## ** Papers **

**The expected area of a filled planar Brownian loop is $\frac \pi 5$**

with José Trujillo Ferreras

*Communications in Mathematical Physics,***264**(2006), n0. 3, 797--810

[More info / Picture] [Arxiv] [Journal]**The Fourier spectrum of critical percolation**

with Gábor Pete and Oded Schramm

*Acta Mathematica,***205,**Number 1 (2010), 19--104

[More info] [Arxiv] [Journal]**Continuity of the SLE trace in simply connected domains**

with Steffen Rohde and Oded Schramm

*Israel Journal of Mathematics,***187,**No 1 (2012), 23--36

[More info] [Arxiv] [Journal]**Oded Schramm's contributions to Noise Sensitivity**

*Annals of probability*,**39,**no 5 (2011), 1702--1767

[More info] [Arxiv] [Journal]**Multi-scale bound on the four-arm event on $\mathbb{Z}^2$**

**Appendix**to the paper "On the scaling limit of planar percolation" by Oded Schramm and Stanislav Smirnov.*Annals of Probability,***39,**no 5,

[Journal]**Exclusion sensitivity of Boolean functions**

with Erik Broman and Jeff Steif

*Probability Theory and Related Fields,***155**(2013), Issue 3-4, 621--663

[More info] [Arxiv] [Journal]**Lectures on noise sensitivity and percolation**

with Jeff Steif

Proceedings of the Clay Mathematics Institute Summer School (Buzios, Brazil), Clay Mathematics Proceedings**15**(2012), 49--154

[Arxiv]**Pivotal, cluster and interface measures for critical planar percolation**

with Gábor Pete and Oded Schramm

*J. Amer. Math. Soc.***26**(2013), 939--1024

[More info] [Arxiv] [Journal]**Quantum gravity and the KPZ formula**

*Bourbaki seminar (after Duplantier-Sheffield),*Astérisque. No.**352**(2013) Exp. No. 1052. 315--354.

[More info] [Arxiv]**The near-critical planar FK-Ising model**

with Hugo Duminil-Copin and Gábor Pete

*Communications in Mathematical Physics.*Volume**326**(2014), Issue 1, 1--35

[More info] [Arxiv] [Journal]**The Ising magnetization exponent is $\frac{1}{15}$**

with Federico Camia and Charles M. Newman

*Probability Theory and Related Fields.*Volume**160**(2014), Issue 1-2, 175--187

[More info] [Arxiv] [Journal]**Planar Ising magnetization field I. Uniqueness of the critical scaling limit**

with Federico Camia and Charles M. Newman

*Annals of Probability.*Volume**43**(2015), Number 2, 528--571.

[More info] [Arxiv] [Journal]**Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits**

with Federico Camia and Charles M. Newman

*Ann. Inst. Henri Poincaré (B)*Volume**52**, Number 1 (2016), 146--161.

[More info] [Arxiv] [Journal]**Liouville Brownian motion**

with Rémi Rhodes and Vincent Vargas

*Annals of Probability.*Volume**44**, Number 4 (2016), 3076-3110.

[More info] [Arxiv] [Journal]**On the heat kernel and the Dirichlet form of Liouville Brownian Motion**

with Rémi Rhodes and Vincent Vargas

*Electron. J. Probab.*Volume**19**(2014) 95, 1--25

[More info] [Arxiv] [Journal]**Coalescing Brownian flows: a new approach**

with Nathanael Berestycki and Arnab Sen

*Annals of Probability.*Volume**43**, Number 6 (2015), 3177-3215.

[More info] [Arxiv] [Journal]**KPZ formula derived from Liouville heat kernel**

with Nathanael Berestycki , Rémi Rhodes and Vincent Vargas

*Journal. of London Math. Soc.*Volume**94**Number 1 (2016), 186--208.

