A bijection for rooted maps on general surfaces

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This general construction requires new ideas and is more delicate than the special orientable case, but it carries the same information. In particular, it leads to a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rooted connected maps of Euler characteristic $2-2h$, and of the algebraicity of their generating functions, similar to the one previously obtained in the orientable case via the Marcus-Schaeffer bijection. It also shows that the renormalization factor $n^{1/4}$ for distances between vertices is universal for maps on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation on any fixed surface converge in distribution when the size $n$ tends to infinity. Finally, we extend the Miermont and Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our construction opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Comment: v2: 55 pages, 22 figure

Gaussian fluctuations of Young diagrams and structure constants of Jack characters

In this paper, we consider a deformation of Plancherel measure linked to Jack polynomials. Our main result is the description of the first and second-order asymptotics of the bulk of a random Young diagram under this distribution, which extends celebrated results of Vershik-Kerov and Logan-Shepp (for the first order asymptotics) and Kerov (for the second order asymptotics). This gives more evidence of the connection with Gaussian $\beta$-ensemble, already suggested by some work of Matsumoto. Our main tool is a polynomiality result for the structure constant of some quantities that we call Jack characters, recently introduced by Lassalle. We believe that this result is also interested in itself and we give several other applications of it.Comment: 71 pages. Minor modifications from version 1. An extended abstract of this work, with significantly fewer results and a different title, is available as arXiv:1201.180

Jack polynomials and orientability generating series of maps

We study Jack characters, which are the coefficients of the power-sum expansion of Jack symmetric functions with a suitable normalization. These quantities have been introduced by Lassalle who formulated some challenging conjectures about them. We conjecture existence of a weight on non-oriented maps (i.e., graphs drawn on non-oriented surfaces) which allows to express any given Jack character as a weighted sum of some simple functions indexed by maps. We provide a candidate for this weight which gives a positive answer to our conjecture in some, but unfortunately not all, cases. In particular, it gives a positive answer for Jack characters specialized on Young diagrams of rectangular shape. This candidate weight attempts to measure, in a sense, the non-orientability of a given map.Comment: v2: change of title, substantial changes of the content v3: substantial changes in the presentatio

A bijection for rooted maps on general surfaces (extended abstract)

International audienceWe extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable maps of Euler characteristic $2-2h$ and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor $n$¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size $n$ on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Nous étendons la bijection de Marcus et Schaeffer entre quadrangulations biparties orientables (de manière équivalente: cartes enracinées) et cartes à une face étiquetées orientables à toutes les surfaces, orientables ou non. Cette construction générale requiert des idées nouvelles et est plus délicate que dans le cas particulier orientable, mais permet des utilisations similaires. Elle donne donc une interprétation combinatoire uniforme de l’exposant de comptage $\frac{5(h-1)}{2}$ pour les cartes orientables et non-orientables de caractéristique d’Euler $2-2h$, et de l’algébricité des fonctions génératrices. Elle montre l’universalité du facteur de normalisation $n$¼ pour la métrique des cartes, sur toutes les surfaces: le profil et le rayon d’une quadrangulation enracinée pointée sur une surface fixée converge en distribution. Enfin, elle ouvre à la voie à l’étude des surfaces Browniennes pour toute 2-variété compacte

Non-orientable branched coverings, bb-Hurwitz numbers, and positivity for multiparametric Jack expansions

We introduce a one-parameter deformation of the 2-Toda tau-function of (weighted) Hurwitz numbers, obtained by deforming Schur functions into Jack symmetric functions. We show that its coefficients are polynomials in the deformation parameter $b$ with nonnegative integer coefficients. These coefficients count generalized branched coverings of the sphere by an arbitrary surface, orientable or not, with an appropriate $b$-weighting that "measures" in some sense their non-orientability. Notable special cases include non-orientable dessins d'enfants for which we prove the most general result so far towards the Matching-Jack conjecture and the "$b$-conjecture" of Goulden and Jackson from 1996, expansions of the $\beta$-ensemble matrix model, deformations of the HCIZ integral, and $b$-Hurwitz numbers that we introduce here and that are $b$-deformations of classical (single or double) Hurwitz numbers obtained for $b=0$. A key role in our proof is played by a combinatorial model of non-orientable constellations equipped with a suitable $b$-weighting, whose partition function satisfies an infinite set of PDEs. These PDEs have two definitions, one given by Lax equations, the other one following an explicit combinatorial decomposition.Comment: 56 pages, 6 figures; v2: definition of generalized branched covers fixed; combinatorial decomposition and corresponding equations now presented for connected objects and duality introduced; proof of piecewise polynomiality changed accordingly; v3: minor correction

