Welcome

• Michel Bauer (IPhT)
• Anne Itzykson
• Raphaël Itzykson

Counting doodles

• Jean-Bernard Zuber (LPTHE, Université Pierre et Marie Curie)

The Potts model on planar maps

• Mireille Bousquet-Mélou (CNRS, LaBRI, Université de Bordeaux)

(joint work with Olivier Bernardi, Brandeis University) Let q be an integer. We address the enumeration of q-colored planar maps (planar graphs embedded in the sphere), counted by the total number of edges and the number of monochromatic edges (those that have the same colour at both ends). In physics terms, we are averaging the partition function of the Potts model over all maps of a given size. We prove that the associated generating function is algebraic when q is of the form 2 + 2 cos(jπ/m), for integers j and m (but distinct from 0 and 4). This includes the two integer values q = 2 and q = 3, for which we give explicit algebraic equations. For a generic value of q, we prove that the generating function satisfies an explicit system of differential equations. Both results hold as well for planar triangulations, with a strikingly similar system of differential equations. The starting point of our approach is a recursive construction of q-coloured maps, in the spirit of what W. Tutte did in the seventies and eighties for properly coloured triangulations. This model has also been addressed by other authors and other methods (Bonnet & Eynard in 1999, and more recently Guionnet, Jones, Shlyakhtenko & Zinn-Justin, and Borot, Bouttier & Guitter), but our results are of a different nature and seem more explicit.

Exponential bounds on the number of 3d causal triangulations

• Thordur Jonsson (University of Iceland)

We discuss the problem of bounding the number of distinct triangulations of the 3-dimensional sphere. We prove that the number of causal triangulations of the 3-sphere is bounded by an exponential function of the number of tetrahedra. We describe how the argument might be extended to 4 dimensions.

Extracting Hidden Hierarchies in Weighted Distribution Networks

• Eleni Katifori (University of Pennsylvania and Max Planck Institute for Dynamics and Self-Organization)

Natural and man-made transport webs are frequently dominated by dense sets of nested cycles. The architecture of these networks, as defined by the topology and edge weights, determines how efficiently the networks perform their function. Yet, the set of tools that can characterize such a weighted cycle-rich architecture in a physically relevant, mathematically compact way is sparse. In order to fill this void, we have developed an algorithm that hierarchically decomposes (filters) the graph and characterizes the graph based on the decomposition process. The algorithm starts by identifying a physically meaningful tiling of the network, proceeds to sequentially remove the weakest links as determined by the edge weight, merge neighboring plaquettes and and finally produces a tree characterizing this merging process. The properties of this characteristic tree can provide the physical and topological data required to describe the architecture of the network and to build physical models. We show how this new algorithm can be used for automated phenotypic characterization of 2D and 3D weighted networks the structure of which is dominated by cycles, such as the vasculature of leaves and complex transportation webs.

Pseudoknots and Knots in RNA

• Henri Orland (IPhT)

We present some algorithms for the prediction of RNA pseudo knots based on their topological classification. In addition, we present an analysis of knots in RNA and show that unlike DNA or proteins where they are abundant, they are probably non-existent in RNA.

Conformal representations of Random Maps and Surfaces

• Steffen Rohde (University of Washington)

Whereas the combinatorial and metric structure of random maps are somewhat understood, a mathematical understanding of their conformal structure is still missing. Motivated by the quest to find a "conformal map" from the Riemann sphere to a Liouville quantum-gravity sphere, I will talk about uniformization of discrete maps (based on joint work with Don Marshall), and discuss an analog of Mario Bonk's carpet uniformization in the setting of the Conformal Loop Ensemble CLE (based on joint work with Brent Werness).

Generalized & Quantum SLE Multifractality

• Bertrand Duplantier (IPhT)

We describe some recent advances in the multifractality of the Schramm-Loewner evolution. This includes a generalized notion of integral means spectrum for unbounded whole-plane SLE, depending on two moments, with a phase transition between a new spectrum and the usual spectrum, originally obtained via quantum gravity. Conversely, we show that any such multifractal spectrum in the plane has a universal analogue in Liouville quantum gravity, with specific applications to SLE. Based on joint works with H. Ho, B. Le & M. Zinsmeister ; and G. Borot & J. Miller.

