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21
Lectures on random planar curves and SchrammLoewner evolutions
, 2003
"... The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some twodimensional random curves. The (distinguished) audience of the SaintFlour summer school consists mainly of probabilist ..."
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Cited by 88 (6 self)
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The goal of these lectures is to review some of the mathematical results that have been derived in the last years on conformal invariance, scaling limits and properties of some twodimensional random curves. The (distinguished) audience of the SaintFlour summer school consists mainly of probabilists and I therefore assume knowledge in stochastic calculus (Itô’s formula etc.), but no special background in basic complex analysis. These lecture notes are neither a book nor a compilation of research papers. While preparing them, I realized that it was hopeless to present all the recent results on this subject, or even to give the complete detailed proofs of a selected portion of them. Maybe this will disappoint part of the audience but the main goal of these lectures will be to try to transmit some ideas and heuristics. As a reader/part of an audience, I often think that omitting details is dangerous, and that ideas are sometimes better understood when the complete proofs are given, but in the present case, partly because the technicalities often use complex analysis tools that the audience might not be so
Optimal Coding and Sampling of Triangulations
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
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Cited by 39 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
Planar Maps as Labeled Mobiles
 Electronic Journal of Combinatorics
"... Gif sur Yvette Cedex, France We extend Schaeffer’s bijection between rooted quadrangulations and welllabeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the c ..."
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Cited by 34 (1 self)
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Gif sur Yvette Cedex, France We extend Schaeffer’s bijection between rooted quadrangulations and welllabeled trees to the general case of Eulerian planar maps with prescribed face valences to obtain a bijection with a new class of labeled trees, which we call mobiles. Our bijection covers all the classes of maps previously enumerated by either the twomatrix model used by physicists or by the bijection with blossom trees used by combinatorists. Our bijection reduces the enumeration of maps to that, much simpler, of mobiles and moreover keeps track of the geodesic distance within the initial maps via the mobiles ’ labels. Generating functions for mobiles are shown to obey systems of algebraic recursion relations.
Conformally invariant scaling limits: an overview and collection of open problems
 Proceedings of the International Congress of Mathematicians, Madrid (M. SanzSolé et
, 2007
"... Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions beco ..."
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Cited by 18 (1 self)
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Abstract. Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a oneparameter family of random curves called Stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been understood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
2D Quantum Gravity, Matrix Models and Graph Combinatorics, preprint, Applications of random matrices in physics
, 2006
"... 1.1. Matrix models per se......................... 2 1.2. A brief history............................ 3 2. Matrix models for 2D quantum gravity................... 4 2.1. Discrete 2D quantum gravity..................... 4 ..."
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Cited by 10 (0 self)
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1.1. Matrix models per se......................... 2 1.2. A brief history............................ 3 2. Matrix models for 2D quantum gravity................... 4 2.1. Discrete 2D quantum gravity..................... 4
Combinatorics of Hard Particles on Planar Graphs, Nucl.Phys
 A: Math.Gen
"... Gif sur Yvette Cedex, France We revisit the problem of hard particles on planar random tetravalent graphs in view of recent combinatorial techniques relating planar diagrams to decorated trees. We show how to recover the twomatrix model solution to this problem in this purely combinatorial language ..."
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Cited by 9 (4 self)
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Gif sur Yvette Cedex, France We revisit the problem of hard particles on planar random tetravalent graphs in view of recent combinatorial techniques relating planar diagrams to decorated trees. We show how to recover the twomatrix model solution to this problem in this purely combinatorial language.
Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices
"... Abstract. In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large N, of the logarithm of the partition function of N × N Hermitian random matrices. These coefficients are generating functions for graphical enumera ..."
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Cited by 7 (3 self)
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Abstract. In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large N, of the logarithm of the partition function of N × N Hermitian random matrices. These coefficients are generating functions for graphical enumeration on Riemann surfaces. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree 2ν. The coupling parameter for this term plays the role of the independent dynamical variable in the differential equations. From these equations one may deduce functional analytic characterizations of the coefficients in the asymptotic expansion. Moreover, this ode system can be solved recursively to explicitly construct these coefficients as functions of the coupling parameter. This analysis of the fine structure of the asymptotic coefficients can be extended to multiple coupling parameters and we present a limited illustration of this for the case of two parameters. 1. Motivation
Combinatorial aspects of matrix models
 ALEA LAT. AM. J. PROBAB. MATH. STAT
, 2005
"... We show that under reasonably general assumptions, the first order asymptotics of the free energy of matrix models are generating functions for colored planar maps. This is based on the fact that solutions of the differential SchwingerDyson equations are, by nature, generating functions for enumera ..."
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Cited by 7 (3 self)
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We show that under reasonably general assumptions, the first order asymptotics of the free energy of matrix models are generating functions for colored planar maps. This is based on the fact that solutions of the differential SchwingerDyson equations are, by nature, generating functions for enumerating planar maps, a remark which bypasses the use of Gaussian calculus.
The threepoint function of planar quadrangulations
, 2008
"... We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete threepoint function converges to a simple universal scaling function, which is the continuous threepoint f ..."
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Cited by 6 (2 self)
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We compute the generating function of random planar quadrangulations with three marked vertices at prescribed pairwise distances. In the scaling limit of large quadrangulations, this discrete threepoint function converges to a simple universal scaling function, which is the continuous threepoint function of pure 2D quantum gravity. We give explicit expressions for this universal threepoint function both in the grandcanonical and canonical ensembles. Various limiting regimes are studied when some of the distances become large or small. By considering the case where the marked vertices are aligned, we also obtain the probability law for the number of geodesic points, namely vertices that lie on a geodesic path between two given vertices, and at prescribed distances from these vertices. 1.
Geodesic Distance in Planar Graphs: An Integrable Approach
 The Ramanujan Journal
"... Gif sur Yvette Cedex, France We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the graph. The corresponding generating functions obey nonlinear recursion relations on the geodesic dist ..."
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Cited by 5 (1 self)
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Gif sur Yvette Cedex, France We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the graph. The corresponding generating functions obey nonlinear recursion relations on the geodesic distance. These are solved by use of stationary multisoliton taufunctions of suitable reductions of the KP hierarchy. We obtain a unified formulation of the (multi) critical continuum limit describing large graphs with marked points at large geodesic distances, and obtain integrable differential equations for the corresponding scaling functions. This provides a continuum formulation of twodimensional quantum gravity, in terms of the geodesic distance.