Abstract
Let \(\{\eta (v): v\in V_N\}\) be a discrete Gaussian free field in a two-dimensional box \(V_N\) of side length N with Dirichlet boundary conditions. We study the Liouville first passage percolation, i.e., the shortest path metric where each vertex is given a weight of \(e^{\gamma \eta (v)}\) for some \(\gamma >0\). We show that for sufficiently small but fixed \(\gamma >0\), with probability tending to 1 as \(N\rightarrow \infty \), all geodesics between vertices of macroscopic Euclidean distances simultaneously have (the conjecturally unique) length exponent strictly larger than 1.
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We warmly thank an anonymous referee for a detailed report, which leads to a significant improvement on exposition.
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Jian Ding: Partially supported by an NSF Grant DMS-1455049, an Alfred Sloan fellowship, and NSF of China 11628101. Fuxi Zhang: Supported by NSF of China 11371040 and 11771027.
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Ding, J., Zhang, F. Liouville first passage percolation: geodesic length exponent is strictly larger than 1 at high temperatures. Probab. Theory Relat. Fields 174, 335–367 (2019). https://doi.org/10.1007/s00440-019-00905-5
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DOI: https://doi.org/10.1007/s00440-019-00905-5
Mathematics Subject Classification
- 60G60
- 60K35