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Stationary Determinantal Processes on $${\mathbb {Z}}^d$$ Z d with N Labeled Objects per Site, Part I: Basic Properties and Full Domination

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  • Justin Cyr

    (Indiana University)

Abstract

We study a class of stationary determinantal processes on configurations of N labeled objects that may be present or absent at each site of $${\mathbb {Z}}^d$$ Z d . Our processes, which include the uniform spanning forest as a principal example, arise from the block Toeplitz matrices of matrix-valued functions on the d-torus. We find the maximum level of uniform insertion tolerance for these processes, extending a result of Lyons and Steif from the $$N = 1$$ N = 1 case to $$N > 1$$ N > 1 . We develop a method for conditioning determinantal processes in the general discrete setting to be as large as possible in a fixed set as an approach to determining uniform insertion tolerance. The method of conditioning on maximality developed here is used in a subsequent paper to study stochastic domination, strong domination and phase uniqueness for the same class of processes.

Suggested Citation

  • Justin Cyr, 2021. "Stationary Determinantal Processes on $${\mathbb {Z}}^d$$ Z d with N Labeled Objects per Site, Part I: Basic Properties and Full Domination," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1321-1365, September.
    • Handle: RePEc:spr:jotpro:v:34:y:2021:i:3:d:10.1007_s10959-020-01062-5
    DOI: 10.1007/s10959-020-01062-5
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    References listed on IDEAS

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    1. Deborah Heicklen & Russell Lyons, 2003. "Change Intolerance in Spanning Forests," Journal of Theoretical Probability, Springer, vol. 16(1), pages 47-58, January.
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