The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by Chekhov, Eynard and Orantin.
We call a map fully simple if it has non self-intersecting disjoint boundaries, and ordinary if such a restriction is not imposed. We study the combinatorial relation between fully simple and ordinary maps with the topology of a disk or a cylinder, which reproduces relations between moments and free cumulants established in the context of free probability. We propose a combinatorial interpretation of the exchange symplectic transformation of TR. We provide a matrix model interpretation for fully simple maps via the formal hermitian matrix model with external field and deduce a universal relation between generating series of fully simple and ordinary maps, which involves double monotone Hurwitz numbers. In particular, we obtain an ELSV-like formula for double $2$-orbifold strictly monotone Hurwitz numbers.
We consider ordinary maps endowed with an $O(\mathsf{n})$ loop model, which is a classical model in statistical physics, and determine which shapes are more likely to occur by looking at the nesting properties of the loops decorating the maps. We want to study the limiting objects when the number of vertices becomes arbitrarily large, which can be done by studying the generating series at dominant singularities. We analyze the nesting statistics in the $O(\mathsf{n})$ loop model on random maps of arbitrary topologies in the presence of large and small boundaries, relying on previous results for disks and cylinders and TR for this model. We study the generating series of maps which realize a fixed nesting graph and characterize their critical behavior in the dense and dilute phases.
The topological recursion of Eynard and Orantin governs a variety of problems in enumerative geometry and mathematical physics. The recursion uses the data of a spectral curve to define an infinite family of multidifferentials. It has been conjectured that, under certain conditions, the spectral curve possesses a non-commutative quantisation whose associated differential operator annihilates the partition function for the spectral curve. In this paper, we determine the quantum curves and partition functions for an infinite sequence of enumerative problems involving generalisations of ribbon gr…
This is a survey article describing the relationship between quantum curves and topological recursion. A quantum curve is a Schrödinger operator-like noncommutative analogue of a plane curve which encodes (quantum) enumerative invariants in a new and interesting way. The Schrödinger operator annihilates a wave function which can be constructed using the WKB method, and conjecturally constructed in a rather different way via topological recursion.
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative geometry like maps, partitions, Hurwitz numbers, intersection numbers, Gromov-Witten invariants... The problem is thus to understand what they count, or in other words, given a spectral curve, construct an enumerative geometry problem. This is what we do in a semi-heuristic approach in this article. Starting from a spectral curve, i.e. an integrable system, we us…
We study topological recursion on the irregular spectral curve $xy^2-xy+1=0$, which produces a weighted count of dessins d'enfant. This analysis is then applied to topological recursion on the spectral curve $xy^2=1$, which takes the place of the Airy curve $x=y^2$ to describe asymptotic behaviour of enumerative problems associated to irregular spectral curves. In particular, we calculate all one-point invariants of the spectral curve $xy^2=1$ via a new three-term recursion for the number of dessins d'enfant with one face.
This manuscript recounts some of the author's contributions to algebraic and enumerative combinatorics. We have focused on two types of generalizations of bipartite maps, which are bipartite graphs embedded on surfaces. Maps are known to appear in many areas of theoretical physics and discrete mathematics, but one key interest for fundamental computer science is how multi-facetted they are in the sense that multiple encodings exist which are not interchangeable, like Tutte's/loop equations, the topological recursion, the KP hierarchy and numerous bijections which made the field so rich. One ge…
In this paper we consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology. Consider a real 2-dimensional compact surface $S$, and fix a number of points $F$ on its boundary. We ask: how many configurations of disjoint arcs are there on $S$ whose boundary is $F$? We find that this enumerative problem, counting curves on surfaces, has a rich structure. For instance, we show that the curve counts obey an effective recursion, in the general framework of topological recursion. Moreover, they exhibit quasi-polynomial behaviour. This "elementary curve-counting…
The paper aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A ge…
Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In this paper, we prove that monotone Hurwitz numbers arise from the topological recursion of Eynard and Orantin applied to a particular spectral curve. We furthermore derive a quantum curve for monotone Hurwitz numbers. These results extend the collection of enumerative problems kn…
This review is an extended version of the Seoul ICM 2014 proceedings.It is a short overview of the "topological recursion", a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall how computing large size asymptotics in random matrices, has allowed to discover some fascinating and ubiquitous geometric invariants. Specializations of this method recover many classical invariants, like Gromov--Witten invariants, or knot polynomials (Jones, HOMFLY,...). In this short review, we give some examples, give definitions, and review some properti…
Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers are defined analogously, but with prescribed ramification over both zero and infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz numbers also arise as intersection numbers on moduli spaces. In t…
Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory,... All these topics share certain relations, called "loop equations" or "Virasoro constraints". In the simplest case, the complete solution of those equations was found recently: it can be expressed in the framework of differential geometry over a certain Riemann surface which depends on the problem : the "spectral curve". This thesis is a contribution to the development of these techniques, and to their applications. Keywords: r…
The second author and Norbury initiated the enumeration of lattice points in the Deligne-Mumford compactifications of moduli spaces of curves. They showed that the enumeration may be expressed in terms of polynomials, whose top and bottom degree coefficients store psi-class intersection numbers and orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$, respectively. Furthermore, they ask whether the enumeration is governed by the topological recursion and whether the intermediate coefficients also store algebro-geometric information. In this paper, we prove that the enumeration does …
We prove the existence of a 1/N expansion in unitary multimatrix models which are Gibbs perturbations of the Haar measure, and express the expansion coefficients recursively in terms of the unique solution of a noncommutative initial value problem. The recursion obtained is closely related to the "topological recursion" which underlies the asymptotics of many random matrix ensembles and appears in diverse enumerative geometry problems, but has not previously appeared in the context of random unitary matrices. Our approach consists of two main ingredients: an asymptotic study of the Schwinger-D…
Topological recursion for spin enumeration of hypermaps
Center for Geometry and Physics, Institute for Basic Science (IBS)（基础科学研究院几何物理中心）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；HSE University（高等经济学院）；Skoltech（斯科尔科沃科学技术学院）；NRC “Kurchatov Institute” – ITEP（库尔恰托夫研究所国家研究中心–理论物理与实验物理研究所）；University of Amsterdam（阿姆斯特丹大学）
This paper is based on the author's talk at the 2012 Workshop on Geometric Methods in Physics held in Bialowieza, Poland. The aim of the talk is to introduce the audience to the Eynard-Orantin topological recursion. The formalism is originated in random matrix theory. It has been predicted, and in some cases it has been proven, that the theory provides an effective mechanism to calculate certain quantum invariants and a solution to enumerative geometry problems, such as open Gromov-Witten invariants of toric Calabi-Yau threefolds, single and double Hurwitz numbers, the number of lattice points…
We derive bulk asymptotics of skew-orthogonal polynomials (sop) $π^{\bt}_{m}$, $β=1$, 4, defined w.r.t. the weight $\exp(-2NV(x))$, $V (x)=gx^4/4+tx^2/2$, $g>0$ and $t<0$. We assume that as $m,N \to\infty$ there exists an $ε> 0$, such that $ε\leq (m/N)\leq λ_{\rm cr}-ε$, where $λ_{\rm cr}$ is the critical value which separates sop with two cuts from those with one cut. Simultaneously we derive asymptotics for the recursive coefficients of skew-orthogonal polynomials. The proof is based on obtaining a finite term recursion relation between sop and orthogonal polynomials (op) and using asymptoti…
The goal of this article is to rederive the connection between the Painlevé $5$ integrable system and the universal eigenvalues correlation functions of double-scaled hermitian matrix models, through the topological recursion method. More specifically we prove, \textbf{to all orders}, that the WKB asymptotic expansions of the $τ$-function as well as of determinantal formulas arising from the Painlevé $5$ Lax pair are identical to the large $N$ double scaling asymptotic expansions of the partition function and correlation functions of any hermitian matrix model around a regular point in the bul…
The random variable $1+z_1+z_1z_2+\dots$ appears in many contexts and was shown by Kesten to exhibit a heavy tail distribution. We consider natural extensions of this variable and its associated recursion to $N \times N$ matrices either real symmetric $β=1$ or complex Hermitian $β=2$. In the continuum limit of this recursion, we show that the matrix distribution converges to the inverse-Wishart ensemble of random matrices. The full dynamics is solved using a mapping to $N$ fermions in a Morse potential, which are non-interacting for $β=2$. At finite $N$ the distribution of eigenvalues exhibits…
The goal of this "Habilitation à diriger des recherches" is to present two different applications, namely computations of certain partition functions in probability and applications to integrable systems, of the topological recursion developed by B. Eynard and N. Orantin in 2007. Since its creation, the range of applications of the topological recursion has been growing and many results in different fields have been obtained. The first aspect that I will develop deals with the historical domain of the topological recursion: random matrix integrals. I will review the formalism of the topologica…
Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of construction developed here involves a novel asymptotic symbol calculus for difference operators based on the relation between spectral asymptotics for Hermitian random matrices and asymptotics of orthogonal polynomials with exponential weights. These closed form expressions have a universal character in the sense that they are independent of the explicit vale…