Bijective proof of Hurwitz formula

• Dominique Poulalhon (LIAFA, Université Paris Diderot)

In 1891, Hurwitz gives a formula for the number of some branched coverings of the sphere by itself, corresponding to transitive \(m\)-tuples of transpositions such that their product has a prescribed cycle type. I will present a combinatorial proof of this quite simple formula, reminiscent of combinatorial constructions for planar maps. This proof explains in particular why, as formulas for planar maps involve numbers of plane trees (aka Catalan trees), Hurwitz formula involves the number of Cayley trees, a fact that the many previous proofs of the formula do not explain. Moreover, the construction can be extended to a more general setting, leading to results about double Hurwitz numbers of any genus. This is a joint work with Enrica Duchi and Gilles Schaeffer.

A bivariate two-point function for planar bicolored maps

• Emmanuel Guitter (IPhT)

I will show how to compute the distance-dependent two-point function of vertex-bicolored planar maps with, in addition to the usual control on the faces degrees, a separate control on the numbers of vertices of both colors. This bivariate two-point function is obtained via a technique of slice decomposition and by use of the Stieltjes-type continued fraction formalism. If time allows, I will also discuss another bivariate two-point function in relation with another type of continued fractions. This is joint work with Éric Fusy.

Graphene Statistical Mechanics

• Mark Bowick (Syracuse University)

Graphene provides an ideal system to test the statistical mechanics of thermally fluctuating elastic membranes. The high Young’s modulus of graphene means that thermal fluctuations over even small length scales significantly stiffen the renormalized bending rigidity. We study the effect of thermal fluctuations on graphene ribbons of width W and length L, pinned at one end, via coarse-grained Molecular Dynamics simulations and compare with analytic predictions of the scaling of width-averaged root-mean-squared height fluctuations as a function of distance along the ribbon. Scaling collapse as a function of W and L also allows us to extract the scaling exponent eta governing the long-wavelength stiffening of the bending rigidity. A full understanding of the geometry-dependent mechanical properties of graphene, including arrays of cuts, may allow the design of a variety of modular elements with desired mechanical properties starting from pure graphene alone.

Scaling laws for large deviations in Voronoi tessellations

• Henk Hilhorst (LPT, Université Paris-Sud)

In 1984 Drouffe and Itzykson asked about the probability that a randomly picked two-dimensional Poisson-Voronoi cell have exactly n edges. I will discuss the answer to this question, obtained in 2005, and the developments that have taken place since, including recent results.

Quantum space time engineering

• Bianca Dittrich (Perimeter Institute)

We will consider lattice gravity approaches such as loop quantum gravity and spin foams, in which quantum geometry is defined via the assignment of quantum geometric variables to a (fixed) lattice. The main problem is then to construct the refinement limit and in this way to loose the dependence on the choice of lattice. I will discuss a renormalization framework in which such a refinement limit can be constructed and present first results for a range of (toy) theories.

Random Tensors

• Vincent Rivasseau (LPT, Université Paris-Sud)

Random tensors generalize random matrices and group field theory.Their Feynman perturbative expansion sums over all manifolds and a restricted class of quasi-manifolds, hence may be used to probe random geometries in higher dimensions. We shall review the 1/N expansion of random tensors. It is indexed by a new parameter (Gurau's degree) which is not a topological invariant of the underlying manifold, hence suggests new questions in enumerative combinatorics. Finally we shall discuss briefly the associated class of non-local tensor quantum field theories which generalize non-commutative field theories.

Fixed-energy harmonic functions and acyclic orientations

• Richard Kenyon (Brown University)

Hurwitz numbers and matrix models

• Leonid Chekhov (Steklov Mathematical Institute)

Hurwitz numbers enumerate combinatorial classes of mapping of genus g Riemann surfaces on the complex projective line with branchings at a fixed number of points (at three points for the case of Belyi pairs and Grothendieck's dessins d'enfant and at n points for hypergeometric Hurwitz numbers). The first variant of a matrix-model description of such mappings was proposed by Itzykson and Di Francesco in 1993. We construct the matrix model describing the general situation of n branching points with ramification data fixed at two of them. All these models are tau functions of the KP hierarchy and upon some constraints on their generating functions their solutions can be attained using the topological recursion technique for special chains of Hermitian matrices. The corresponding systems are conjecturally related to TQFTs. (Based on joint papers with Jan Ambjørn, NBI, Copenhagen)

