The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions

Jason Miller; Wei Qian

doi:10.1007/s00440-019-00949-7

The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions

Jason Miller
, Wei Qian^*

^*Corresponding author for this work

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

19 Citations (Scopus)

53 Downloads (CityUHK Scholars)

Abstract

We prove that the geodesics associated with any metric generated from Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are necessarily singular with respect to the law of any type of SLE_κ. These hypotheses are satisfied by the LQG metric for γ = √8/3 constructed by the first author and Sheffield, and subsequent work by Gwynne and the first author has shown that there is a unique metric which satisfies these hypotheses for each γ ∈ (0 , 2). As a consequence of our analysis, we also establish certain regularity properties of LQG geodesics which imply, among other things, that they are conformally removable.

Original language	English
Pages (from-to)	677-709
Journal	Probability Theory and Related Fields
Volume	177
Issue number	3-4
Online published	21 Dec 2019
DOIs	https://doi.org/10.1007/s00440-019-00949-7
Publication status	Published - Aug 2020
Externally published	Yes

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abstract = "We prove that the geodesics associated with any metric generated from Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are necessarily singular with respect to the law of any type of SLEκ. These hypotheses are satisfied by the LQG metric for γ = √8/3 constructed by the first author and Sheffield, and subsequent work by Gwynne and the first author has shown that there is a unique metric which satisfies these hypotheses for each γ ∈ (0 , 2). As a consequence of our analysis, we also establish certain regularity properties of LQG geodesics which imply, among other things, that they are conformally removable.",

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AB - We prove that the geodesics associated with any metric generated from Liouville quantum gravity (LQG) which satisfies certain natural hypotheses are necessarily singular with respect to the law of any type of SLEκ. These hypotheses are satisfied by the LQG metric for γ = √8/3 constructed by the first author and Sheffield, and subsequent work by Gwynne and the first author has shown that there is a unique metric which satisfies these hypotheses for each γ ∈ (0 , 2). As a consequence of our analysis, we also establish certain regularity properties of LQG geodesics which imply, among other things, that they are conformally removable.

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SP - 677

EP - 709

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -

The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions

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