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Conformal Random Geometry

Bertrand Duplantier

TL;DR

The notes establish a comprehensive, unifying framework that uses 2D quantum gravity and the KPZ map to compute the critical exponents of conformally invariant curves in the plane, including Brownian paths, SAWs, percolation, and Potts/O($N$ models, as well as SLE traces). By translating plane exponents to gravity exponents and applying additivity rules for mutually-avoiding path configurations, the work derives a vast array of multifractal spectra (harmonic, mixed, and double) and reveals dualities such as $ extrm{SLE}_{ extrm{kappa}} o extrm{SLE}_{16/ extrm{kappa}}$ and hull-EP correspondences. The framework unifies results across probabilistic and field-theoretic approaches, providing exact spectra parameterized by central charge $c$ (or $ extrm{kappa}$) and rigorously connecting to SLE through dual KPZ relations. These exponents, spectra, and dualities have deep implications for the geometry of CI fractals, offering a powerful toolkit for predicting fractal dimensions, harmonic measures, and winding properties in two-dimensional critical systems. The methods link random matrices, CFT, and SLE into a coherent description of CI geometries with broad applicability to statistical mechanics and mathematical physics.

Abstract

In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents associated with interacting random paths, by exploiting their underlying quantum gravity structure. The latter relates exponents in the plane to those on a random lattice, i.e., in a fluctuating metric, using the so-called Knizhnik, Polyakov and Zamolodchikov (KPZ) map. This is accomplished within the framework of random matrix theory and conformal field theory, with applications to geometrical critical models, like Brownian paths, self-avoiding walks, percolation, and more generally, the O(N) or Q-state Potts models and, last but not least, Schramm's Stochastic Loewner Evolution (SLE_kappa). These Notes can be considered as complementary to those by Wendelin Werner (2006 Fields Medalist!), ``Some Recent Aspects of Random Conformally Invariant Systems,'' arXiv:math.PR/0511268.

Conformal Random Geometry

TL;DR

The notes establish a comprehensive, unifying framework that uses 2D quantum gravity and the KPZ map to compute the critical exponents of conformally invariant curves in the plane, including Brownian paths, SAWs, percolation, and Potts/O( models, as well as SLE traces). By translating plane exponents to gravity exponents and applying additivity rules for mutually-avoiding path configurations, the work derives a vast array of multifractal spectra (harmonic, mixed, and double) and reveals dualities such as and hull-EP correspondences. The framework unifies results across probabilistic and field-theoretic approaches, providing exact spectra parameterized by central charge (or ) and rigorously connecting to SLE through dual KPZ relations. These exponents, spectra, and dualities have deep implications for the geometry of CI fractals, offering a powerful toolkit for predicting fractal dimensions, harmonic measures, and winding properties in two-dimensional critical systems. The methods link random matrices, CFT, and SLE into a coherent description of CI geometries with broad applicability to statistical mechanics and mathematical physics.

Abstract

In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents associated with interacting random paths, by exploiting their underlying quantum gravity structure. The latter relates exponents in the plane to those on a random lattice, i.e., in a fluctuating metric, using the so-called Knizhnik, Polyakov and Zamolodchikov (KPZ) map. This is accomplished within the framework of random matrix theory and conformal field theory, with applications to geometrical critical models, like Brownian paths, self-avoiding walks, percolation, and more generally, the O(N) or Q-state Potts models and, last but not least, Schramm's Stochastic Loewner Evolution (SLE_kappa). These Notes can be considered as complementary to those by Wendelin Werner (2006 Fields Medalist!), ``Some Recent Aspects of Random Conformally Invariant Systems,'' arXiv:math.PR/0511268.

Paper Structure

This paper contains 47 sections, 322 equations, 45 figures, 1 table.

Table of Contents

  1. Preamble
  2. Introduction
  3. A Brief Conformal History
  4. Conformal Geometrical Structures
  5. Quantum Gravity
  6. Stochastic Löwner Evolution
  7. Recent Developments
  8. Synopsis
  9. Intersections of Random Walks
  10. Non-Intersection Probabilities
  11. Quantum Gravity
  12. Random Walks on a Random Lattice
  13. Non-Intersections of Packets of Walks
  14. Mixing Random & Self-Avoiding Walks
  15. General Star Configurations
  16. ...and 32 more sections

Figures (45)

  • Figure 1: A planar Brownian path and its external frontier.
  • Figure 2: A planar self-avoiding walk (Courtesy of T. Kennedy).
  • Figure 3: Non-intersecting planar random walks crossing an annulus from $r$ to $R$, or a half-annulus in the half-plane ${\mathbb H}$.
  • Figure 4: A random planar triangulated lattice. (Courtesy of Ivan Kostov.)
  • Figure 5: A set of random lines on the regular triangular lattice and its counterpart on the random triangular lattice. (Courtesy of I. K.)
  • ...and 40 more figures