Quantum gravity and the KPZ formula

Abstract

This text is a survey (Bourbaki seminar) on the paper "Liouville quantum gravity and KPZ" By B.Duplantier and S.Sheffield. The study of statistical physics models in two dimensions (d=2) at their critical point is in general a significantly hard problem (not to mention the d=3 case). In the eighties, three physicists, Knizhnik, Polyakov et Zamolodchikov (KPZ) came up in \cite{\KPZ} with a novel and far-reaching approach in order to understand the critical behavior of these models. Among these, one finds for example random walks, percolation as well as the Ising model. The main underlying idea of their approach is to study these models along a two-step procedure as follows: a/ First of all, instead of considering the model on some regular lattice of the plane (such as ${\mathbb{Z}}^2$ for example), one defines it instead on a well-chosen "random planar lattice". Doing so corresponds to studying the model in its {\it quantum gravity} form. In the case of percolation, the appropriate choice of random lattice matches with the so-called planar maps. b/ Then it remains to get back to the actual {\it Euclidean} setup. This is done thanks to the celebrated {\bf KPZ formula} which gives a very precise correspondence between the geometric properties of models in their quantum gravity formulation and their analogs in the Euclidean case. The nature and the origin of such a powerful correspondence remained rather mysterious for a long time. In fact, the KPZ formula is still not rigorously established and remains a conjectural correspondence. The purpose of this survey is to explain how the recent work of Duplantier and Sheffield enables to explain some of the mystery hidden behind this KPZ formula. To summarize their contribution in one sentence, their work implies a beautiful interpretation of the KPZ correpondence through a uniformization of the random lattice, seen as a Riemann surface.

Bourbaki seminar (after Duplantier-Sheffield), 38 pages

References in corpus (4)

• Uniqueness and universality of the Brownian map
• Scaling limits of random trees and planar maps
• The Brownian map is the scaling limit of uniform random plane quadrangulations
• Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points

Cited by in corpus (8)

• KPZ formula derived from Liouville heat kernel
• Liouville heat kernel: regularity and bounds
• Equivalence of Liouville measure and Gaussian free field
• The scaling limit of random simple triangulations and random simple quadrangulations
• Extreme boundary conditions and random tilings
• A glimpse of the conformal structure of random planar maps
• Diffusion in planar Liouville quantum gravity
• Tensor Field Theories: Renormalization and Random Geometry