Stationary determinantal processes: $\psi$-mixing property and $L^q$-dimensions

TL;DR

This paper characterizes the $$-mixing property of stationary determinantal processes, establishes the existence of their $L^q$-dimensions, and analyzes the growth rate of longest common substrings in typical pairs.

Contribution

It provides necessary and sufficient conditions for $$-mixing, proves the existence of $L^q$-dimensions, and links these to the growth rate of common substrings in determinantal processes.

Findings

Characterization of $$-mixing for determinantal processes

Existence of $L^q$-dimensions under certain conditions

Determination of the growth rate of longest common substrings

Abstract

The results of this paper are 3-folded. Firstly, for any stationary determinantal process on the integer lattice, induced by strictly positive and strictly contractive involution kernel, we obtain the necessary and sufficient condition for the $\psi$-mixing property. Secondly, we obtain the existence of the $L^q$-dimensions of the stationary determinantal measure on symbolic space $\{0, 1\}^\mathbb{N}$ under appropriate conditions. Thirdly, the previous two results together imply the precise increasing rate of the longest common substring of a typical pair of points in $\{0, 1\}^\mathbb{N}$.