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15
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 38 (4 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.
TwoDimensional Scaling Limits via Marked Nonsimple Loops
, 2006
"... We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE6 and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of ..."
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Cited by 12 (5 self)
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We postulate the existence of a natural Poissonian marking of the double (touching) points of SLE6 and hence of the related continuum nonsimple loop process that describes macroscopic cluster boundaries in 2D critical percolation. We explain how these marked loops should yield continuum versions of nearcritical percolation, dynamical percolation, minimal spanning trees and related plane filling curves, and invasion percolation. We show that this yields for some of the continuum objects a conformal covariance property that generalizes the conformal invariance of critical systems. It is an open problem to rigorously construct the continuum objects and to prove that they are indeed the scaling limits of the corresponding lattice objects.
Geodesics in large planar maps and in the brownian map, preprint available on arxiv
, 2009
"... We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the ..."
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Cited by 6 (2 self)
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We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a noncompact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps. 1
SchrammLoewner equations driven by symmetric stable processes
 Ann. Probab
"... We consider shape, size and regularity of the hulls Kt of the chordal SchrammLoewner evolution driven by a symmetric αstable process. We obtain derivative estimates, show that the domains H \ Kt are Hölder domains, prove that Kt has Hausdorff dimension 1, and show that the trace is rightcontinuou ..."
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Cited by 4 (2 self)
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We consider shape, size and regularity of the hulls Kt of the chordal SchrammLoewner evolution driven by a symmetric αstable process. We obtain derivative estimates, show that the domains H \ Kt are Hölder domains, prove that Kt has Hausdorff dimension 1, and show that the trace is rightcontinuous with left limits almost surely.
Spacetime percolation
"... The contact model for the spread of disease may be viewed as a directed percolation model on Z×R in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The c ..."
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Cited by 3 (0 self)
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The contact model for the spread of disease may be viewed as a directed percolation model on Z×R in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the timeaxis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a randomcluster model on Z × R, with associated continuum Ising and Potts models. These models are of independent interest, in addition to providing a pathintegral representation of the quantum Ising model with transverse field. This representation may be used to obtain a bound on the entanglement of a finite set of spins in the quantum Ising model on Z, where this entanglement is measured via the entropy of the reduced density matrix. The meanfield version of the quantum Ising model gives rise to a randomcluster model on Kn × R, thereby extending the Erdős–Rényi random graph on the complete graph Kn. 1
Conformal Invariance for Certain Models of the Bond–Triangular Type
"... Abstract: Convergence to SLE6 of the percolation exploration process for a correlated bond–triangular type model studied in [5] is established, which puts the said model in the same universality class as the standard site percolation model on the triangular lattice [12]. The result is proven for all ..."
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Cited by 1 (0 self)
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Abstract: Convergence to SLE6 of the percolation exploration process for a correlated bond–triangular type model studied in [5] is established, which puts the said model in the same universality class as the standard site percolation model on the triangular lattice [12]. The result is proven for all domains with boundary (upper) Minkowski dimension less than 2, following the general streamlined approach outlined in [11].
Random even graphs and the Ising model
 Electronic J. Combin
"... We explore the relationship between the Ising model with inverse temperature β, the q = 2 randomcluster model with edgeparameter p = 1 − e −2β, and the random even subgraph with edgeparameter 1 2 p. For a planar graph G, the boundary edges of the + clusters of the Ising model on the planar dual o ..."
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Cited by 1 (1 self)
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We explore the relationship between the Ising model with inverse temperature β, the q = 2 randomcluster model with edgeparameter p = 1 − e −2β, and the random even subgraph with edgeparameter 1 2 p. For a planar graph G, the boundary edges of the + clusters of the Ising model on the planar dual of G forms a random even subgraph of G. A coupling of the random even subgraph of G and the q = 2 randomcluster model on G is presented, thus extending the above observation to general graphs. A random even subgraph of a planar lattice undergoes a phase transition at the parametervalue 1 2 pc, where pc is the critical point of the q = 2 randomcluster model on the dual lattice. These results are motivated in part by an exploration of the socalled randomcurrent method utilised by Aizenman, Barsky, Fernández and others to solve the Ising model on the ddimensional hypercubic lattice. 1
Three theorems in discrete random geometry
, 2011
"... Abstract: These notes are focused on three recent results in discrete random geometry, namely: the proof by DuminilCopin √ and Smirnov that the connective constant of the hexagonal lattice is 2 + 2; the proof by the author and Manolescu of the universality of inhomogeneous bond percolation on the s ..."
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Cited by 1 (1 self)
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Abstract: These notes are focused on three recent results in discrete random geometry, namely: the proof by DuminilCopin √ and Smirnov that the connective constant of the hexagonal lattice is 2 + 2; the proof by the author and Manolescu of the universality of inhomogeneous bond percolation on the square, triangular, and hexagonal lattices; the proof by Beffara and DuminilCopin that the critical point of the randomcluster model on Z2 is √ q/(1 + √ q). Background information on the relevant random processes is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs. AMS 2000 subject classifications: Primary 60K35; secondary 82B43. Keywords and phrases: Selfavoiding walk, connective constant, percolation, randomcluster model, Ising model, star–triangle transformation,
The Brownian cactus I. scaling limits of discrete cactuses
 Z. Wahrsch. Verw. Gebiete
, 1982
"... The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space E, one can associate an Rtree called the continuous cactus of E. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzma ..."
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The cactus of a pointed graph is a discrete tree associated with this graph. Similarly, with every pointed geodesic metric space E, one can associate an Rtree called the continuous cactus of E. We prove under general assumptions that the cactus of random planar maps distributed according to Boltzmann weights and conditioned to have a fixed large number of vertices converges in distribution to a limiting space called the Brownian cactus, in the GromovHausdorff sense. Moreover, the Brownian cactus can be interpreted as the continuous cactus of the socalled Brownian map. Résumé Le cactus d’un graphe pointé est un certain arbre discret associé à ce graphe. De façon similaire, à tout espace métrique géodésique pointé E, on peut associer un Rarbre appelé cactus continu de E. Sous des hypothèses générales, nous montrons que le cactus de cartes planaires aléatoires — dont la loi est déterminée par des poids de Boltzmann, et qui sont conditionnées à avoir un grand nombre fixé de sommets — converge en loi vers un espace limite appelé cactus brownien, au sens de la topologie de GromovHausdorff. De plus, le cactus brownien peut être interprété comme le cactus continu de la carte brownienne.