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Random point fields associated with certain Fredholm determinants. I: Fermion, Poisson and Boson point processes. (English) Zbl 1051.60052
Locally finite point fields on a Polish space \(R\) are considered. The class of \(\alpha\)-determinant point fields associated with an integral operator \(K\) is defined. The measure \(\mu_{\alpha, K}\) of such a field is determined by its generalized Laplace transformation, i.e. the integral of the function \(\exp(-\sum^\infty_{i=1} f(x_i))\) with respect to the measure \(\mu_{\alpha K}\) where \((x_1,x_2,\dots)\) is the configuration of the field, and \(f\) is a nonnegative test function (parameter of the Laplace transformation). In the case considered the Laplace image is the Fredholm determinant of the operator \(-\alpha K_\varphi\) raised to the power \(-1/\alpha\) (where \(K_\varphi(x, y)= \sqrt{\varphi(x)} K(x,y)\sqrt{\varphi(y)}\), \(\varphi= 1-e^{-f}\)). The Fredholm determinant itself is defined as the series of some averaged generalized algebraic determinants \(\text{det}_\alpha\) of the matrix \(K_\varphi(x, y)\), where \[ \text{det}_\alpha A= \sum_\sigma \alpha^{n-\nu(\sigma)} \prod^n_{i=1} a_{i\sigma(i)}, \] \(A= (a_{ij})_{n\times n}\), \(\sigma\) is a permutation of numbers \(\{1,\dots, n\}\), \(\nu(\sigma)\) is the number of cycles in \(\sigma\). Thus \(\text{det}_{-1}A\) is the usual determinant of \(A\), \(\text{det}_1A\) is the permanent of \(A\). If \(\alpha=-1\), the corresponding field is called Fermion field, and if \(\alpha= 1\), it is said to be Boson field. It can be proved that the case \(\alpha= 0\) corresponds to Poisson point field (the limit as \(\alpha\to 0\)).
Given \(\alpha\) and \(K\) the theorem for the \(\alpha\)-determinant point field to exist is proved. Some properties of such a point field are derived. For example: for a.s. \(x_0\in R\) the Palm measure \(\mu^{x_0}\) of a Fermion point field is proved to be the measure of the Fermion point field with some operator \(K^{x_0}\) (of determined form). Some analogies of the law of large numbers and the central limit theorem for the \(\alpha\)-determinant point fields are proved.
[For part II, Ann. Probab. 31, No. 3, 1533–1564 (2003; Zbl 1051.60053), see below.]

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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