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Documents  05C30 | enregistrements trouvés : 5

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Multi angle  Vertex degrees in planar maps
Drmota, Michael (Auteur de la Conférence) | CIRM (Editeur )

We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner
We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even inf...

05A19 ; 05A16 ; 05C10 ; 05C30

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We show how recent results of the authors on equidistribution of square-tiled surfaces of given combinatorial type allow to compute approximate values of Masur-Veech volumes of the strata in the moduli spaces of Abelian and quadratic differentials by Monte Carlo method.
We also show how similar approach allows to count asymptotical number of meanders of fixed combinatorial type in various settings in all genera. Our formulae are particularly efficient for classical meanders in genus zero.
We construct a bridge between flat and hyperbolic worlds giving a formula for the Masur-Veech volume of the moduli space of quadratic differentials in terms of intersection numbers of $\mathcal{M}_{g,n}$ (in the spirit of Mirzakhani's formula for Weil-Peterson volume of the moduli space of pointed curves).
Joint work with V. Delecroix, E. Goujard, P. Zograf.
We show how recent results of the authors on equidistribution of square-tiled surfaces of given combinatorial type allow to compute approximate values of Masur-Veech volumes of the strata in the moduli spaces of Abelian and quadratic differentials by Monte Carlo method.
We also show how similar approach allows to count asymptotical number of meanders of fixed combinatorial type in various settings in all genera. Our formulae ...

32G15 ; 05C30 ; 05Axx

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Multi angle  Covering spaces and spanning trees
Cimasoni, David (Auteur de la Conférence) | CIRM (Editeur )

The aim of this talk is to show how basic notions traditionally used in the study of "knotted embeddings in dimensions $3$ and $4$", such as covering spaces and representation theory, can have non-trivial applications in combinatorics and statistical mechanics. For example, we will show that for any finite covering $G'$ of a finite edge-weighted graph $G$, the spanning tree partition function on $G$ divides the spanning tree partition function on $G'$ (in the polynomial ring with variables given by the weights). Setting all the weights equal to $1$, this implies a theorem known since 30 years: the number of spanning trees on $G$ divides the number of spanning trees on $G'$. Other examples of such results will be presented.
Joint work (in progress) with Adrien Kassel.
The aim of this talk is to show how basic notions traditionally used in the study of "knotted embeddings in dimensions $3$ and $4$", such as covering spaces and representation theory, can have non-trivial applications in combinatorics and statistical mechanics. For example, we will show that for any finite covering $G'$ of a finite edge-weighted graph $G$, the spanning tree partition function on $G$ divides the spanning tree partition function ...

57M12 ; 05C30 ; 82B20

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Angel and Schramm ont étudié en 2003 la limite locale des triangulations uniformes. La loi limite, appelée UIPT (pour Uniform Infinite planar Triangulation) a depuis été pas mal étudiée et est plutôt bien comprise. Dans cet exposé, je vais expliquer comment on peut obtenir un résultat analogue à celui d’Angel et Schramm mais lorsque les triangulations ne sont plus uniformes mais distribuées selon un modèle d’Ising. Une partie importante de la preuve consiste à étudier une équation sur des séries génératrices à deux variables catalytiques et repose sur la méthode des invariants de Tutte (introduite par Tutte et popularisée par Bernardi et Bousquet-Mélou). L’objet limite est pour le moment très mal compris et soulève un grand nombre de questions ouvertes !
Angel and Schramm ont étudié en 2003 la limite locale des triangulations uniformes. La loi limite, appelée UIPT (pour Uniform Infinite planar Triangulation) a depuis été pas mal étudiée et est plutôt bien comprise. Dans cet exposé, je vais expliquer comment on peut obtenir un résultat analogue à celui d’Angel et Schramm mais lorsque les triangulations ne sont plus uniformes mais distribuées selon un modèle d’Ising. Une partie importante de la ...

05C30 ; 05C10 ; 05C81 ; 60D05 ; 60B10

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Let $X_{n}$ be an ensemble of combinatorial structures of size $N$, equipped with a measure. Consider the algorithmic problem of exactly sampling from this measure. When this ensemble has a ‘combinatorial specification, the celebrated Boltzmann sampling algorithm allows to solve this problem with a complexity which is, typically, of order $N(3/2)$. Here, a factor $N$ is inherent to the problem, and implied by the Shannon bound on the average number of required random bits, while the extra factor $N$.
Let $X_{n}$ be an ensemble of combinatorial structures of size $N$, equipped with a measure. Consider the algorithmic problem of exactly sampling from this measure. When this ensemble has a ‘combinatorial specification, the celebrated Boltzmann sampling algorithm allows to solve this problem with a complexity which is, typically, of order $N(3/2)$. Here, a factor $N$ is inherent to the problem, and implied by the Shannon bound on the average ...

05A15 ; 05A05 ; 05A18 ; 05C30

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