2019 Volume 73 Issue 1 Pages 43-67
The asymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field at the infimum of the spectrum is investigated. The random point field is taken from a subclass of the class given by Shirai and Takahashi (J. Funct. Anal. 205 (2003), 414-463) in terms of the Fredholm determinant. In the subclass, the obtained leading orders are the same as the well known results for the Poisson point fields, and the character of the random field appears in the leading constants. The random point fields associated with the sine kernel and the Ginibre random point field are well studied examples not included in the above subclass, though they are included in the class by Shirai and Takahashi. By applying the results on asymptotics of the hole probability for these random fields, the corresponding asymptotic behaviors of the densities of states are also investigated in the case where the single site potentials have compact supports. The same method also applies to another well studied example, the zeros of a Gaussian random analytic function.