[More info] [Arxiv] [Journal]**A dissipative random velocity field for fully developed fluid turbulence**

with Laurent Chevillard and Rodrigo Pereira

*Journal of Fluid Mechanics.*Volume**794**(May 2016), 369--408

[More info] [Arxiv] [Journal]**The scaling limits of near-critical and dynamical percolation**

with Gábor Pete and Oded Schramm

To appear in*Journal of Eur. Math. Soc.*

[More info] [Arxiv] [Journal]**The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane**

with Gábor Pete and Oded Schramm

To appear in*Annals of Probability.*

[More info] [Arxiv] [Journal]**Exceptional times for percolation under exclusion dynamics**

with Hugo Vanneuville

To appear in*Annales scientifiques de l'École normale supérieure.*

[More info] [Arxiv]

Let $B_t$ be a planar Brownian loop of time duration 1 (a Brownian motion conditioned so that $B_0 = B_1$). We consider the compact hull obtained by filling in all the holes, i.e. the complement of the unique unbounded component of $R^2 \setminus B[0,1]$. We show that the expected area of this hull is $\pi/5$. The proof uses, perhaps not surprisingly, the Schramm Loewner Evolution (SLE). Also, using the result of Yor about the law of the index of a Brownian loop, we show that the expected areas of the regions of non-zero index $n$ equal $1/(2 \pi n^2)$. As a consequence, we find that the expected area of the region of index zero inside the loop is $\pi/30$; this value could not be obtained directly using Yor's index description.

Consider the indicator function $f$ of a two-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 situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension 31/36 a.s., and the corresponding dimension in the half-plane is 5/9. It is also proved that critical bond percolation on the square grid has exceptional times a.s. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.

We prove that the $SLE_\kappa$ trace in any simply connected domain $G$ is continuous (except possibly near its endpoints) if $\kappa < 8$. We also prove an SLE analog of Makarov's Theorem about the support of harmonic measure.

We survey in this paper the main contributions of Oded Schramm related to noise sensitivity. We will describe in particular his various works which focused on the "spectral analysis" of critical percolation (and more generally of Boolean functions), his work on the shape-fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation.

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 different class of perturbations, namely those arising from running the symmetric exclusion process for a short amount of time. In this study, the case of monotone Boolean functions will turn out to be of particular interest. We show that for this class of functions, ordinary noise sensitivity and noise sensitivity with respect to the complete graph exclusion process are equivalent. We also show this equivalence with respect to stability.

After obtaining these fairly general results, we study "exclusion sensitivity" of critical percolation in more detail with respect to medium-range dynamics. The exclusion dynamics, due to its conservative nature, is in some sense more physical than the classical i.i.d dynamics. Interestingly, we will see that in order to obtain a precise understanding of the exclusion sensitivity of percolation, we will need to describe how typical spectral sets of percolation diffuse under the underlying exclusion process.

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 and the Minimal Spanning Tree. We show here that the counting measure on the set of pivotal points of critical site percolation on the triangular grid, normalized appropriately, has a scaling limit, which is a function of the scaling limit of the percolation configuration. We also show that this limit measure is conformally covariant, with exponent 3/4. Similar results hold for the counting measure on macroscopic open clusters (the area measure), and for the counting measure on interfaces (length measure).

Since the aforementioned processes are very much governed by pivotal sites, the construction and properties of the "local time"-like pivotal measure are key results in this project. Another application is that the existence of the limit length measure on the interface is a key step towards constructing the so-called natural time-parametrization of the SLE(6) curve.

The proofs make extensive use of coupling arguments, based on the separation of interfaces phenomenon. This is a very useful tool in planar statistical physics, on which we included a self-contained Appendix. Simple corollaries of our methods include ratio limit theorems for arm probabilities and the rotational invariance of the two-point function.

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 $Z^2$ 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.

