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

We investigate the ground state energy of the random Schrodinger operator -1/2Delta + beta(logt)V-2/d 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 t\o 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(beta) with probability 1 as t --> infinity. 1(beta) 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 less than or equal to 4 and we investigate the large-beta-behavior, which asymptotically approaches a similar picture as for the unsealed Poissonian potential considered by Sznitman [9]. (C) 2002 Editions scientifiques et medicales Elsevier SAS.