Rapport sur les contributions - 20ème conférence Claude Itzykson - Random Surfaces and Random Geometry - IN2P3
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20ème conférence Claude
Itzykson - Random Surfaces
and Random Geometry
Rapport sur les
contributions
https://indico.in2p3.fr/e/itzykson201520ème conférenc … / Rapport sur les contributions Random Tensors
ID de Contribution: 0 Type: Invited talk
Random Tensors
jeudi 11 juin 2015 15:15 (45 minutes)
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 general-
ize non-commutative field theories.
Auteur principal: RIVASSEAU, Vincent (LPT, Université Paris-Sud)
Orateur: RIVASSEAU, Vincent (LPT, Université Paris-Sud)
December 28, 2021 Page 120ème conférenc … / Rapport sur les contributions Scaling laws for large deviations in …
ID de Contribution: 1 Type: Invited talk
Scaling laws for large deviations in Voronoi
tessellations
jeudi 11 juin 2015 12:15 (45 minutes)
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.
Auteur principal: HILHORST, Henk (LPT, Université Paris-Sud)
Orateur: HILHORST, Henk (LPT, Université Paris-Sud)
December 28, 2021 Page 220ème conférenc … / Rapport sur les contributions The Potts model on planar maps
ID de Contribution: 2 Type: Invited talk
The Potts model on planar maps
mercredi 10 juin 2015 11:30 (45 minutes)
(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.
Auteur principal: BOUSQUET-MÉLOU, Mireille (CNRS, LaBRI, Université de Bordeaux)
Orateur: BOUSQUET-MÉLOU, Mireille (CNRS, LaBRI, Université de Bordeaux)
December 28, 2021 Page 320ème conférenc … / Rapport sur les contributions Graphene Statistical Mechanics
ID de Contribution: 3 Type: Invited talk
Graphene Statistical Mechanics
jeudi 11 juin 2015 11:30 (45 minutes)
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 rib-
bon. 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.
Auteur principal: BOWICK, Mark (Syracuse University)
Orateur: BOWICK, Mark (Syracuse University)
December 28, 2021 Page 420ème conférenc … / Rapport sur les contributions Hurwitz numbers and matrix models
ID de Contribution: 4 Type: Invited talk
Hurwitz numbers and matrix models
jeudi 11 juin 2015 17:15 (45 minutes)
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 Hur-
witz 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 func-
tions 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)
Auteur principal: CHEKHOV, Leonid (Steklov Mathematical Institute)
Orateur: CHEKHOV, Leonid (Steklov Mathematical Institute)
December 28, 2021 Page 520ème conférenc … / Rapport sur les contributions Random planar geometry
ID de Contribution: 5 Type: Invited talk
Random planar geometry
vendredi 12 juin 2015 17:15 (45 minutes)
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.
Auteur principal: LE GALL, Jean-François (Université Paris-Sud et Institut universitaire de France)
Orateur: LE GALL, Jean-François (Université Paris-Sud et Institut universitaire de France)
December 28, 2021 Page 620ème conférenc … / Rapport sur les contributions Liouville quantum gravity on Riem …
ID de Contribution: 6 Type: Invited talk
Liouville quantum gravity on Riemann surfaces
vendredi 12 juin 2015 10:15 (45 minutes)
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.
Auteur principal: RHODES, Rémi (Université Paris-Est)
Orateur: RHODES, Rémi (Université Paris-Est)
December 28, 2021 Page 720ème conférenc … / Rapport sur les contributions Extracting Hidden Hierarchies in …
ID de Contribution: 7 Type: Invited talk
Extracting Hidden Hierarchies in Weighted
Distribution Networks
mercredi 10 juin 2015 14:30 (45 minutes)
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.
Auteur principal: KATIFORI, Eleni (University of Pennsylvania and Max Planck Institute for Dy-
namics and Self-Organization)
Orateur: KATIFORI, Eleni (University of Pennsylvania and Max Planck Institute for Dynamics and
Self-Organization)
December 28, 2021 Page 820ème conférenc … / Rapport sur les contributions Conformal representations of Ran …
ID de Contribution: 8 Type: Invited talk
Conformal representations of Random Maps and
Surfaces
mercredi 10 juin 2015 16:30 (45 minutes)
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).
Auteur principal: ROHDE, Steffen (University of Washington)
Orateur: ROHDE, Steffen (University of Washington)
December 28, 2021 Page 920ème conférenc … / Rapport sur les contributions Peeling of infinite Boltzmann plan …
ID de Contribution: 9 Type: Invited talk
Peeling of infinite Boltzmann planar maps
vendredi 12 juin 2015 16:30 (45 minutes)
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.
Auteur principal: BUDD, Timothy (Niels Bohr Institute)
Orateur: BUDD, Timothy (Niels Bohr Institute)
December 28, 2021 Page 1020ème conférenc … / Rapport sur les contributions Quantum space time engineering
ID de Contribution: 10 Type: Invited talk
Quantum space time engineering
jeudi 11 juin 2015 14:30 (45 minutes)
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.
Auteur principal: DITTRICH, Bianca (Perimeter Institute)
Orateur: DITTRICH, Bianca (Perimeter Institute)
December 28, 2021 Page 1120ème conférenc … / Rapport sur les contributions Exponential bounds on the numbe …
ID de Contribution: 11 Type: Invited talk
Exponential bounds on the number of 3d causal
triangulations
mercredi 10 juin 2015 12:15 (45 minutes)
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 expo-
nential function of the number of tetrahedra. We describe how the argument might be extended
to 4 dimensions.
