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Bounds for distances and geodesic dimension in Liouville first passage percolation. (English) Zbl 1459.60022
Summary: For \(\xi \geq 0\), Liouville first passage percolation (LFPP) is the random metric on \(\varepsilon \mathbb{Z} ^2\) obtained by weighting each vertex by \(\varepsilon e^{\xi h_{\varepsilon }(z)}\), where \(h_{\varepsilon }(z)\) is the average of the whole-plane Gaussian free field \(h\) over the circle \(\partial B_{\varepsilon }(z)\). J. Ding and the first author [Commun. Math. Phys. 374, No. 3, 1877–1934 (2020; Zbl 1436.83024)] showed that for \(\gamma \in (0,2)\), LFPP with parameter \(\xi = \gamma /d_{\gamma }\) is related to \(\gamma \)-Liouville quantum gravity (LQG), where \(d_{\gamma }\) is the \(\gamma \)-LQG dimension exponent. For \(\xi > 2/d_2\), LFPP is instead expected to be related to LQG with central charge greater than 1. We prove several estimates for LFPP distances for general \(\xi \geq 0\). For \(\xi \leq 2/d_2\), this leads to new bounds for \(d_{\gamma }\) which improve on the best previously known upper (resp. lower) bounds for \(d_{\gamma }\) in the case when \(\gamma > \sqrt{8/3} \) (resp. \(\gamma \in (0.4981, \sqrt{8/3} ))\). These bounds are consistent with the Y. Watabiki [“Analytic study of fractal structure of quantized surface in two-dimensional quantum gravity”, Progr. Theor. Phys. Suppl. 114, 1–17 (1993)]. prediction for \(d_{\gamma }\). However, for \(\xi > 1/\sqrt{3} \) (or equivalently for LQG with central charge larger than 17) our bounds are inconsistent with the analytic continuation of Watabiki’s prediction to the \(\xi >2/d_2\) regime. We also obtain an upper bound for the Euclidean dimension of LFPP geodesics.

MSC:
60D05 Geometric probability and stochastic geometry
60G15 Gaussian processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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