Conditional measures of determinantal point processes

TL;DR

This paper investigates the conditional measures of certain one-dimensional determinantal point processes, showing they form orthogonal polynomial ensembles with explicit weights, including examples like the sine-process and Bessel kernel process.

Contribution

It establishes that the conditional measures of these processes are orthogonal polynomial ensembles with explicitly determined weights, extending understanding of their structure.

Findings

Conditional measures are orthogonal polynomial ensembles.

Explicit weights are derived for these ensembles.

Includes examples like sine-process and Bessel kernel process.

Abstract

For a class of one-dimensional determinantal point processes including those induced by orthogonal projections with integrable kernels satisfying a 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 the Bessel kernel. The argument uses the quasi-invariance, established in [1], of our point processes under the group of piecewise isometries of the real line.