The \(\varvec{d_\gamma /2}\)-Variation of Distance Profiles in \(\varvec{\gamma }\)-Liouville Quantum Gravity

IF 2.2 1区 物理与天体物理 Q1 PHYSICS, MATHEMATICAL
Manan Bhatia
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引用次数: 0

Abstract

For Brownian surfaces with boundary and an interior marked point, a natural observable to consider is the distance profile, defined as the process of distances from the marked point to a variable point lying on the boundary. When the boundary is parametrized by the natural length measure on it, this distance profile turns out to be locally absolutely continuous to Brownian motion, and as a result, the boundary length measure itself has a natural interpretation as the quadratic variation process of the distance profile. In this paper, we extend this interpretation to \(\gamma \)-Liouville quantum gravity (\(\gamma \)-LQG), a one-parameter family of models of random geometry which is known to specialize to the case of Brownian geometry for the case \(\gamma =\sqrt{8/3}\). With \(d_\gamma \) denoting the Hausdorff dimension of \(\gamma \)-LQG, we show that for a \(\gamma \)-LQG surface with boundary, the natural boundary length measure can be interpreted (up to a constant factor) as the \(d_\gamma /2\)-variation process of the distance profile from an interior point.

\(\varvec{\gamma }\) -Liouville量子引力中\(\varvec{d_\gamma /2}\) -距离分布的变化
对于具有边界和内部标记点的布朗曲面,要考虑的一个自然观察值是距离轮廓,定义为从标记点到位于边界上的可变点的距离过程。当边界被其上的自然长度度量参数化时,该距离轮廓局部对布朗运动是绝对连续的,因此,边界长度度量本身可以自然地解释为距离轮廓的二次变分过程。在本文中,我们将这种解释扩展到\(\gamma \) -Liouville量子引力(\(\gamma \) -LQG),这是一种单参数随机几何模型族,已知它专门用于\(\gamma =\sqrt{8/3}\)情况下布朗几何的情况。用\(d_\gamma \)表示\(\gamma \) -LQG的Hausdorff维数,我们证明了对于具有边界的\(\gamma \) -LQG曲面,自然边界长度测量可以解释为(直到一个常数因子)距离剖面到内部点的\(d_\gamma /2\) -变化过程。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Communications in Mathematical Physics
Communications in Mathematical Physics 物理-物理:数学物理
CiteScore
4.70
自引率
8.30%
发文量
226
审稿时长
3-6 weeks
期刊介绍: The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.
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