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 FK-Ising is highlighted, which is completely missing from the case of standard percolation: in any monotone coupling of FK configurations $\omega_p$ (e.g., in the one introduced in [Gri95]), as one raises $p$ near $p_c$, 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

We prove that for the Ising model defined on the plane $Z^2$ at $\beta=\beta_c$, the average magnetization under an external magnetic field $h>0$ behaves exactly like $${\sigma_0}_{\beta_c, h} \asymp h^{\frac 1 {15}}\,. $$ The proof, which is surprisingly simple compared to an analogous result for percolation (i.e. that $\theta(p)=(p-p_c)^{5/36+o(1)}$ on the triangular lattice \cite{\SmirnovWerner,\KestenScaling}) relies on the GHS inequality as well as the RSW theorem for FK percolation from \cite{\RSWfk}. The use of GHS to obtain inequalities involving critical exponents is not new; in this paper we show how it can be combined with RSW to obtain matching upper and lower bounds for the average magnetization.

The aim of this paper is to prove the following result. Consider the critical Ising model on the rescaled grid $a\, Z^2$, then the renormalized magnetization field $$\Phi^a:= a^{15/8} \, \sum_{x\in a\, Z^2} \sigma_x \delta_x,$$ seen as a random distribution (i.e., generalized function) on the plane, has a unique scaling limit as the mesh size $a\searrow 0$. The limiting field is conformally covariant.

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 denoted by $\Phi^\infty$. The purpose of this paper is to establish some fundamental properties satisfied by this $\Phi^\infty$ and the near-critical fields $\Phi^{\infty,h}$. More precisely, we obtain the following results:

(i) If $A\subset \mathbb{C}$ is a smooth bounded domain and if $m=m_A := \langle\Phi^\infty, 1_A\rangle$ denotes the limiting rescaled magnetization in $A$, then there is a constant $c=c_A>0$ such that
$$\log \mathbb{P}[m > x] \underset{x\to \infty}{\sim} -c \; x^{16}$$
In particular, this provides an alternative proof that the field $\Phi^\infty$ is non-Gaussian (another proof of this fact would use the $n$-point correlation functions established in \cite{CHI} which do not satisfy Wick's formula).

(ii) The random variable $m=m_A$ has a smooth {\it density} and one has more precisely the following bound on its Fourier transform: $|\mathbb{E}[e^{i\,t m}] |\le e^{- \tilde{c}\, |t|^{16/15}}$.

(iii) There exists a one-parameter family $\Phi^{\infty,h}$ of near-critical scaling limits for the magnetization field in the plane with vanishingly small external magnetic field.

We construct a stochastic process, called the {\bf Liouville Brownian motion}, which is the Brownian motion associated to the metric $e^{\gamma X(z)}dz^2$, $\gamma <\gamma_c=2$ and $X$ is a Gaussian Free Field. Such a process is conjectured to be the scaling limit of random walks on large planar maps eventually weighted by a model of statistical physics which are embedded in the Euclidean plane or in the sphere in a conformal manner.

The construction amounts to changing the speed of a standard two-dimensional Brownian motion $B_t$ depending on the local behaviour of the Liouville measure "$M_\gamma(dz) = e^{\gamma X(z)} dz$". We prove that the associated Markov process is a Feller diffusion for all $\gamma<\gamma_c=2$ and that for all $\gamma<\gamma_c$, the Liouville measure $M_\gamma$ is invariant under $P_t$. This Liouville Brownian motion enables us to introduce a whole set of tools of stochastic analysis in Liouville quantum gravity, which will be hopefully useful in analyzing the geometry of Liouville quantum gravity.

In \cite{GRV}, a Feller process called Liouville Brownian motion on $R^2$ has been introduced. It can be seen as a Brownian motion evolving in a random geometry given formally by the exponential of a (massive) Gaussian Free Field $e^{\gamma X}$ and is the right diffusion process to consider regarding 2d-Liouville quantum gravity. In this note, we discuss the construction of the associated Dirichlet form, following essentially \cite{fuku} and the techniques introduced in \cite{GRV}. Then we carry out the analysis of the Liouville resolvent. In particular, we prove that it is strong Feller, thus obtaining the existence of the Liouville heat kernel via a non-trivial theorem of Fukushima and al.

