Divide and color representations for threshold Gaussian and stable vectors
Journal article, 2020

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 stable vectors

threshold Gaus- sian vectors

Divide and color representations


Malin Palö Forsström

Royal Institute of Technology (KTH)

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Jeffrey Steif

Chalmers, Mathematical Sciences, Analysis and Probability Theory

Electronic Journal of Probability

1083-6489 (ISSN)

Vol. 25 54

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

Knut and Alice Wallenberg Foundation (KAW2012,0067), 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) (2016-03835), 2017-01-01 -- 2020-12-31.


Basic sciences

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Probability Theory and Statistics



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