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Vertices from replica in a random matrix theory. (English) Zbl 1132.15020

Authors’ abstract: M. Kontsevich’s work on Airy matrix integrals [Commun. Math. Phys. 147, No. 1, 1–23 (1992; Zbl 0756.35081)] has led to explicit results for the intersection numbers of the moduli space of curves. In a subsequent work A. Okounkov [Int. Math. Res. Not. 2002, No. 18, 933–957 (2002; Zbl 1048.14012)] rederived these results from the edge behavior of a Gaussian matrix integral. In our work we consider the correlation functions of vertices in a Gaussian random matrix theory, with an external matrix source. We deal with operator products of the form \(\langle \prod_{i=1}^n\frac{1}{N} \text{tr}\;M^{k_i}\rangle\), in a \(\frac{1}{N}\) expansion. For large values of the powers \(k_i\), in an appropriate scaling limit relating large \(k\)’s to large \(N\), universal scaling functions are derived. Furthermore, we show that the replica method applied to characteristic polynomials of the random matrices, together with a duality exchanging \(N\) and the number of points, provides a new way to recover Kontsevich’s results on these intersection numbers.

MSC:

15B52 Random matrices (algebraic aspects)
82B05 Classical equilibrium statistical mechanics (general)
62H20 Measures of association (correlation, canonical correlation, etc.)
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