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Divide and color representations for threshold Gaussian and stable vectors
Preprint, 2019

We study the question of when a {0, 1}-valued threshold process asso- ciated to a mean zero Gaussian or a symmetric stable vector corresponds to a divide and color (DC) process. This means that the process corre- sponding 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 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 α 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.

threshold Gaus- sian vectors

threshold stable vectors

Divide and color representations

## Author

### Malin Palo Forsstrom

### Jeffrey Steif

Chalmers, Mathematical Sciences, Analysis and Probability Theory

### Stochastics for big data and big systems - bridging local and global

Knut and Alice Wallenberg Foundation, 2013-01-01 -- 2018-09-01.

### Färgning av slumpmässiga ekvivalensrelationer, slumpvandringar på dynamisk perkolation och bruskänslighet för gränsgrafen i den Erdös-Renyi-slumpgrafsmodellen

Swedish Research Council (VR), 2017-01-01 -- 2020-12-31.

### Roots

Basic sciences

### Subject Categories

Probability Theory and Statistics