**The Prize-winners 2012-2013**

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

Abstract

We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy–Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.

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

Abstract (part I)

We study trajectories of d-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential V is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii ν has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by Wühtrich (Ann. Probab. 26 (1998) 1000–1015, Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 279–308): the superdiffusivity phenomenon is enhanced by the presence of correlation.

Abstract (part II)

This paper continues a study on trajectories of Brownian Motion in a field of soft trap whose radius distribution is unbounded. We show here that for both point-to-point and point-to-plane model the volume exponent (the exponent associated to transversal fluctuation of the trajectories) ξ is strictly less than 1 and give an explicit upper bound that depends on the parameters of the problem. In some specific cases, this upper bound matches the lower bound proved in the first part of this work and we get the exact value of the volume exponent.

**2011**

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

**2010 **

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

**2009**

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

Summary

We consider the random progress to the closest neighbors within ℤ^{d}, *d*≥2, of which the transitions are given by a field of random confined conductance *ω*_{xy}∈[0, 1]. The conductance law is iid on the edges, and such that the probability that *ω _{xy}*>0 be superior to the percolation threshold (by edge) on ℤ

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

Summary

We consider a model of polymer formation presented by Durett and Rogers (*Probab. Theory Related Fields ***92 **(1992) 337-349). We prove their conjecture about asymptotic behavior of the associated continuous process *X _{t}* (that corresponds to the place of the polymer extremity at a given moment) for a particular type of function of repulsive interaction without solid support.

*Last updated 06/13/2014*