# Divide and color representations for threshold Gaussian and stable vectors Artikel i vetenskaplig tidskrift, 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

## Författare

Kungliga Tekniska Högskolan (KTH)

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

### Jeffrey Steif

Chalmers, Matematiska vetenskaper, Analys och sannolikhetsteori

1083-6489 (ISSN)

Vol. 25 54

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

Knut och Alice Wallenbergs Stiftelse, 2013-01-01 -- 2018-09-01.

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

Vetenskapsrådet (VR), 2017-01-01 -- 2020-12-31.

### Fundament

Grundläggande vetenskaper

### Ämneskategorier

Sannolikhetsteori och statistik

### DOI

10.1214/20-EJP459

2021-01-28