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Pivotal, cluster, and interface measures for critical planar percolation

- C. Garban, G. Pete, O. Schramm
- Mathematics
- 8 August 2010

This work is the first in a series of papers devoted to the construction and study of scaling limits of dynamical and near-critical planar percolation and related objects like invasion percolation… Expand

The Fourier spectrum of critical percolation

- C. Garban, G. Pete, O. Schramm
- Mathematics
- 26 March 2008

Consider the indicator function f of a 2-dimensional percolation crossing event. In this paper, the Fourier transform of f is studied and sharp bounds are obtained for its lower tail in several… Expand

Liouville Brownian motion

We construct a stochastic process, called the Liouville Brownian motion which we conjecture to be the scaling limit of random walks on large planar maps which are embedded in the euclidean plane or… Expand

Noise Sensitivity of Boolean Functions and Percolation

This is a graduate-level introduction to the theory of Boolean functions, an exciting area lying on the border of probability theory, discrete mathematics, analysis, and theoretical computer science.… Expand

A dissipative random velocity field for fully developed fluid turbulence

- R. Pereira, C. Garban, L. Chevillard
- PhysicsJournal of Fluid Mechanics
- 2 October 2015

We investigate the statistical properties, based on numerical simulations and analytical calculations, of a recently proposed stochastic model for the velocity field (Chevillard et al., Europhys.… Expand

The scaling limits of near-critical and dynamical percolation

- C. Garban, G. Pete, O. Schramm
- Mathematics
- 23 May 2013

We prove that near-critical percolation and dynamical percolation on the triangular lattice $\eta \mathbb{T}$ have a scaling limit as the mesh $\eta \to 0$, in the "quad-crossing" space $\mathcal{H}$… Expand

Exclusion sensitivity of Boolean functions

Recently the study of noise sensitivity and noise stability of Boolean functions has received considerable attention. The purpose of this paper is to extend these notions in a natural way to a… Expand

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 $a\mathbb{Z}^2$, then the renormalized magnetization field \[\Phi^a:=a^{15/8}\sum_{x\in… Expand

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

In [CGN12], we proved that the renormalized critical Ising magnetization fields $\Phi^a:= a^{15/8} \sum_{x\in a\, \Z^2} \sigma_x \, \delta_x$ converge as $a\to 0$ to a random distribution that we… Expand

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