Stationary determinantal processes on \({\mathbb{Z}}^d\) with \(N\) labeled objects per site. I: Basic properties and full domination (Q2042039)

In this part of his study on stationary determinantal processes on \(\mathbb{Z}^d\) with \(N\) labeled objects per site, the author presents basic properties and full domination, following to continue the study for stochastic domination, strong domination and phase uniqueness for the same class of processes. In this paper, a result of \textit{R. Lyons} and \textit{J. E. Steif} [Duke Math. J. 120, No. 3, 515--575 (2003; Zbl 1068.82010)] is extended from \(N=1\) to \(N>1\), finding the maximum level of uniform insertion tolerance. To do this, the paper is organized in six sections, the first one introducing the reader into the topic and giving the main notions and definitions about determinantal subset of a countable set, uniform spanning tree measures, uniform spanning forest, and presenting the framework for conditioning the discrete determinantal measures such that the extension to \(N>1\) to be done. In the second section, stationary determinantal probability measures \(\textbf{P}^F\) for processes that arise from measurable Hermitian matrix-valued functions \(F:\mathbb{T}^d\rightarrow \mathbb{C}^{N\times N}\) are recalled in the discrete setting, where \(\mathbb{C}^{N\times N}\) denotes the space of complex-valued \(N\times N\) matrices, with the property that \(0<F<I\) a.e. with respect to Lebesgue measure on \(\mathbb{T}^d\). In the third section, the results of Lyons and Steif on \(2^{\mathbb{Z}^d}\)-invariant determinantal probability measures are extended on \(2^{[N]\times\mathbb{Z}^d}\), where \([N]:=\{1,\dots,N\}\), and various properties of \(\textbf{P}^F\) are analyzed. In the fourth section, a formula for conditioning \(\textbf{P}^F\) to contain a maximal subset of a given set from \([N]\times\mathbb{Z}^d\) is given. Maximality on finite and infinite sets is analyzed. The obtained results are applied in the fifth section to condition \(\textbf{P}^F\) to be maximal on \([N]\times B\) where \(B\) is a proper subset of \(\mathbb{Z}^d\). Also reductions to the \(N=1\) case is exemplified. In the sixth section of the paper, the main result is established (Theorem 6.8) which gives a straightforward formula for the matrix which induces the determinantal probability measure in terms of the symbol \(F\) that extends the harmonic mean formula obtained by Lions and Steif to the matrix case. The section concludes with applications of Theorem 6.8 and examples related to the uniform spanning forest on \(\mathbb{Z}^d\).