Published on *IHP | CNRS | Sorbonne Université* (http://ihp.fr)

** The Prize-winners** 2016-2017

for : "Height fluctuations in interacting dimers". Read the article

Ann. Instit. H. Poincaré (B), Volume 53, Number 1 (2017), 98-168.

Abstract

We consider a non-integrable model for interacting dimers on the two-dimensional square lattice. Configurations are perfect matchings of ℤ2

, i.e. subsets of edges such that each vertex is covered exactly once (“close-packing” condition). Dimer configurations are in bijection with discrete height functions, defined on faces ξ of ℤ2. The non-interacting model is “integrable” and solvable via Kasteleyn theory; it is known that all the moments of the height difference hξ−hη converge to those of the massless Gaussian Free Field (GFF), asymptotically as |ξ−η|→∞. We prove that the same holds for small non-zero interactions, as was conjectured in the theoretical physics literature. Remarkably, dimer-dimer correlation functions are instead not universal and decay with a critical exponent that depends on the interaction strength. Our proof is based on an exact representation of the model in terms of lattice interacting fermions, which are studied by constructive field theory methods. In the fermionic language, the height difference hξ−hη takes the form of a non-local operator, consisting of a sum of monomials along an arbitrary path connecting ξ and η . As in the non-interacting case, this path-independence plays a crucial role in the proof.

AND

fot : "Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions". Read the article.

Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 2, 503–574.

Abstract

Let V=ℝd be the Euclidean d-dimensional space, μ (resp. λ) a probability measure on the linear (resp. affine) group G=GL(V) (resp. H=Aff(V)) and assume that μ is the projection of λ on G. We study asymptotic properties of the iterated convolutions μn∗δv (resp. λn∗δv) if v∈V, i.e. asymptotics of the random walk on V defined by μ (resp. λ), if the subsemigroup T⊂G (resp. Σ⊂H) generated by the support of μ (resp. λ) is “large.” We show spectral gap properties for the convolution operator defined by μ on spaces of homogeneous functions of degree s≥0 on V, which satisfy Hölder type conditions. As a consequence of our analysis we get precise asymptotics for the potential kernel ∑∞0μk∗δv, which imply its asymptotic homogeneity. Under natural conditions the H-space V is a λ-boundary; then we use the above results and radial Fourier Analysis on V∖{0} to show that the unique λ-stationary measure ρ on V is “homogeneous at infinity” with respect to dilations v→tv (for t>0), with a tail measure depending essentially of μ and Σ. Our proofs are based on the simplicity of the dominant Lyapunov exponent for certain products of Markov-dependent random matrices, on the use of renewal theorems for “tame” Markov walks, and on the dynamical properties of a conditional λ-boundary dual to V.

**The Prize-winners** 2014-2015

for : "Maximum of a log-correlated Gaussian field". Read the article

Abstract

We study the maximum of a Gaussian field on [0,1]d(d≥1) whose correlations decay logarithmically with the distance. Kahane (Ann. Sci. Math. Québec 9 (1985) 105–150) introduced this model to construct mathematically the Gaussian multiplicative chaos in the subcritical case. Duplantier, Rhodes, Sheffield and Vargas (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint, Renormalization of critical Gaussian multiplicative chaos and KPZ formula (2012) Preprint) extended Kahane’s construction to the critical case and established the KPZ formula at criticality. Moreover, they made in (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint) several conjectures on the supercritical case and on the maximum of this Gaussian field. In this paper we resolve Conjecture 12 in (Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012) Preprint): we establish the convergence in law of the maximum and show that the limit law is the Gumbel distribution convoluted by the limit of the derivative martingale.

for : "New insights into Approximate bayesian Computation". Read the article

Abstract

Approximate Bayesian Computation (ABC for short) is a family of computational techniques which offer an almost automated solution in situations where evaluation of the posterior likelihood is computationally prohibitive, or whenever suitable likelihoods are not available. In the present paper, we analyze the procedure from the point of view of k-nearest neighbor theory and explore the statistical properties of its outputs. We discuss in particular some asymptotic features of the genuine conditional density estimate associated with ABC, which is an interesting hybrid between a k-nearest neighbor and a kernel method.

**The Prize-winners 2012-2013**

for : "Universality for certain Hermitian Wigner matrices under weak moment conditions". Read the article

for "Superdiffusivity for Brownian Motion in a Poissonian potential with long range correlation I & II ". Read the articles : part I, part II

**The Prize-winners **2011

for : "Giant vacant component left by a random walk in a random d-regular graph". Read article here.

**The Prize-winners **2010

for : "Behavior near the extinction time in self-similar fragmentations I : The stable case". Read article here.

**The Prize-winners **2009

for "Anomalous heat-kernel decay for random walk among bounded random conductances." Read article here.

for "An asymptotic result for Brownian polymers". Read article here.

*06/20/2018*