Divide and color representations for threshold Gaussian and stable vectors

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 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 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 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.