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.

中文翻译：

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每个站点有 N 个标记对象的 $${\mathbb {Z}}^d$$ 上的平稳行列式过程，第一部分：基本属性和完全支配

我们在 $${\mathbb {Z}}^d$$Z d 的每个位置上可能存在或不存在 N 标记对象的配置上研究一类静止行列式过程。我们的过程，包括统一生成森林作为主要示例，来自 d 环上矩阵值函数的块 Toeplitz 矩阵。我们找到了这些过程的最大统一插入容差水平，将 Lyons 和 Steif 的结果从 $$N = 1$$ N = 1 情况扩展到 $$N > 1$$ N > 1 。我们开发了一种方法，用于将一般离散设置中的行列式过程调节为在固定设置中尽可能大，作为确定均匀插入容差的方法。在随后的论文中使用了此处开发的最大条件调节方法来研究随机支配，