Infinite volume asymptotics of the ground state energy in a scaled Poissonian potential

Abstract

We investigate the ground state energy of the random Schrödinger operator -1 2 + β(log t) -2/d V on the box (-t, t) d with Dirichlet boundary conditions.V denotes the Poissonian potential which is obtained by translating a fixed non-negative compactly supported shape function to all the particles of a d-dimensional Poissonian point process.The scaling (log t) -2/d is chosen to be of critical order, i.e. it is determined by the typical size of the largest hole of the Poissonian cloud in the box (-t, t) d .We prove that the ground state energy (properly rescaled) converges to a deterministic limit I (β) with probability 1 as t → ∞.I (β) can be expressed by a (deterministic) variational principle.This approach leads to a completely different method to prove the phase transition picture developed in [4].Further we derive critical exponents in dimensions d 4 and we investigate the large-β-behavior, which asymptotically approaches a similar picture as for the unscaled Poissonian potential considered by Sznitman [9].

Locations

• Annales de l Institut Henri Poincaré Probabilités et Statistiques - View
• Radboud Repository (Radboud University) - View - PDF
• Data Archiving and Networked Services (DANS) - View - PDF
• French digital mathematics library (Numdam) - View - PDF