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Random point fields associated with certain Fredholm determinants. II: Fermion shifts and their ergodic and Gibbs properties. (English) Zbl 1051.60053

[For Part I, J. Funct. Anal. 205, No. 2, 414–463 (2003; Zbl 1051.60052), see above.]
Random point fields form an interesting class of random measures and have been studied from many angles. Processes that do not admit multiple points are quite amenable for the study of Fermion point processes. The authors of the present paper focus their attention on the case when \(R\) is a countable space and \(\lambda\) is the counting measure with \(L^{2}(R,\lambda)=\ell^{2}(R).\) They construct a family of probability measures on the configuration space over countable discrete space associated with nonnegative symmetric operators via determinants. Using this theory they establish that
(i) the measure structure is discrete and determinantal;
(ii) the metric entropy of a Fermion shift is positive except in trivial cases when the operator in question is zero or identity; and
(iii) the Fermion process is tail trivial under certain mild constraints on the measure.
Besides the paper also deals with the ergodic properties of the process in question.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G57 Random measures
60G60 Random fields
82B10 Quantum equilibrium statistical mechanics (general)
28D20 Entropy and other invariants

Citations:

Zbl 1051.60052
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References:

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