One of the motivations which led to introduce the Liouville Brownian motion in \cite{GRV} was to investigate the puzzling Liouville metric through the eyes of this new stochastic process. One possible approach was to use the theory developed for example in \cite{stollmann,sturm1,sturm2}, whose aim is to capture the "geometry" of the underlying space out of the Dirichlet form of a process living on that space. More precisely, under some mild hypothesis on the regularity of the Dirichlet form, they provide an intrinsic metric which is interpreted as an extension of Riemannian geometry applicable to non differential structures. We prove that the needed mild hypotheses are satisfied but that the associated intrinsic metric unfortunately vanishes, thus showing that renormalization theory remains out of reach of the metric aspect of Dirichlet forms.

The coalescing Brownian flow on $R$ is a process which was introduced by Arratia (1979) and Tóth and Werner (1997), and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. This result holds under a finite variance assumption and is thus optimal. In previous works by Fontes et al. (2004), Newman et al. (2005), the topology and state-space required a moment of order $3-\epsilon$ for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work -- in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the non-crossing property.

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.

We investigate the statistical properties, based on numerical simulations and analytical calculations, of a recently proposed stochastic model for the velocity field of an incompressible, homogeneous, isotropic and fully developed turbulent flow. A key step in the construction of this model is the introduction of some aspects of the vorticity stretching mechanism that governs the dynamics of fluid particles along their trajectory. An additional further phenomenological step aimed at including the long range correlated nature of turbulence makes this model depending on a single free parameter γ that can be estimated from experimental measurements. We confirm the realism of the model regarding the geometry of the velocity gradient tensor, the power-law behaviour of the moments of velocity increments (i.e. the structure functions), including the intermittent corrections, and the existence of energy transfers across scales. We quantify the dependence of these basic properties of turbulent flows on the free parameter γ and derive analytically the spectrum of exponents of the structure functions in a simplified non dissipative case. A perturbative expansion in power of γ shows that energy transfers, at leading order, indeed take place, justifying the dissipative nature of this random field.

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 $\mathcal{H}$ of percolation configurations introduced by Schramm and Smirnov. The proof essentially proceeds by "perturbing" the scaling limit of the critical model, using the pivotal measures studied in our earlier paper. Markovianity and conformal covariance of these new limiting objects are also established.

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.

We analyse in this paper a conservative analogue of the celebrated model of dynamical percolation introduced by Häggström, Peres and Steif in [HPS97]. It is simply defined as follows: start with an initial percolation configuration $\omega(t=0)$. Let this configuration evolve in time according to a simple exclusion process with symmetric kernel $K(x,y)$. We start with a general investigation (following [HPS97]) of this dynamical process $t \mapsto \omega_K(t)$ which we call $K$-exclusion dynamical percolation. We then proceed with a detailed analysis of the planar case at the critical point (both for the triangular grid and the square lattice $Z^2$) where we consider the power-law kernels $K^\alpha$ $$ K^{\alpha}(x,y) \propto \frac 1 {\|x-y\|_2^{2+\alpha}} \, .$$ We prove that if $\alpha > 0$ is chosen small enough, there exist exceptional times $t$ for which an infinite cluster appears in $\omega_{K^{\alpha}}(t)$. (On the triangular grid, we prove that it holds for all $\alpha < \alpha_0 = \frac {217}{816}$.) The existence of such exceptional times for standard i.i.d. dynamical percolation (where sites evolve according to independent Poisson point processes) goes back to the work by Schramm-Steif in [SS10]. In order to handle such a $K$-exclusion dynamics, we push further the spectral analysis of exclusion noise sensitivity which had been initiated in [BGS13]. (The latter paper can be viewed as a conservative analogue of the seminal paper by Benjamini-Kalai-Schramm [BKS99] on i.i.d. noise sensitivity.) The case of a nearest-neighbour simple exclusion process, corresponding to the limiting case $\alpha = +\infty$, is left widely open.

