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Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields
Ist Teil von
Probability theory and related fields, 2018-08, Vol.171 (3-4), p.1157-1188
Ort / Verlag
Berlin/Heidelberg: Springer Berlin Heidelberg
Erscheinungsjahr
2018
Link zum Volltext
Quelle
EBSCOhost Business Source Ultimate
Beschreibungen/Notizen
We consider first passage percolation (FPP) where the vertex weight is given by the exponential of two-dimensional log-correlated Gaussian fields. Our work is motivated by understanding the discrete analog for the random metric associated with
Liouville quantum gravity
(LQG), which roughly corresponds to the exponential of a two-dimensional Gaussian free field (GFF). The particular focus of the present paper is an aspect of universality for such FPP among the family of log-correlated Gaussian fields. More precisely, we construct a family of log-correlated Gaussian fields, and show that the FPP distance between two typically sampled vertices (according to the LQG measure) is
N
1
+
O
(
ε
)
, where
N
is the side length of the box and
ε
can be made arbitrarily small if we tune a certain parameter in our construction. That is, the exponents can be arbitrarily close to 1. Combined with a recent work of the first author and Goswami on an upper bound for this exponent when the underlying field is a GFF, our result implies that such exponent is
not
universal among the family of log-correlated Gaussian fields.