The random pseudo-metric on a graph defined via the zero-set of the Gaussian free field on its metric graph
Open access
Date
2018-08Type
- Journal Article
Abstract
We further investigate properties of the Gaussian free field (GFF) on the metric graph associated to a discrete weighted graph (where the edges of the latter are replaced by continuous line-segments of appropriate length) that has been introduced by the first author. On such a metric graph, the GFF is a random continuous function that generalises one-dimensional Brownian bridges so that one-dimensional techniques can be used. In the present paper, we define and study the pseudo-metric defined on the metric graph (and therefore also on the discrete graph itself), where the length of a path on the metric graph is defined to be the local time at level zero accumulated by the Gaussian free field along this path. We first derive a pathwise transformation that relates the GFF on the metric graph with the reflected GFF on the metric graph via the pseudo-distance defined by the latter. This is a generalisation of Paul Lévy’s result relating the local time at zero of Brownian motion to the supremum of another Brownian motion. We also compute explicitly the distribution of certain functionals of this pseudo-metric and of the GFF. In particular, we point out that when the boundary consists of just two points, the law of the pseudo-distance between them depends solely on the resistance of the network between them. We then discuss questions related to the scaling limit of this pseudo-metric in the two-dimensional case, which should be the conformally invariant way to measure distances between CLE(4) loops introduced and studied by the second author with Wu, and by Sheffield, Watson and Wu. Our explicit laws on metric graphs also lead to new conjectures for related functionals of the continuum GFF on fairly general Riemann surfaces. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000192290Publication status
publishedExternal links
Journal / series
Probability Theory and Related FieldsVolume
Pages / Article No.
Publisher
SpringerSubject
Random distances; Gaussian free field; Metric graph; Local time; Conformal loop ensembleOrganisational unit
09453 - Werner, Wendelin (ehemalig) / Werner, Wendelin (former)
02889 - ETH Institut für Theoretische Studien / ETH Institute for Theoretical Studies
Funding
155922 - Exploring two-dimensional continuous structures (SNF)
Related publications and datasets
Is new version of: http://hdl.handle.net/20.500.11850/126914
Notes
It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.More
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