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27
Towards conformal invariance of 2D lattice models
 Proceedings of the international congress of mathematicians (ICM
"... Abstract. Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, selfavoiding polymers,...This has led to numerous exact (but nonrigorous) predictions of their scaling exponents and dimensions. We will discuss how to pro ..."
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Cited by 106 (11 self)
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Abstract. Many 2D lattice models of physical phenomena are conjectured to have conformally invariant scaling limits: percolation, Ising model, selfavoiding polymers,...This has led to numerous exact (but nonrigorous) predictions of their scaling exponents and dimensions. We will discuss how to prove the conformal invariance conjectures, especially in relation to SchrammLoewner Evolution.
Critical percolation exploration path and SLE6: a proof of convergence
, 2006
"... It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE6. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scal ..."
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Cited by 55 (13 self)
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It was argued by Schramm and Smirnov that the critical site percolation exploration path on the triangular lattice converges in distribution to the trace of chordal SLE6. We provide here a detailed proof, which relies on Smirnov’s theorem that crossing probabilities have a conformally invariant scaling limit (given by Cardy’s formula). The version of convergence to SLE6 that we prove suffices for the SmirnovWerner derivation of certain critical percolation crossing exponents and for our analysis of the critical percolation full scaling limit as a process of continuum nonsimple loops.
Conformally invariant scaling limits: an overview and collection of open problems
, 2006
"... 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 acces ..."
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Cited by 51 (2 self)
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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.
Discrete complex analysis and probability
 PROC. INT. CONGRESS OF MATHEMATICIANS (ICM) (HYDERABAD, INDIA) PP 565–621 (ARXIV:1009.6077
, 2010
"... We discuss possible discretizations of complex analysis and some of their applications to probability and mathematical physics, following our recent work with Dmitry Chelkak, Hugo DuminilCopin and Clément Hongler. ..."
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Cited by 16 (1 self)
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We discuss possible discretizations of complex analysis and some of their applications to probability and mathematical physics, following our recent work with Dmitry Chelkak, Hugo DuminilCopin and Clément Hongler.
Conformal radii for conformal loop ensembles
, 2009
"... The conformal loop ensembles CLEκ, defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point. Our results agree w ..."
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Cited by 16 (4 self)
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The conformal loop ensembles CLEκ, defined for 8/3 ≤ κ ≤ 8, are random collections of loops in a planar domain which are conjectured scaling limits of the O(n) loop models. We calculate the distribution of the conformal radii of the nested loops surrounding a deterministic point. Our results agree with predictions made by Cardy and Ziff and by Kenyon and Wilson for the O(n) model. We also compute the expectation dimension of the CLEκ gasket, which consists of points not surrounded by any loop, to be (8 − κ)(3κ − 8) 2 −,
Critical percolation and conformal invariance
 In: XIVth International Congress on Mathematical Physics (Lissbon
, 2003
"... Many 2D critical lattice models are believed to have conformally invariant scaling limits. This belief allowed physicists to predict (unrigorously) many of their properties, including exact values of various dimensions and scaling exponents. We describe some of the recent progress in the mathematic ..."
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Cited by 13 (2 self)
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Many 2D critical lattice models are believed to have conformally invariant scaling limits. This belief allowed physicists to predict (unrigorously) many of their properties, including exact values of various dimensions and scaling exponents. We describe some of the recent progress in the mathematical understanding of these models, using critical percolation as an example. 1.
Ising (conformal) fields and cluster area measures
, 2008
"... We provide a representation for the scaling limit of the d = 2 critical Ising magnetization field as a (conformal) random field using SLE (SchrammLoewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (FortuinKast ..."
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Cited by 8 (3 self)
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We provide a representation for the scaling limit of the d = 2 critical Ising magnetization field as a (conformal) random field using SLE (SchrammLoewner Evolution) clusters and associated renormalized area measures. The renormalized areas are from the scaling limit of the critical FK (FortuinKasteleyn) clusters and the random field is a convergent sum of the area measures with random signs. Extensions to offcritical scaling limits, to d = 3 and to Potts models are also considered.
Parafermionic observables and their applications to planar statistical physics models
, 2013
"... This volume is based on the PhD thesis of the author. Through the examples of the selfavoiding walk, the randomcluster model, the Ising model and others, the book explores in details two important techniques: 1. Discrete holomorphicity and parafermionic observables, which have been used in the pa ..."
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Cited by 7 (2 self)
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This volume is based on the PhD thesis of the author. Through the examples of the selfavoiding walk, the randomcluster model, the Ising model and others, the book explores in details two important techniques: 1. Discrete holomorphicity and parafermionic observables, which have been used in the past few years to study planar models of statistical physics (in particular their conformal invariance), such as randomcluster models and loop O(n)models. 2. The RussoSeymourWelsh theory for percolationtype models with dependence. This technique was initially available for Bernoulli percolation only. Recently, it has been extended to models with dependence, thus opening the way to a deeper study of their critical regime. The book is organized as follows. The first part provides a general introduction to planar statistical physics, as well as a first example of the parafermionic observable and its application to the computation of the connective constant for the selfavoiding walk on the hexagonal lattice. The second part deals with the family of randomcluster models. It studies the RussoSeymourWelsh theory of crossing probabilities for these models. As an application, the critical point of the randomcluster model is computed on the square lattice. Then, the parafermionic observable is introduced and two of its applications are described in detail. This part contains a chapter describing basic properties of the randomcluster model. The third part is devoted to the Ising model and its randomcluster representation, the FKIsing model. After a first chapter gathering the basic properties of the Ising model, the theory of sholomorphic functions as well as Smirnov and ChelkakSmirnov’s proofs of conformal invariance (for these two models) are presented. Conformal invariance paves the way to a better understanding of the critical phase and the two next chapters are devoted to the study of the geometry of the critical phase, as well as the relation between the critical and nearcritical phases. The last part presents possible directions of future research by describing other models and several open questions.
Quantum gravity and inventory accumulation. ArXiv eprints
, 2011
"... We begin by studying inventory accumulation at a LIFO (lastinfirstout) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory t ..."
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Cited by 6 (2 self)
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We begin by studying inventory accumulation at a LIFO (lastinfirstout) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on Z2. In more interesting versions, a p fraction of customers orders the “freshest available” product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on p. We then turn our attention to the critical FortuinKastelyn random planar map model, which gives, for each q> 0, a probability measure on random (discretized) twodimensional surfaces decorated by loops, related to the qstate Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loopdecorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformal loop ensemble, with parameters depending on q. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at p = 1/2, q = 4.