Phase transition of the principal Dirichlet eigenvalue in a scaled Poissonian potential

We consider d-dimensional Brownian motion in a scaled Poissonian potential and the principal Dirichlet eigenvalue (ground state energy) of the corresponding Schrodinger operator. The scaling is chosen to be of critical order, i.e. it is determined by the typical size of large holes in the Poissonian cloud. We prove existence of a phase transition in dimensions d greater than or equal to 4: There exists a critical scaling constant for the potential. Below this constant the scaled infinite volume limit of the corresponding principal Dirichlet eigenvalue is Linear in the scale. On the other hand, for large values of the scaling constant this limit is strictly smaller than the linear bound. For d < 4 we prove that this phase transition does not take place on that scale. Further we show that the analogous picture holds true for the partition sum of the underlying motion process.