Claude, the early days

• Edouard Brézin (Ecole normale supérieure)

Liouville quantum gravity on Riemann surfaces

• Rémi Rhodes (Université Paris-Est)

I will present a generic way to construct rigorously Liouville quantum field theory on Riemann surfaces with emphasis on the case of the Riemann sphere. The construction is based on Polyakov’s functional integral and yield non trivial conformal field theories. Then I will explain its main properties, the relation with the uniformization theorem for 2d Riemann surfaces and relate it via precise conjectures to the scaling limit of random planar maps conformally embedded onto the Riemann sphere. Based on joint works with F. David, Y. Huang, A. Kupiainen, H. Lacoin, V. Vargas.

Causal Dynamical Triangulations in 4D - the plot thickens

• Renate Loll (Radboud University, Nijmegen)

Causal Dynamical Triangulations (CDT) is a framework for defining a nonperturbative path integral for quantum gravity, based on random geometries with a built-in space-time anisotropy, related to the presence of a local causal (Lorentzian) structure. In four dimensions, several nontrivial results highlight that CDT is a serious contender for THE theory of quantum gravity, including the presence of second-order phase transitions and the demonstration that a renormalization group analysis can be performed despite the absence of a fixed background. I will report on recent results.

Some practical problems about planar graphs: time evolution and typology.

• Marc Barthelemy (IPhT)

Planar graphs pervade many aspects of science: they are the subject of numerous studies in graph theory, in combinatorics, in quantum gravity, and in biology and botanics. Planar networks are also extensively used to represent various infrastructure networks. In particular, transportation networks and streets patterns are the subject of many studies that are trying to characterize both topological (degree distribution, clustering, etc.) and geometrical (angles, segment length, face area distribution, etc.) aspects of these networks. I will illustrate in this talk some of the problems that are encountered in these studies such as characterizing the structure of simplest paths, how to describe the time evolution of road networks and the possibility of a typology of street patterns.

The field theory of avalanches

• Kay Wiese (Ecole normale supérieure)

When elastic systems like contact lines on a rough substrate, domain walls in disordered magnets, or tectonic plates are driven slowly, they remain immobile most of the time, before responding with strong intermittent motion, termed avalanche. I will describe the field theory behind these phenomena, explain why its effective action has a cusp, and how such intricate objects as the temporal shape of an avalanche can be obtained.

The circle packing of random hyperbolic triangulations

• Asaf Nachmias (University of British Columbia and Tel Aviv University)

We study random hyperbolic planar triangulations via their circle packing embedding in order to connect their geometry to that of the hyperbolic plane. This leads to several results: Identification of the Poisson and geometric boundaries, a connection between hyperbolicity and a form of non-amenability, and a new proof of the Benjamini-Schramm recurrence result. Based on works with subsets of Omer Angel, Martin Barlow, Ori Gurel-Gurevich, Tom Hutchcroft and Gourab Ray.

Peeling of infinite Boltzmann planar maps

• Timothy Budd (Niels Bohr Institute)

For a long time it has been known that distances in random surfaces can be conveniently studied by considering associated peeling processes. Inspired by recent results by Curien and Le Gall in the case of random triangulations, I will give a very simple description of a particular peeling process, and its scaling limit, in the general setting of infinite Boltzmann planar maps (IBPM), where one includes independent Boltzmann weights on the faces of different degrees. I will show how using this description one can (at least heuristically) derive many explicit asymptotic relations between various quantities associated to the IBPM, including graph distance, dual graph distance, first-passage time and hop count.

Random planar geometry

• Jean-François Le Gall (Université Paris-Sud et Institut universitaire de France)

We will survey recent results showing that the random metric space called the Brownian map appears as the continuous limit of various classes of large discrete random graphs embedded in the plane. These results indicate that the Brownian map is a universal model of random geometry in two dimensions, which has fractal dimension four although it has the topology of the sphere. If time permits, we will also discuss a recent work with Nicolas Curien, which shows that the Brownian map still appears when one considers local perturbations of the graph distance, for instance when one assigns random lengths to the edges of the graphs.