arXiv subjects

Séverin Charbonnier

Publications and source records attributed to Séverin Charbonnier.

11 recordsLinked to original sources

arXiv2024

Shifted Witten classes and topological recursion

The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold using the Givental--Teleman reconstruction theorem. We show that the $R$-matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Using this, we prove that the descendant intersection theory of the shifted Witten classes satisfies topological recursion on two $1$-parameter families of spectral curves. By taking the limit as the parameter goes to zero for these families of spectral curves, we prove that the descendant intersection theory of the Witten $r$-spin class can be computed by topological recursion on the $r$-Airy spectral curve. We finally show that this proof suffices to deduce Witten's $r$-spin conjecture, already proved by Faber, Shadrin and Zvonkine, which claims that the generating series of $r$-spin intersection numbers is the tau function of the $r$-KdV hierarchy that satisfies the string equation.

math.AG↗

arXiv2023

Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers

Kontsevich introduced certain ribbon graphs as cell decompositions for combinatorial models of moduli spaces of complex curves with boundaries in his proof of Witten's conjecture. In this work, we define four types of generalised Kontsevich graphs and find combinatorial relations among them. We call the main type ciliated maps and use the auxiliary ones to show they satisfy a Tutte recursion that we turn into a combinatorial interpretation of the loop equations of topological recursion for a large class of spectral curves. It follows that ciliated maps, which are Feynman graphs for the Generalised Kontsevich matrix Model (GKM), are computed by topological recursion. Our particular instance of the GKM relates to the r-KdV integrable hierarchy and since the string solution of the latter encodes intersection numbers with Witten's $r$-spin class, we find an identity between ciliated maps and $r$-spin intersection numbers, implying that they are also governed by topological recursion. In turn, this paves the way towards a combinatorial understanding of Witten's class. This new topological recursion perspective on the GKM provides concrete tools to explore the conjectural symplectic invariance property of topological recursion for large classes of spectral curves.

math.CO↗

arXiv2023

Functional relations for higher-order free cumulants

We establish the functional relations between generating series of higher-order free cumulants and moments in higher-order free probability, solving an open problem posed fifteen years ago by Collins, Mingo, Śniady and Speicher. We propose an extension of free probability theory, which governs the all-order topological expansion in unitarily invariant matrix ensembles, with a corresponding notion of free cumulants and give as well their relation to moments via functional relations. Our approach is based on the study of a master transformation involving double monotone Hurwitz numbers via semi-infinite wedge techniques, building on the recent advances of the last-named author with Bychkov, Dunin-Barkowski and Kazarian. We illustrate our formulas by computing the first few decaying terms of the correlation functions of an ensemble of spiked GUE matrices, going beyond the law of large numbers and the central limit theorem.

math.OA↗

arXiv2022

Topological recursion for Orlov-Scherbin tau functions, and constellations with internal faces

We study the correlators $W_{g,n}$ arising from Orlov-Scherbin 2-Toda tau functions with rational content-weight $G(z)$, at arbitrary values of the two sets of time parameters. Combinatorially, they correspond to generating functions of weighted Hurwitz numbers and $(m,r)$-factorisations of permutations. When the weight function is polynomial, they are generating functions of constellations on surfaces in which two full sets of degrees (black/white) are entirely controlled, and in which internal faces are allowed in addition to boundaries. We give the spectral curve (the "disk" function $W_{0,1}$, and the "cylinder" function $W_{0,2}$) for this model, generalising Eynard's solution of the 2-matrix model which corresponds to $G(z)=1+z$, by the addition of arbitrarily many free parameters. Our method relies both on the Albenque-Bouttier combinatorial proof of Eynard's result by slice decompositions, which is strong enough to handle the polynomial case, and on algebraic arguments. Building on this, we establish the topological recursion (TR) for the model. Our proof relies on the fact that TR is already known at time zero (or, combinatorially, when the underlying graphs have only boundaries, and no internal faces) by work of Bychkov-Dunin-Barkowski-Kazarian-Shadrin (or Alexandrov-Chapuy-Eynard-Harnad for the polynomial case), and on the general idea of deformation of spectral curves due to Eynard and Orantin, which we make explicit in this case. As a result of TR, we obtain strong structure results for all fixed-genus generating functions.

math.CO↗

arXiv2022

Around the combinatorial unit ball of measured foliations on bordered surfaces

The volume $\mathscr{B}_Σ^{\rm comb}(\mathbb{G})$ of the unit ball -- with respect to the combinatorial length function $\ell_{\mathbb{G}}$ -- of the space of measured foliations on a stable bordered surface $Σ$ appears as the prefactor of the polynomial growth of the number of multicurves on $Σ$. We find the range of $s \in \mathbb{R}$ for which $(\mathscr{B}_Σ^{\rm comb})^{s}$, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depends on the topology of $Σ$, in contrast with the situation for hyperbolic surfaces where Arana-Herrera and Athreya (arXiv:1907.06287) recently proved an optimal square-integrability.

