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...

We construct a class of conformally invariant measures on sets (or paths) and we study the critical exponents called intersection exponents associated to these measures. We show that these exponents exist and that they correspond to intersection exponents between planar Brownian motions. More precisely, using the definitions and results of our paper [27], we show that any set defined under such a conformal invariant measure behaves exactly as a pack (containing maybe a non-integer number) of Brownian...

There has been much success in describing the limiting spatial fluctuations of growth models in the Kardar–Parisi–Zhang (KPZ) universality class. A proper rescaling of time should introduce a non-trivial temporal dimension to these limiting fluctuations. In one-dimension, the KPZ class has the dynamical scaling exponent z = 3/2, that means one should find a universal space–time limiting process under the scaling of time as tT, space like t2/3X and fluctuations like t1/3 as t → ∞. In this paper we...

In this paper we construct upper bounds for families of functionals of the form$$E_{\epsilon }\left(\phi \right):={\int }_{\Omega }\left({\epsilon \left|\nabla \phi \right|}^2+\frac{1}{\epsilon }W\left(\phi \right)\right)\mathrm{d}x+\frac{1}{\epsilon }{\int }_{{ℝ}^N}{\left|\nabla {\overline{H}}_{F\left(\phi \right)}\right|}^2\mathrm{d}x$$where Δ${\overline{H}}_u$ = div {${\chi }_{\Omega }$u}. Particular cases of such functionals arise in Micromagnetics. We also use our technique to construct upper bounds for functionals that appear in a variational formulation of the method of vanishing viscosity for conservation laws.