Roughness of geodesics in Liouville quantum gravity

被引:0
|
作者
Fan, Zherui [1 ]
Goswami, Subhajit [2 ]
机构
[1] Peking Univ, Sch Math Sci, 5 Yiheyuan Rd, Beijing 100871, Peoples R China
[2] Tata Inst Fundamental Res, Sch Math, 1 Homi Bhabha Rd, Mumbai 400005, India
来源
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES | 2024年 / 60卷 / 03期
关键词
Liouville quantum gravity (LQG); Liouville first passage percolation (LFPP); Gaussian free field (GFF); Random metrics; Random curves; Hausdorff dimension; PERCOLATION; GEOMETRY; BOUNDS;
D O I
10.1214/23-AIHP1377
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
The metric associated with the Liouville quantum gravity (LQG) surface has been constructed through a series of recent works and several properties of its associated geodesics have been studied. In the current article we confirm the folklore conjecture that the Euclidean Hausdorff dimension of LQG geodesics is strictly greater than 1 for all values of the so-called Liouville first passage percolation (LFPP) parameter xi. We deduce this from a general criterion due to Aizenman and Burchard ( Duke Math. J. 99 (1999), 419-453) which in our case amounts to near-geometric bounds on the probabilities of certain crossing events for LQG geodesics in the number of crossings. We obtain such bounds using the axiomatic characterization of the LQG metric after proving a special regularity property for the Gaussian free field (GFF). We also prove an analogous result for the LFPP geodesics.
引用
收藏
页码:2194 / 2210
页数:17
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