Divide and color representations for threshold Gaussian and stable vectors
Journal article, 2020
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
Author
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
10836489 (eISSN)
Vol. 25 54Stochastics 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.
Roots
Basic sciences
Subject Categories
Probability Theory and Statistics
DOI
10.1214/20-EJP459