Blossoming bijection for bipartite pointed maps and parametric rationality of general maps on any surface

We construct an explicit bijection between bipartite pointed maps on an arbitrary surface $\mathcal{S}$, and specific unicellular blossoming maps on the same surface. Our bijection gives access to the degrees of all the faces, and distances from the pointed vertex in the initial map. The main construction generalizes recent work of the second author which covered the case of orientable surface. Our bijection gives rise to a first combinatorial proof of a parametric rationality result concerning the bivariate generating series of maps on a given surface with respect to their numbers of faces and vertices. In particular, it provides a combinatorial explanation of the structural difference between the aforementioned bivariate parametric generating series in the case of orientable and non-orientable maps.Comment: 41 pages, 2 algorithms, 15 figure

Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures

This paper establishes universal formulas describing the global asymptotics of two different models of discrete $\beta$-ensembles in high, low and fixed temperature regimes. Our results affirmatively answer a question posed by the second author and \'Sniady. We first consider the Jack measures on Young diagrams of arbitrary size, which depend on the inverse temperature parameter $\beta>0$ and specialize to Schur measures when $\beta=2$. We introduce a class of Jack measures of Plancherel-type and prove a law of large numbers and central limit theorem in the three regimes. In each regime, we provide explicit formulas for polynomial observables of the limit shape and Gaussian fluctuations around the limit shape. These formulas have surprising positivity properties and are expressed in terms of weighted lattice paths. We also establish connections between these measures and the work of Kerov-Okounkov-Olshanski on Jack-positive specializations and show that this is a rich class of measures parametrized by the elements in the Thoma cone. Second, we show that the formulas from limits of Plancherel-type Jack measures are universal: they also describe the limit shape and Gaussian fluctuations for the second model of random Young diagrams of a fixed size defined by Jack characters with the approximate factorization property (AFP) studied by the second author and \'Sniady. Finally, we discuss the limit shape in the high/low-temperature regimes and show that, contrary to the continuous case of $\beta$-ensembles, there is a phase transition phenomenon in passing from the fixed temperature regime to the high/low temperature regimes. We note that the relation we find between the two different models of random Young diagrams appears to be new, even in the special case of $\beta=2$ that relates Schur measures to the character measures with the AFP studied by Biane and \'Sniady.Comment: 64 pages, 7 figure

A bijection for rooted maps on general surfaces (extended abstract)

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable maps of Euler characteristic $2-2h$ and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor $n$¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size $n$ on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold

Enumeration of non-oriented maps via integrability

In this note, we examine how the BKP structure of the generating series of several models of maps on non-oriented surfaces can be used to obtain explicit and/or efficient recurrence formulas for their enumeration according to the genus and size parameters. Using techniques already known in the orientable case (elimination of variables via Virasoro constraints or Tutte equations), we naturally obtain recurrence formulas with non-polynomial coefficients. This non-polynomiality reflects the presence of shifts of the charge parameter in the BKP equation. Nevertheless, we show that it is possible to obtain non-shifted versions, meaning pure ODEs for the associated generating functions, from which recurrence relations with polynomial coefficients can be extracted. We treat the cases of triangulations, general maps, and bipartite maps. These recurrences with polynomial coefficients are conceptually interesting but bigger to write than those with non-polynomial coefficients. However they are relatively nice-looking in the case of one-face maps. In particular we show that Ledoux's recurrence for non-oriented one-face maps can be recovered in this way, and we obtain the analogous statement for the (bivariate) bipartite case.Comment: 31 page