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Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields

Abstract

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 \(N^{1+ O(\varepsilon )}\), where N is the side length of the box and \(\varepsilon \) 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.

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Acknowledgements

We warmly thank Ofer Zeitouni, Pascal Maillard, Steve Lalley, Marek Biskup, Rémi Rhodes and Vincent Vargas for many helpful discussions.

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Correspondence to Jian Ding.

Additional information

Jian Ding: Partially supported by NSF Grant DMS-1313596, and NSF of China 11628101.

Fuxi Zhang: Supported by NSF of China 11371040.

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Ding, J., Zhang, F. Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields. Probab. Theory Relat. Fields 171, 1157–1188 (2018). https://doi.org/10.1007/s00440-017-0811-z

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  • DOI: https://doi.org/10.1007/s00440-017-0811-z

Mathematics Subject Classification

  • 60G60
  • 60K35