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Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness
- Part of
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- Journal:
- Journal of Applied Probability / Volume 58 / Issue 2 / June 2021
- Published online by Cambridge University Press:
- 23 June 2021, pp. 469-483
- Print publication:
- June 2021
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- Article
- Export citation
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For a determinantal point process (DPP) X with a kernel K whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process
$X^u$ at a point u with 
$K(u,u)>0$ so that, almost surely, 
$X^u$ is obtained by removing a finite number of points from X. We sharpen this result, assuming weaker conditions and establishing that 
$X^u$ can be obtained by removing at most one point from X, where we specify the distribution of the difference 
$\xi_u: = X\setminus X^u$. This is used to discuss the degree of repulsiveness in DPPs in terms of 
$\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.
