Conditional Measures of Determinantal Point Processes
- PDF / 230,810 Bytes
- 14 Pages / 612 x 792 pts (letter) Page_size
- 60 Downloads / 286 Views
nditional Measures of Determinantal Point Processes A. I. Bufetov Received May 8, 2019; in final form, September 20, 2019; accepted September 22, 2019
Abstract. Given one-dimensional determinantal point processes induced by orthogonal projections with integrable kernels satisfying a certain growth condition, it is proved that their conditional measures with respect to the configuration in the complement of a compact interval are orthogonal polynomial ensembles with explicitly found weights. Examples include the sine-process and the process with Bessel kernel. The main rˆole in the argument is played by the quasi-invariance, established in [2], of our point processes under the group of piecewise isometries of R. Key words: determinantal point processes, Gibbs property, Palm measures. DOI: 10.1134/S0016266320010025
1. Formulation of the Main Result 1.1. Conditional measures. Let E be a locally compact complete metric space, and let Conf(E) be the space of configurations on E. Given a configuration X ∈ Conf(E) and a subset C ⊂ E, we use X|C to denote the restriction of X to the subset C. A point process on E is a Borel probability measure on Conf(E). For such a measure P, the measure P( · |X; C) on Conf(E \ C) is defined as the conditional measure of P given that the restriction of our random configuration to C coincides with X|C . More formally, consider the surjective restriction mapping X → X|C from Conf(E) to Conf(C). The fibers of this mapping can be identified with Conf(E\C), and the conditional measures in the sense of Rohlin [7] are precisely the measures P( · |X; C). If a point process P has correlation measures of order up to l, then, given distinct points q1 , . . . , ql ∈ E, we let Pq1 ,...,ql stand for the lth reduced Palm measure of P conditioned at the points q1 , . . . , ql (here and below, we follow the conventions of [2] in working with Palm measures). The main results of this note can informally be summarized as follows. If a measure P( · |X; C) is supported on the subspace of configurations with precisely l particles and the reduced Palm measures conditioned at different l-tuples of points are equivalent, then, under certain additional assumptions (see Proposition 3.1 below), the conditional measure P( · |X; C) has the form Z −1 (q1 , . . . , ql )
dPp1 ,...,pl (X|C )dρl (p1 , . . . , pl ), dPq1 ,...,ql
where q1 , . . . , ql is almost any fixed l-tuple, ρl is the lth correlation measure of P, and Z(q1 , . . . , ql ) is the normalization constant. In particular, it is proved that if P is a one-dimensional determinantal processes induced by projections with integrable kernels satisfying a growth condition and C is the complement of a compact interval, then P( · |X; C) is an orthogonal polynomial ensemble with weight found explicitly. We proceed to precise formulations. Given a compact subset B ⊂ E and a configuration X ∈ Conf(E), let #B (X) stand for the number of particles of X lying in B. Given a Borel subset C ⊂ E, by FC we denote the σ-algebra generated by all random variables of the form #B , B ⊂ C. We w
Data Loading...