math.GT↗

arXiv2021

Topological recursion for fully simple maps from ciliated maps

Ordinary maps satisfy topological recursion for a certain spectral curve $(x, y)$. We solve a conjecture from arXiv:1710.07851 that claims that fully simple maps, which are maps with non self-intersecting disjoint boundaries, satisfy topological recursion for the exchanged spectral curve $(y, x)$, making use of the topological recursion for ciliated maps arXiv:2105.08035.

math.CO↗

arXiv2021

On the Kontsevich geometry of the combinatorial Teichmüller space

For bordered surfaces S, we develop a complete parallel between the geometry of the combinatorial Teichmüller space $T_S^{comb}$ equipped with Kontsevich symplectic form $ω_K$, and then the usual Weil-Petersson geometry of Teichmüller space $T_S$. The basis for this is an identification of $T_S^{comb}$ with a space of measured foliations with transverse boundary conditions. We equip $T_S^{comb}$ with an analog of the Fenchel-Nielsen coordinates (defined similarly as Dehn-Thurston coordinates) and show they are Darboux for $ω_K$ (analog of Wolpert formula). We then set up the geometric recursion of Andersen-Borot-Orantin to produce mapping class group invariants functions on $T_S^{comb}$ whose integration with respect to Kontsevich volume form satisfy topological recursion. Further we establish an analog of Mirzakhani-McShane identities, and provide applications to the study of the enumeration of multicurves with respect to combinatorial lengths and Masur-Veech volumes. The formalism allows us to provide uniform and completely geometric proofs of Witten's conjecture/Kontsevich theorem and Norbury's topological recursion for lattice point count in the combinatorial moduli space, parallel to Mirzakhani's proof of her recursion for Weil-Petersson volumes. We strengthen results of Mondello and Do on the convergence of hyperbolic geometry to combinatorial geometry along the rescaling flow, allowing us to flow systematically natural constructions on the usual Teichmüller space to their combinatorial analogue, such as a new derivation of the piecewise linear structure of $T_S^{comb}$ originally obtained in the work of Penner, as the limit under the flow of the smooth structure of $T_S$.

math.DG↗

arXiv2020

Topological recursion for Masur-Veech volumes

We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a family of polynomials in $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of \cite{Delecroix} proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain an expression of the area Siegel--Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur--Veech volumes, and thus of area Siegel--Veech constants, for low $g$ and $n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

math.GT↗

arXiv2019

Relating ordinary and fully simple maps via monotone Hurwitz numbers

A direct relation between the enumeration of ordinary maps and that of fully simple maps first appeared in the work of the first and last authors. The relation is via monotone Hurwitz numbers and was originally proved using Weingarten calculus for matrix integrals. The goal of this paper is to present two independent proofs that are purely combinatorial and generalise in various directions, such as to the setting of stuffed maps and hypermaps. The main motivation to understand the relation between ordinary and fully simple maps is the fact that it could shed light on fundamental, yet still not well-understood, problems in free probability and topological recursion.

math.CO↗

arXiv2018

Local properties of the random Delaunay triangulation model and topological 2D gravity

Delaunay triangulations provide a bijection between a set of $N+3$ points in the complex plane, and the set of triangulations with given circumcircle intersection angles. The uniform Lebesgue measure on these angles translates into a Kähler measure for Delaunay triangulations, or equivalently on the moduli space $\mathcal M_{0,N+3}$ of genus zero Riemann surfaces with $N+3$ marked points. We study the properties of this measure. First we relate it to the topological Weil-Petersson symplectic form on the moduli space $\mathcal M_{0,N+3}$. Then we show that this measure, properly extended to the space of all triangulations on the plane, has maximality properties for Delaunay triangulations. Finally we show, using new local inequalities on the measures, that the volume $\mathcal{V}_N$ on triangulations with $N+3$ points is monotonically increasing when a point is added, $N\to N+1$. We expect that this can be a step towards seeing that the large $N$ limit of random triangulations can tend to the Liouville conformal field theory.

math-ph↗

arXiv2017

Large Strebel graphs and $(3,2)$ Liouville CFT

2D quantum gravity is the idea that a set of discretized surfaces (called map, a graph on a surface), equipped with a graph measure, converges in the large size limit (large number of faces) to a conformal field theory (CFT), and in the simplest case to the simplest CFT known as pure gravity, also known as the gravity dressed (3,2) minimal model. Here we consider the set of planar Strebel graphs (planar trivalent metric graphs) with fixed perimeter faces, with the measure product of Lebesgue measure of all edge lengths, submitted to the perimeter constraints. We prove that expectation values of a large class of observables indeed converge towards the CFT amplitudes of the (3,2) minimal model.

math-ph↗