# Rigid stationary determinantal processes in non-Archimedean fields

## TL;DR

This paper constructs stationary determinantal point processes over non-Archimedean fields and identifies a unique geometric condition on subsets that ensures the process's rigidity, contrasting with Euclidean cases.

## Contribution

It introduces a new geometric criterion for rigidity of determinantal processes in non-Archimedean fields, expanding understanding beyond Euclidean settings.

## Key findings

- Established existence of determinantal processes with specific kernels
- Identified a novel geometric condition for process rigidity
- Demonstrated differences from Euclidean case geometries

## Abstract

Let $F$ be a non-discrete non-Archimedean local field. For any subset $S\subset F$ with finite Haar measure, there is a stationary determinantal point process on $F$ with correlation kernel $\widehat{\mathbb{1}}_S(x-y)$, where $\widehat{\mathbb{1}}_S$ is the Fourier transform of the indicator function $\mathbb{1}_S$. In this note, we give a geometrical condition on the subset $S$, such that the associated determinantal point process is rigid in the sense of Ghosh and Peres. Our geometrical condition is very different from the Euclidean case.