On the NLS dynamics for infinite energy vortex configurations on the plane.
On the non-convexity of the time constant in first-passage percolation.
On the number of fully packed loop configurations with a fixed associated matching.
On the number of ground states of the Edwards–Anderson spin glass model
Ground states of the Edwards–Anderson (EA) spin glass model are studied on infinite graphs with finite degree. Ground states are spin configurations that locally minimize the EA Hamiltonian on each finite set of vertices. A problem with far-reaching consequences in mathematics and physics is to determine the number of ground states for the model on for any . This problem can be seen as the spin glass version of determining the number of infinite geodesics in first-passage percolation or the number...
On the number of matchings in regular graphs.
On the number of solutions of some integral equations arising in radiative transfer.
On the occupation measure of super-Brownian motion.
On the one-dimensional Boltzmann equation for granular flows
We consider a Boltzmann equation for inelastic particles on the line and prove existence and uniqueness for the solutions.
On the one-dimensional Boltzmann equation for granular flows
We consider a Boltzmann equation for inelastic particles on the line and prove existence and uniqueness for the solutions.
On the Periodic Lorentz Gas and the Lorentz Kinetic Equation
We prove that the Boltzmann-Grad limit of the Lorentz gas with periodic distribution of scatterers cannot be described with a linear Boltzmann equation. This is at variance with the case of a Poisson distribution of scatterers, for which the convergence to the linear Boltzmann equation was proved by Gallavotti [Phys. Rev. (2)185, 308 (1969)]. The arguments presented here complete the analysis in [Golse-Wennberg, M2AN Modél. Math. et Anal. Numér.34, 1151 (2000)], where the impossibility of a kinetic...
On the Plasma-Charge problem
This short report is a review on recent results of S. Caprino, C. Marchioro, E. Miot and the author on the initial value problem associated to the evolution of a continuous distribution of charges (plasma) in presence of a finite number of point charges.
On the proof of the Parisi formula by Guerra and Talagrand
The Parisi formula is an expression for the limiting free energy of the Sherrington-Kirkpatrick spin glass model, which had first been derived by Parisi using a non-rigorous replica method together with an hierarchical ansatz for the solution of the variational problem. It had become quickly clear that behind the solution, if correct, lies an interesting mathematical structure. The formula has recently been proved by Michel Talagrand based partly on earlier ideas and results by Francesco Guerra....
On the regularity of a free boundary for a nonlinear obstacle problem arising in superconductor modelling
On the small maximal flows in first passage percolation
We consider the standard first passage percolation on : with each edge of the lattice we associate a random capacity. We are interested in the maximal flow through a cylinder in this graph. Under some assumptions Kesten proved in 1987 a law of large numbers for the rescaled flow. Chayes and Chayes established that the large deviations far away below its typical value are of surface order, at least for the Bernoulli percolation and cylinders of certain height. Thanks to another approach we extend...
On the smallest eigenvalue of the Stokes operator in a domain with fine-grained random boundary.
On the spatially homogeneous Boltzmann equation
On the stationary Boltzmann equation
For stationary kinetic equations, entropy dissipation can sometimes be used in existence proofs similarly to entropy in the time dependent situation. Recent results in this spirit obtained in collaboration with A. Nouri, are here presented for the nonlinear stationary Boltzmann equation in bounded domains of with given indata and diffuse reflection on the boundary.
On the thermodynamic limit for Hartree–Fock type models
On the time constant in a dependent first passage percolation model
We pursue the study of a random coloring first passage percolation model introduced by Fontes and Newman. We prove that the asymptotic shape of this first passage percolation model continuously depends on the law of the coloring. The proof uses several couplings, particularly with greedy lattice animals.
On the transient and recurrent parts of a quantum Markov semigroup
We define the transient and recurrent parts of a quantum Markov semigroup 𝓣 on a von Neumann algebra 𝓐 and we show that, when 𝓐 is σ-finite, we can write 𝓣 as the sum of such semigroups. Moreover, if 𝓣 is the countable direct sum of irreducible semigroups each with a unique faithful normal invariant state ρₙ, we find conditions under which any normal invariant state is a convex combination of ρₙ's.