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Добірка наукової літератури з теми "Énumération asymptotique"
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Статті в журналах з теми "Énumération asymptotique"
Gilmore, Tomack. "Enumerating some symmetry classes of rhombus tilings of holey hexagons." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 27th..., Proceedings (January 1, 2015). http://dx.doi.org/10.46298/dmtcs.2473.
Повний текст джерелаFang, Wenjie. "A generalization of the quadrangulation relation to constellations and hypermaps." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AS,..., Proceedings (January 1, 2013). http://dx.doi.org/10.46298/dmtcs.12789.
Повний текст джерелаДисертації з теми "Énumération asymptotique"
Yakovlev, Ivan. "Graphes en rubans métriques." Electronic Thesis or Diss., Bordeaux, 2024. http://www.theses.fr/2024BORD0143.
Повний текст джерелаThis thesis presents several contributions to the study of counting functions for metric ribbon graphs. Ribbon graphs, also known as combinatorial maps, are cellular embeddings of graphs in surfaces modulo homeomorphisms. They are combinatorial objects that can be represented as gluings of polygons or factorizations of permutations. Metric on a ribbon graph is an assignment of positive lengths to its edges. The counting functions give the number of integral metric ribbon graphs with fixed combinatorics (genus of the surface, degrees of vertices, number of boundaries) as a function of the perimeters of the boundaries. Our approach to their study is purely combinatorial and relies on bijections and surgeries for ribbon graphs. Firstly, we show that these functions are piecewise (quasi-)polynomials, specifying exactly the regions of (quasi-)polynomiality. We then study the cases when their top-degree terms are honest polynomials. Our interest in such cases comes from the fact that the corresponding polynomials can be used for refined enumeration of square-tiled surfaces, which correspond to integer points in the strata of (half-)translations surfaces (equivalently, strata of differentials on Riemann surfaces). Consequently, one can give refined/alternative formulas for Masur-Veech volumes of strata. One known example are the Kontsevich polynomials, counting trivalent metric ribbon graphs of given genus and perimeters of boundaries. They were recently used by Delecroix, Goujard, Zograf and Zorich to give a combinatorial formula for the volumes of principal strata of quadratic differentials. We concentrate on face-bipartite metric ribbon graphs, which appear in the study of Abelian differentials. We show that in the case of one-vertex graphs the top-degree terms of the counting functions on certain subspaces are in fact (explicit) polynomials. As a consequence, we deduce the generating function for the contributions of n-cylinder square-tiled surfaces to the volumes of minimal strata of Abelian differentials, refining a previous result of Sauvaget. We then present a similar polynomiality result for the two subfamilies of graphs corresponding to even/odd spin connected components of the minimal strata. This also gives a refinement of a formula for the corresponding volume differences previously obtained by Chen, Möller, Sauvaget and Zagier. Next we conjecture that the polynomiality phenomenon holds for families of graphs with several vertices, if each graph is weighted by the count of certain spanning trees. We prove the conjecture in the planar case. In the process, we construct families of plane trees which correspond to certain triangulations of the product of two simlpices, which are interesting from the point of view of the theory of polytopes. Finally, we present a contribution to a joint work with Duryev and Goujard, where the combinatorial formula of Delecroix, Goujard, Zograf and Zorich is generalized to all strata of quadratic differentials with odd singularities. The contribution is a combinatorial proof of the formula for coefficients counting certain degenerations of (non-face-bipartite) metric ribbon graphs
Curien, Nicolas. "Etude asymptotique de grands objets combinatoires aléatoires." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00607721.
Повний текст джерелаBorot, Gaetan. "Quelques problèmes de géométrie énumérative, de matrices aléatoires, d'intégrabilité, étudiés via la géométrie des surfaces de Riemann." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00625776.
Повний текст джерелаBorot, Gaëtan. "Quelques problèmes de géométrie énumérative, de matrices aléatoires, d'intégrabilité, étudiés via la géométrie des surfaces de Riemann." Thesis, Paris 11, 2011. http://www.theses.fr/2011PA112092/document.
Повний текст джерелаComplex 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.First, we consider all order large N asymptotics in some N-dimensional integrals coming from random matrix theory, or more generally from "log gases" problems. We shall explain how to use loop equations to establish those asymptotics in beta matrix models within a one cut regime. This can be applied in the study of large fluctuations of the maximum eigenvalue in beta matrix models, and lead us to heuristic predictions about the asymptotics of Tracy-Widom beta law to all order, and for all positive beta. Second, we study the interplay between integrability and loop equations. As a corollary, we are able to prove the previous prediction about the asymptotics to all order of Tracy-Widom law for hermitian matrices.We move on with the solution of some combinatorial problems in all topologies. In topological string theory, a conjecture from Bouchard, Klemm, Mariño and Pasquetti states that certain generating series of Gromov-Witten invariants in toric Calabi-Yau threefolds, are solutions of loop equations. We have proved this conjecture in the simplest case, where those invariants coincide with the "simple Hurwitz numbers". We also explain recent progress towards the general conjecture, in relation with our work. In statistical physics on the random lattice, we have solved the trivalent O(n) model introduced by Kostov, and we explain the method to solve more general statistical models.Throughout the thesis, the computation of some "generalized matrices integrals" appears to be increasingly important for future applications, and this appeals for a general theory of loop equations
Courtiel, Julien. "Combinatoire du polynôme de Tutte et des cartes planaires." Thesis, Bordeaux, 2014. http://www.theses.fr/2014BORD0083/document.
Повний текст джерелаThis thesis deals with the Tutte polynomial, studied from different points of view. In the first part, we address the enumeration of planar maps equipped with a spanning forest, here called forested maps, with a weight z per face and a weight u per non-root component of the forest. Equivalently, we count (with respect to the number of faces) the planar maps C weighted by TC(u + 1; 1), where TC is the Tutte polynomial of C.We begin by a purely combinatorial characterization of the corresponding generating function, denoted by F(z; u). We deduce from this that F(z; u) is differentially algebraic in z, that is, satisfies a polynomial differential equation in z. Finally, for u ≥ -1, we study the asymptotic behaviour of the nth coefficient of F(z; u).We observe a phase transition at 0, with a very unusual regime in n-3 ln-2(n) for u ϵ [-1; 0[, which testifiesa new universality class for planar maps. In the second part, we propose a framework unifying the notions of activity used in the literature to describe the Tutte polynomial. The new notion of activity thereby defined is called Δ-activity. It gathers all the notions of activities that were already known and has nice properties, as Crapo’s property that defines a partition of the lattice of the spanning subgraphs into intervals with respect to the activity. Lastly we conjecture that every activity that describes the Tutte polynomial and that satisfies Crapo’s property can be defined in terms of Δ-activity