# Divide and color representations for threshold Gaussian and stable vectors

## TL;DR

This paper investigates when thresholded Gaussian and stable vectors produce divide and color processes, revealing dimension-dependent behaviors, threshold effects, and a phase transition at stability index 1/2.

## Contribution

It characterizes conditions under which threshold processes from Gaussian and stable vectors are divide and color, highlighting new phase transition phenomena and dimension-specific results.

## Key findings

- Gaussian free fields yield DC processes at zero threshold
- For Gaussian vectors with nonnegative covariances, DC property holds for n=3 but not for n=4
- Stable vectors exhibit a phase transition at stability index 1/2

## Abstract

We study the question of when a (\{0,1\})-valued threshold process associated to a mean zero Gaussian or a symmetric stable vector corresponds to a {\it divide and color (DC) process}. This means that the process corresponding to fixing a threshold level $h$ and letting a 1 correspond to the variable being larger than $h$ arises from a random partition of the index set followed by coloring {\it all} elements in each partition element 1 or 0 with probabilities $p$ and $1-p$, independently for different partition elements. While it turns out that all discrete Gaussian free fields yield a DC process when the threshold is zero, for general $n$-dimensional mean zero, variance one Gaussian vectors with nonnegative covariances, this is true in general when $n=3$ but is false for $n=4$. The behavior is quite different depending on whether the threshold level $h$ is zero or not and we show that there is no general monotonicity in $h$ in either direction. We also show that all constant variance discrete Gaussian free fields with a finite number of variables yield DC processes for large thresholds. In the stable case, for the simplest nontrivial symmetric stable vector with three variables, we obtain a phase transition in the stability exponent $\alpha$ at the surprising value of $1/2$; if the index of stability is larger than $1/2$, then the process yields a DC process for large $h$ while if the index of stability is smaller than $1/2$, then this is not the case.