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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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