Christophe Garban

On the scaling limits of planar percolation

We prove Tsirelsons conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.

Planar Ising magnetization field I. Uniqueness of the critical scaling limit

The aim of this paper is to prove the following result. Consider the critical Ising model on the rescaled grid

The Near-Critical Planar FK-Ising Model

a\mathbb{Z}^2

KPZ formula derived from Liouville heat kernel

, then the renormalized magnetization field \[\Phi^a:=a^{15/8}\sum_{x\in a\mathbb{Z}^2}\sigma_x\delta_x,\] seen as a random distribution (i.e., generalized function) on the plane, has a unique scaling limit as the mesh size

Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits

a\searrow0

Oded Schramm’s contributions to Noise Sensitivity

. The limiting field is conformally covariant.

A dissipative random velocity field for fully developed fluid turbulence

We study the near-critical FK-Ising model. First, a determination of the correlation length defined via crossing probabilities is provided. Second, a phenomenon about the near-critical behavior of the FK-Ising is highlighted, which is completely missing from the case of standard percolation: in any monotone coupling of FK configurations ωp (e.g., in the one introduced in Grimmett (Ann Probab 23(4):1461–1510, 1995)), as one raises p near pc, the new edges arrive in a self-organized way, so that the correlation length is not governed anymore by the number of pivotal edges at criticality.

Coalescing Brownian flows: A new approach

In this paper, we establish the Knizhnik--Polyakov--Zamolodchikov (KPZ) formula of Liouville quantum gravity, using the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. We also provide a careful study of the (no)-doubling behaviour of the Liouville measures in the appendix, which is of independent interest.

The Fourier spectrum of critical percolation

In [CGN12], we proved that the renormalized critical Ising magnetization fields

Liouville Brownian motion

\Phi^a:= a^{15/8} \sum_{x\in a\, \Z^2} \sigma_x \, \delta_x