Liouville first passage percolation: geodesic length exponent is strictly larger than 1 at high temperatures. (English) Zbl 1412.60072
Summary: 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.
MSC:
| 60G60 | Random fields |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
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