Determinantal (a.k.a. fermion) random point processes were introduced in probability theory by Macchi about thirty years ago ([13], [14], [3]). In the

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ABSTRACT. The paper ‘Modern statistics for spatial point processes ’ by Jesper Møller and Rasmus P. Waagepetersen is based on a special invited lecture given by the authors at the 21st

An α-permanental random field is briefly speaking a model for a collection of random variables with positive associations, where α is a positive number and the probability generating function is given in terms of a covariance or more general function so that density and moment expressions are given by certain α-permanents. Though such models possess many appealing probabilistic properties, many statisticians seem unaware of α-permanental random fields and their potential applications. The purpose of this paper is first to summarize useful probabilistic results using the simplest possible setting, and second to study stochastic constructions and simulation techniques, which should provide a useful basis for discussing the statistical aspects in future work. The paper also discusses some examples of α-permanental random fields.

This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian Unitary Ensemble. In particular, the kth smallest gap, normalized by a factor n 4/3, has a limiting density proportional to x 3k−1 e −x3. Concerning the largest gaps, normalized by n / √ log n, they converge in ̷L p to a constant for all p> 0. These results are compared with the extreme gaps between zeros of the Riemann zeta function.

Given a positive definite, bounded linear operator A on the Hilbert space H0: = l 2 (E), we consider a reproducing kernel Hilbert space H+ with a reproducing kernel A(x, y). Here E is any countable set and A(x, y), x, y ∈ E, is the representation of A w.r.t. the usual basis of H0. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space H− with a kernel function B(x, y), which is the representation of the inverse of A in a sense, so that H − ⊃ H0 ⊃ H+ becomes a rigged Hilbert space. We investigate a relationship between the ratios of determinants of some partial matrices related to A and B and the suitable projections in H − and H+. We also get a variational principle on the limit ratios of these values. We apply this relation to show the Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I + A) −1 on the set E. It turns out that the class of determinantal point processes that can be recognized as Gibbs measures for suitable interactions is much bigger than that obtained by Shirai and Takahashi.

We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space X, for which certain fermion point processes are invariant. The Glauber dynamics is a birth-and-death process in X, while in the case of the Kawasaki dynamics interacting particles randomly hop over X. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.

Determinantal (a.k.a. fermion) random point processes were introduced in probability theory by Macchi about thirty years ago ([13], [14], [3]). In the