Centrally symmetric convex polyhedra with regular polygonal faces

First we prove that the class $C_{I}$ of centrally symmetric convex polyhedra with regular polygonal faces consists of 4 of the 5 Platonic, 9 of the 13 Archimedean, 13 of the 92 Johnson solids and two infinite families of $2n$-prisms and $(2n+1)$-antiprisms. Then we show how the presented maps of their halves (obtained by identification of all pairs of antipodal points) in the projective plane can be used for obtaining their flag graphs and symmetry-type graphs. Finally, we study some linear dependence relations between polyhedra of the class $C_{I}$

Asymptotic study of regular planar graphs

The central topic of this dissertation is the study of some families of regular planar graphs and maps. We are in particular interested in their asymptotic enumeration in order to understand of the associated uniform random model. In a first part, we give both an exact and an asymptotic enumeration of labelled cubic planar graphs, multigraphs and simple maps, via a recursive scheme following the iterative decompositon of a graph in smaller components of higher connecttivity. In the second part, we apply those results to the study a the uniform random labelled cubic planar graph. We compute for instance the probability of connectivity, and prove that some significant parameters are distributed following a Gaussian limit law: the numbers of cut-vertices, isthmuses, blocks, cherries, near-bricks, and triangles. In the third and last part, we develop the first recursive combinatorial scheme to enumerate 4-regular labelled planar graphs. This scheme is based on a decomposition in terms of connectivity, similar to that of cubic planar graphs, which leads to the exact enumeration of 4-regular planar graphs and simple maps.Das zentrale Thema dieser Dissertation sind Familien von regulären planaren Graphen und Karten. Insbesondere sind wir an daran interessiert, diese zu zählen und die Zusammenhänge zu deren zufälligen Gegenstücken zu erforschen. Im ersten Teil geben wir sowohl eine rekursive als auch eine asymptotische Abzählung von kubischen, planaren Graphen, Multigraphen und einfachen Karten, durch eine Dekomposition entlang deren Komponenten. Im zweiten Teil wenden wir diese Resultate auf zufällige kubische planare Graphen an. Insbesondere berechnen wir die Wahrscheinlichkeit von Zusammenhängigkeit, und beweisen das einige bedeutende Parameter normalverteilt sind: die Anzahl der cut-vertices, isthmuses, Blöcke, cherries, near-bricks und Dreiecke. Im dritten und letzten Teil entwickeln wir das erste kombinatorisches Schema, basierend auf einem Dekompositionsschema das ähnlich zu dem im Kontext von kubischen planaren Graphen ist, das zur rekursiven Abzählung von 4-regulären planaren Graphen und einfachen Karten führt

Gibbs and Quantum Discrete Spaces

Gibbs measure is one of the central objects of the modern probability, mathematical statistical physics and euclidean quantum field theory. Here we define and study its natural generalization for the case when the space, where the random field is defined is itself random. Moreover, this randomness is not given apriori and independently of the configuration, but rather they depend on each other, and both are given by Gibbs procedure; We call the resulting object a Gibbs family because it parametrizes Gibbs fields on different graphs in the support of the distribution. We study also quantum (KMS) analog of Gibbs families. Various applications to discrete quantum gravity are given.Comment: 37 pages, 2 figure

The Graphicahedron

The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph of the symmetric group S_p and then construct a vertex-transitive simple polytope of rank q, called the graphicahedron, whose 1-skeleton (edge graph) is the Cayley graph. The graphicahedron of a graph G is a generalization of the well-known permutahedron; the latter is obtained when the graph is a path. We also discuss symmetry properties of the graphicahedron and determine its structure when G is small.Comment: 21 pages (European Journal of Combinatorics, to appear

Quantum ergodicity for quantum graphs without back-scattering

We give an estimate of the quantum variance for $d$-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random $d$-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds.Comment: 28 pages, 5 figure

Asymptotic enumeration of non-crossing partitions on surfaces

We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface S and introduce the number CS(n) of noncrossing partitions of a set of n points laying on the boundary of SPostprint (author's final draft