Local limits of uniform triangulations in high genus

We prove a conjecture of Benjamini and Curien stating that the local limits of uniform random triangulations whose genus is proportional to the number of faces are the Planar Stochastic Hyperbolic Triangulations (PSHT) defined in arXiv:1401.3297. The proof relies on a combinatorial argument and the Goulden--Jackson recurrence relation to obtain tightness, and probabilistic arguments showing the uniqueness of the limit. As a consequence, we obtain asymptotics up to subexponential factors on the number of triangulations when both the size and the genus go to infinity. As a part of our proof, we also obtain the following result of independent interest: if a random triangulation of the plane $T$ is weakly Markovian in the sense that the probability to observe a finite triangulation $t$ around the root only depends on the perimeter and volume of $t$, then $T$ is a mixture of PSHT.Comment: 36 pages, 10 figure

The generating function of planar Eulerian orientations

37 pp.International audienceThe enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in bijection with properly 3-coloured quadrangulations, while in physics they correspond to configurations of the ice model.We solve both problems -- namely the enumeration of planarEulerian orientations and of 4-valent planar Eulerian orientations --by expressing the associated generating functions as the inverses (for the composition of series) of simple hypergeometric series. Using these expressions, we derive the asymptotic behaviour of the number of planar Eulerian orientations, thus proving earlier predictions of Kostov, Zinn-Justin, Elvey Price and Guttmann. This behaviour, $\mu^n /(n \log n)^2$, prevents the associated generating functions from being D-finite. Still, these generating functions are differentially algebraic, as they satisfy non-linear differential equations of order 2. Differential algebraicity has recently been proved for other map problems, in particular for maps equipped with a Potts model.Our solutions mix recursive and bijective ingredients. In particular, a preliminary bijection transforms our oriented maps into maps carrying a height function on their vertices.In the 4-valent case, we also observe an unexpected connection with theenumeration of maps equipped with a spanning tree that is internallyinactive in the sense of Tutte. This connection remains to beexplained combinatorially

Random cubic planar graphs converge to the Brownian sphere

In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere. The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances. Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere.Comment: 55 page

B-spline-like bases for C2C^2 cubics on the Powell-Sabin 12-split

For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a B\'ezier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for $C^0$-, $C^1$-, and $C^2$-smoothness are derived

Enumerative Combinatorics

Enumerative Combinatorics focusses on the exact and asymptotic counting of combinatorial objects. It is strongly connected to the probabilistic analysis of large combinatorial structures and has fruitful connections to several disciplines, including statistical physics, algebraic combinatorics, graph theory and computer science. This workshop brought together experts from all these various fields, including also computer algebra, with the goal of promoting cooperation and interaction among researchers with largely varying backgrounds

The cartography of time-changing phenomena: the animated map

This research examines the role of the animated film in the portrayal of time series data, specifically change in the British population. It concentrates on cartographic animation and first reviews techniques developed thus far for computer-animated generation of maps for films. In order to generate an animated film, time; series data is first needed. Existing sources of time series data are shown to contain serious deficiencies for this purpose, and thus a new set of population data is generated for Britain throughout the period 1901 - 1971, and based on the Census. Ways of presenting change in this data set are then examined. Conventional methods of measuring change in the population, whilst satisfactory in static cartography, have definite limitations when used in animated cartography. Two methods, based on population density and on expected change in the population, are developed and the results mapped. As with conventional methods of measuring change, standard cartographic techniques may not be used in animated filming with any degree of success, and the resultant film shows significant departures from accepted cartographic theory. The method of film production is then examined, from the compilation of the maps themselves, through the use of the microfilm plotter in generation of the film, to the final sound tracking. The resultant film is enclosed with the thesis; the final chapter examines the success of this film. Whilst significant imperfections are shown in this example, it is concluded that the animated film has a role to play in the portrayal of time series data