We prove a large deviations principle for the number of intersections of two independent infinite-time ranges in dimension 5 and greater, improving upon the moment bounds of Khanin, Mazel, Shlosman, and Sinaï [9]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen, and den Hollander [15], who analyzed this question for the Wiener sausage in the finite-time horizon. The proof builds on their result (which was adapted in the discrete setting by Phetpradap [12]), and combines it with a series of tools that were developed in recent works of the authors [2, 3, 5]. Moreover, we show that most of the intersection occurs in a single box where both walks realize an occupation density of order 1. © 2022 Wiley Periodicals, Inc.

中文翻译：

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随机游走交叉点的大偏差

我们证明了维度 5 和更大维度中两个独立无限时间范围的交点数量的大偏差原理，改进了 Khanin、Mazel、Shlosman 和 Sinaï [9] 的力矩边界。在离散设置中，这解决了 van den Berg、Bolthausen 和 den Hollander [15] 的猜想，他们在有限时间范围内分析了维纳香肠的这个问题。该证明建立在他们的结果（由 Phetpradap [12] 在离散设置中改编）的基础上，并将其与作者最近的作品 [2、3、5] 中开发的一系列工具相结合。此外，我们表明大部分交叉点都出现在一个盒子中，其中两条步行道的占用密度为 1。 © 2022 Wiley Periodicals, Inc.