## ** Preprints **

**On a skewed and multifractal uni-dimensional random field, as a probabilistic representation of Kolmogorov's views on turbulence**

with Laurent Chevillard, Rémi Rhodes and Vincent Vargas

[More info] [Arxiv]**On the convergence of FK-Ising Percolation to $SLE(16/3,16/3−6)$**

with Hao Wu

[More info] [Arxiv]

(This also this earlier Draft of this paper which provides a different way to analyze the discrete dust.)

We construct, for the first time to our knowledge, a one-dimensional stochastic field ${u(x)}_{x\in \mathbb{R}}$ which satisfies the following axioms which are at the core of the phenomenology of turbulence mainly due to Kolmogorov: (i) Homogeneity and isotropy: $u(x) \overset{\mathrm{law}}= -u(x) \overset{\mathrm{law}}=u(0)$ (ii) Negative skewness (i.e. the $4/5^{\mbox{\tiny th}}$-law): \\ $\mathbb{E}{(u(x+\ell)-u(x))^3} \sim_{\ell \to 0+} - C \, \ell\,,$ \, for some constant $C>0$ (iii) Intermittency: $\mathbb{E}{|u(x+\ell)-u(x) |^q} \asymp_{\ell \to 0} |\ell|^{\xi_q}\,,$ for some non-linear spectrum $q\mapsto \xi_q$ Since then, it has been a challenging problem to combine axiom (ii) with axiom (iii) (especially for Hurst indexes of interest in turbulence, namely $H<1/2$). In order to achieve simultaneously both axioms, we disturb with two ingredients a underlying fractional Gaussian field of parameter $H\approx \frac 1 3 $. The first ingredient is an independent Gaussian multiplicative chaos (GMC) of parameter $\gamma$ that mimics the intermittent, i.e. multifractal, nature of the fluctuations. The second one is a field that correlates in an intricate way the fractional component and the GMC without additional parameters, a necessary inter-dependence in order to reproduce the asymmetrical, i.e. skewed, nature of the probability laws at small scales.

We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK-Ising percolation to chordal SLEκ(κ−6) with κ=16/3. Our proof follows the classical excursion-construction of SLEκ(κ−6) processes in the continuum and we are thus lead to introduce suitable cut-off stopping times in order to analyse the behaviour of the driving function of the discrete system when Dobrushin boundary conditions collapse to a single point. Our proof is very different from [KS15, KS16] as it only relies on the convergence to the chordal SLEκ process in Dobrushin boundary conditions and does not require the introduction of a new observable. Still, it relies crucially on several ingredients: a) the powerful topological framework developed in [KS17] as well as its follow-up paper [CDCH+14], b) the strong RSW Theorem from [CDCH16], c) the proof is inspired from the appendix A in [BC16]. One important emphasis of this paper is to carefully write down some properties which are often considered {\em folklore} in the literature but which are only justified so far by hand-waving arguments. The main examples of these are: 1) the convergence of natural discrete stopping times to their continuous analogues. (The usual hand-waving argument destroys the spatial Markov property). 2) the fact that the discrete spatial Markov property is preserved in the the scaling limit. (The enemy being that 𝔼[Xn|Yn] does not necessarily converge to 𝔼[X|Y] when (Xn,Yn)→(X,Y)). We end the paper with a detailed sketch of the convergence to radial SLEκ(κ−6) when κ=16/3 as well as the derivation of Onsager's one-arm exponent 1/8.

## ** PhD and HDR memoirs**

- Manuscript of my PhD thesis (December, 5th 2008) : link to file
- Manuscript of my Habilitation (HDR) (December, 9th 2013) : link to file

## ** Coauthors **

Nathanael Berestycki, Erik Broman, Federico Camia, Hugo Duminil-Copin, Laurent Chevillard , Charles M. Newman, Rodrigo Pereira , Gábor Pete, José Trujillo Ferreras, Rémi Rhodes, Steffen Rohde, Oded Schramm, Arnab Sen , Jeff Steif, Hugo Vanneuville, Vincent Vargas, Hao Wu