arXiv subjects

Morris Ang

Publications and source records attributed to Morris Ang.

25 records · Page 2Linked to original sources

arXiv2022

Integrability of SLE via conformal welding of random surfaces

We demonstrate how to obtain integrable results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called $\mathrm{SLE}_κ(ρ_-;ρ_+)$. Our proof is built on two connections between SLE, LCFT, and mating-of-trees. Firstly, LCFT and mating-of-trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embedding of an LQG surface, we extend earlier equivalence results by allowing fewer marked points and more generic singularities. Secondly, the conformal welding of these random surfaces produces SLE curves as their interfaces. In particular, we rely on the conformal welding results proved in our companion paper [AHS20]. Our paper is an essential part of a program proving integrability results for SLE, LCFT, and mating-of-trees based on these two connections.

math.PR↗

arXiv2021

Conformal welding of quantum disks

Two-pointed quantum disks with a weight parameter $W > 0$ are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with an independent chordal $\mathrm{SLE}_κ(ρ_-;ρ_+)$ curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees theory. Our results can be used to give unified proofs of the mating-of-trees theorems for the quantum disk and the quantum sphere, in addition to a mating-of-trees description of the weight $W = \frac{γ^2}{2}$ quantum disk. Moreover, it serves as a key ingredient in our companion work [AHS21], which proves an exact formula for $\mathrm{SLE}_κ(ρ_-;ρ_+)$ using conformal welding of random surfaces and a conformal welding result giving the so-called SLE loop.

math.PR↗

arXiv2020

Volume of metric balls in Liouville quantum gravity

We study the volume of metric balls in Liouville quantum gravity (LQG). For $γ\in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of Euclidean balls has finite moments exactly for $p \in (-\infty, 4/γ^2)$. Here, we prove that the LQG volume of LQG metric balls admits all finite moments. This answers a question of Gwynne and Miller and generalizes a result obtained by Le Gall for the Brownian map, namely, the $γ= \sqrt{8/3}$ case. We use this moment bound to show that on a compact set the volume of metric balls of size $r$ is given by $r^{d_γ+o_r(1)}$, where $d_γ$ is the dimension of the LQG metric space. Using similar techniques, we prove analogous results for the first exit time of Liouville Brownian motion from a metric ball. Gwynne-Miller-Sheffield (2020) prove that the metric measure space structure of $γ$-LQG a.s. determines its conformal structure when $γ=\sqrt{8/3}$; their argument and our estimate yield the result for all $γ\in (0,2)$.

math.PR↗

arXiv2020

Large deviations of radial SLE$_{\infty}$

We derive the large deviation principle for radial Schramm-Loewner evolution ($\operatorname{SLE}$) on the unit disk with parameter $κ\rightarrow \infty$. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures $\{ϕ_t^2 (ζ)\, dζ\}_{t \in [0,1]}$ on the unit circle and equals $\int_0^1 \int_{S^1} |ϕ_t'|^2/2\,dζ\,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.

math.PR↗

arXiv2020

Brownian loops and the central charge of a Liouville random surface

We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace-Beltrami operator on a compact surface, with or without boundary. We relate the $(-c/2)$-th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity $c$ on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity $c$ corresponds to weighting the law of the surface by the $(-c/2)$-th power of the determinant of the Laplacian. Next, we introduce a method of regularizing a Liouville quantum gravity (LQG) surface (with some matter central charge parameter $\mathbf{c}$) to produce a smooth surface. And we show that weighting the law of this random surface by the $( -\mathbf{c}'/ 2)$-th power of the Laplacian determinant has precisely the effect of changing the matter central charge from $\mathbf{c}$ to $\mathbf{c} + \mathbf{c}'$. Taken together with the earlier results, this provides a way of interpreting an LQG surface of matter central charge $\mathbf{c}$ as a pure LQG surface decorated by a Brownian loop soup of intensity $\mathbf{c}$. Building on this idea, we present several open problems about random planar maps and their continuum analogs. Although the original construction of LQG is well-defined only for $\mathbf{c}\leq 1$, some of the constructions and questions also make sense when $\mathbf{c}>1$.

math.PR↗

arXiv2020

Liouville quantum gravity surfaces with boundary as matings of trees

For $γ\in (0,2)$, the quantum disk and $γ$-quantum wedge are two of the most natural types of Liouville quantum gravity (LQG) surfaces with boundary. These surfaces arise as scaling limits of finite and infinite random planar maps with boundary, respectively. We show that the left/right quantum boundary length process of a space-filling SLE$_{16/γ^2}$ curve on a quantum disk or on a $γ$-quantum wedge is a certain explicit conditioned two-dimensional Brownian motion with correlation $-\cos(πγ^2/4)$. This extends the mating of trees theorem of Duplantier, Miller, and Sheffield (2014) to the case of quantum surfaces with boundary (the disk case for $γ\in (\sqrt 2 , 2)$ was previously treated by Duplantier, Miller, Sheffield using different methods). As an application, we give an explicit formula for the conditional law of the LQG area of a quantum disk given its boundary length by computing the law of the corresponding functional of the correlated Brownian motion.

math.PR↗

arXiv2019

Comparison of discrete and continuum Liouville first passage percolation

Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural $γ$-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the circle average regularization of the continuum GFF respectively. We show that these two models can be coupled so that with high probability distances in these models agree up to $o(1)$ errors in the exponent, and thus have the same distance exponent. Ding and Gwynne (2018) give a formula for the continuum LFPP distance exponent in terms of the $γ$-LQG dimension exponent $d_γ$. Using results of Ding and Li (2018) on the level set percolation of the discrete GFF, we bound the DLFPP distance exponent and hence obtain a new lower bound $d_γ\geq 2 + \frac{γ^2}2$. This improves on previous lower bounds for $d_γ$ for the regime $γ\in (γ_0, 0.576)$, for some small nonexplicit $γ_0 > 0$.

math.PR↗