Statistics of Extreme Spasings in Determinantal Random Point Processes

Determinantal (a.k.a. fermion) random point processes were introduced in probability theory by Macchi about thirty years ago ([13], [14], [3]). In the last ten years the subject has attracted a considerable attention due to its rich connections to Random Matrix Theory, Combinatorics, Representation Theory, Random Growth Models, Number Theory and several other areas of mathematics. We refer the reader to the recent surveys ([20], [9], [8]), and research papers on the subject ([1], [4], [5], [6], [7], [12], [10], [15], [17], [18], [19], [22], [23], [24]). In this paper we shall consider determinantal random point processes on the real line with the translation-invariant correlation kernel. In other words, a one particle space X is given as X = R, and the space of elementary outcomes Ω consists of the countable, locally finite particle configurations on the real line