Filters

2015 - 20162

More filters

20192
Reset filters

Settings

Author: Bolthausen, Erwin ×

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    Exponential decay of covariances for the supercritical membrane model
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Bolthausen, Erwin; Cipriani, Alessandra; Kurt, Noemi
    We consider the membrane model, that is the centered Gaussian field on Zd whose covariance matrix is given by the inverse of the discrete Bilaplacian. We impose a delta-pinning condition, giving a reward of strength " for the field to be 0 at any site of the lattice. In this paper we prove that in dimensions d ≥ 5 covariances of the pinned field decay at least exponentially, as opposed to the field without pinning, where the decay is polynomial. The proof is based on estimates for certain discrete weighted norms, a percolation argument and on a Bernoulli domination result.
  • Thumbnail Image
    Item
    Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Bolthausen, Erwin; König, Wolfgang; Mukherjee, Chiranjib
    We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the itPekar process, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the ``mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]