Auteur principal: JONSSON, Thordur (University of Iceland)
Orateur: JONSSON, Thordur (University of Iceland)
December 28, 2021 Page 1220ème conférenc … / Rapport sur les contributions Fixed-energy harmonic functions a …
ID de Contribution: 12 Type: Invited talk
Fixed-energy harmonic functions and acyclic
orientations
jeudi 11 juin 2015 16:30 (45 minutes)
Auteur principal: KENYON, Richard (Brown University)
Orateur: KENYON, Richard (Brown University)
December 28, 2021 Page 1320ème conférenc … / Rapport sur les contributions Some practical problems about pla …
ID de Contribution: 13 Type: Invited talk
Some practical problems about planar graphs: time
evolution and typology.
vendredi 12 juin 2015 12:15 (45 minutes)
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.
Auteur principal: BARTHELEMY, Marc (IPhT)
Orateur: BARTHELEMY, Marc (IPhT)
December 28, 2021 Page 1420ème conférenc … / Rapport sur les contributions Claude, the early days
ID de Contribution: 14 Type: Invited talk
Claude, the early days
vendredi 12 juin 2015 09:30 (45 minutes)
Auteur principal: BRÉZIN, Edouard (Ecole normale supérieure)
Orateur: BRÉZIN, Edouard (Ecole normale supérieure)
December 28, 2021 Page 1520ème conférenc … / Rapport sur les contributions The circle packing of random hype …
ID de Contribution: 15 Type: Invited talk
The circle packing of random hyperbolic
triangulations
vendredi 12 juin 2015 15:15 (45 minutes)
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.
Auteur principal: NACHMIAS, Asaf (University of British Columbia and Tel Aviv University)
Orateur: NACHMIAS, Asaf (University of British Columbia and Tel Aviv University)
December 28, 2021 Page 1620ème conférenc … / Rapport sur les contributions Counting doodles
ID de Contribution: 16 Type: Invited talk
Counting doodles
mercredi 10 juin 2015 10:15 (45 minutes)
Auteur principal: ZUBER, Jean-Bernard (LPTHE, Université Pierre et Marie Curie)
Orateur: ZUBER, Jean-Bernard (LPTHE, Université Pierre et Marie Curie)
December 28, 2021 Page 1720ème conférenc … / Rapport sur les contributions Causal Dynamical Triangulations i …
ID de Contribution: 17 Type: Invited talk
Causal Dynamical Triangulations in 4D - the plot
thickens
vendredi 12 juin 2015 11:30 (45 minutes)
Causal Dynamical Triangulations (CDT) is a framework for defining a nonperturbative path inte-
gral for quantum gravity, based on random geometries with a built-in space-time anisotropy, re-
lated 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.
Auteur principal: LOLL, Renate (Radboud University, Nijmegen)
Orateur: LOLL, Renate (Radboud University, Nijmegen)
December 28, 2021 Page 1820ème conférenc … / Rapport sur les contributions The field theory of avalanches
ID de Contribution: 18 Type: Invited talk
The field theory of avalanches
vendredi 12 juin 2015 14:30 (45 minutes)
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.
Auteur principal: WIESE, Kay (Ecole normale supérieure)
Orateur: WIESE, Kay (Ecole normale supérieure)
December 28, 2021 Page 1920ème conférenc … / Rapport sur les contributions Pseudoknots and Knots in RNA
ID de Contribution: 19 Type: Invited talk
Pseudoknots and Knots in RNA
mercredi 10 juin 2015 15:15 (45 minutes)
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.
Auteur principal: ORLAND, Henri (IPhT)
Orateur: ORLAND, Henri (IPhT)
December 28, 2021 Page 2020ème conférenc … / Rapport sur les contributions A bivariate two-point function for …
ID de Contribution: 20 Type: Invited talk
A bivariate two-point function for planar bicolored
maps
jeudi 11 juin 2015 10:15 (45 minutes)
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.
Auteur principal: GUITTER, Emmanuel (IPhT)
Orateur: GUITTER, Emmanuel (IPhT)
December 28, 2021 Page 2120ème conférenc … / Rapport sur les contributions Bijective proof of Hurwitz formula
ID de Contribution: 21 Type: Invited talk
Bijective proof of Hurwitz formula
jeudi 11 juin 2015 09:30 (45 minutes)
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 pre-
scribed 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. More-
over, 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.
Auteur principal: POULALHON, Dominique (LIAFA, Université Paris Diderot)
Orateur: POULALHON, Dominique (LIAFA, Université Paris Diderot)
December 28, 2021 Page 2220ème conférenc … / Rapport sur les contributions Generalized & Quantum SLE Mult …
ID de Contribution: 22 Type: Invited talk
Generalized & Quantum SLE Multifractality
mercredi 10 juin 2015 17:15 (45 minutes)
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, depend-
ing on two moments, with a phase transition between a new spectrum and the usual spectrum, orig-
inally 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.
Auteur principal: DUPLANTIER, Bertrand (IPhT)
Orateur: DUPLANTIER, Bertrand (IPhT)
December 28, 2021 Page 2320ème conférenc … / Rapport sur les contributions Welcome
ID de Contribution: 23 Type: Invited talk
Welcome
mercredi 10 juin 2015 09:45 (30 minutes)
Auteurs principaux: ITZYKSON, Anne; BAUER, Michel (IPhT); ITZKYSON, Raphaël
Orateurs: ITZYKSON, Anne; BAUER, Michel (IPhT); ITZYKSON, Raphaël
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