≥1
xk22 −1 tr X k1 tr X −k2 c , k1 +1 x k1 ,k2 ,≥1 1 xk11 −1 xk22 −1 tr X −k1 tr X −k2 c ,
C1,1 (x1 , x2 ) := − C0,2 (x1 , x2 ) :=
,
(1.7)
k1 ,k2 ≥1
and, in general, r−
Cr+ ,r− (x1 , . . . , xr ) := (−1)
k1 ,...,kr ≥1
tr X σ1 k1 · · · tr X σr kr xσ1 1 k1 +1 · · · xrσr kr +1
c
,
(1.8)
Laguerre Ensemble
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