- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:04:49
Annales de l’Institut Henri Poincaré D
Ann. Inst. Henri Poincaré Comb. Phys. Interact.
AIHPD
2308-5827
2308-5835
General
Combinatorics
Quantum theory
10.4171/AIHPD
http://www.ems-ph.org/doi/10.4171/AIHPD
subscribers
European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
1
2014
2
Formal multidimensional integrals, stuffed maps, and topological recursion
Gaëtan
Borot
für Mathematik, BONN, GERMANY
Map enumeration, matrix models, 2D quantum gravity, loop equations, Tutte equation, topological recursion
We show that the large $N$ expansion in the multi-trace 1 formal hermitian matrix model is governed by a topological recursion with initial conditions. In terms of a 1$d$ gas of eigenvalues, this model includes – on top of the squared Vandermonde – multilinear interactions of any order between the eigenvalues. In this problem, the initial data ($\omega_1^0, \omega_2^0$) of the topological recursion is characterized: for $\omega_1^0$, by a non-linear, non-local Riemann-Hilbert problem on the discontinuity locus $\Gamma$ to determine; for $\omega_2^0$, by a related but linear, non-local Riemann-Hilbert problem on the discontinuity locus $\Gamma$. In combinatorics, this model enumerates discrete surfaces (maps) whose elementary 2-cells can have any topology - $\omega_1^0$ being the generating series of disks, $\omega_2^0$ that of cylinders. In particular, by substitution one may consider maps whose elementary cells are themselves maps, for which we propose the name ”stuffed maps”. In a sense, our results complete the program of the ”moment method” initiated in the 90s to compute the formal 1/$N$ in the one hermitian matrix model.
Combinatorics
Linear and multilinear algebra; matrix theory
Functions of a complex variable
225
264
10.4171/AIHPD/7
http://www.ems-ph.org/doi/10.4171/AIHPD/7