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

Justin Cyr ()
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Justin Cyr: Indiana University

Journal of Theoretical Probability, 2021, vol. 34, issue 3, 1321-1365

Abstract: 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.

Keywords: Determinantal probability; Uniform spanning forest; Toeplitz matrix; Insertion tolerance; Harmonic mean; Primary 60B15; 60G10; Secondary 60G60; 60C05; 15B05; 15B48 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10959-020-01062-5

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