相关论文: Geodesics and metric ball boundaries in Liouville …
Let $\gamma \in (0,2)$, let $h$ be the planar Gaussian free field, and consider the $\gamma$-Liouville quantum gravity (LQG) metric associated with $h$. We show that the essential supremum of the Hausdorff dimension of the boundary of a…
We prove that for any metric which one can associate with a Liouville quantum gravity (LQG) surface for $\gamma \in (0,2)$ satisfying certain natural axioms, its geodesics exhibit the following confluence property. For any fixed point $z$,…
Let $h$ be the planar Gaussian free field and let $D_h$ be a supercritical Liouville quantum gravity (LQG) metric associated with $h$. Such metrics arise as subsequential scaling limits of supercritical Liouville first passage percolation…
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…
Let $\gamma\in (0,2)$, let $h$ be the planar Gaussian free field, and let $D_h$ be the associated $\gamma$-Liouville quantum gravity (LQG) metric. We prove that for any random Borel set $X \subset \mathbb{C}$ which is independent from $h$,…
We study the volume of metric balls in Liouville quantum gravity (LQG). For $\gamma \in (0,2)$, 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…
Physics considerations suggest that a theory of Liouville quantum gravity (LQG) should exist for all values of matter central charge $\mathbf{c} \in (-\infty,25)$. Probabilists have rigorously defined LQG as a random metric measure space…
Recent work has shown that for $\gamma \in (0,2)$, a Liouville quantum gravity (LQG) surface can be endowed with a canonical metric. We prove several results concerning geodesics for this metric. In particular, we completely classify the…
For $\gamma \in (0,2)$, we define a weak $\gamma$-Liouville quantum gravity (LQG) metric to be a function $h\mapsto D_h$ which takes in an instance of the planar Gaussian free field (GFF) and outputs a metric on the plane satisfying a…
We show that for each $\gamma \in (0,2)$, there is a unique metric (i.e., distance function) associated with $\gamma$-Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) $h$, there is…
Liouville quantum gravity (LQG) is a one-parameter family of models of random fractal surfaces which first appeared in the physics literature in the 1980s. Recent works have constructed a metric (distance function) on an LQG surface. We…
We endow the $\sqrt{8/3}$-Liouville quantum gravity sphere with a metric space structure and show that the resulting metric measure space agrees in law with the Brownian map. Recall that a Liouville quantum gravity sphere is a priori…
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$_\kappa$. These hypotheses…
In order to study the quantum geometry of random surfaces in Liouville gravity, we propose a definition of geodesic distance associated to a Gaussian free field on a regular lattice. This geodesic distance is used to numerically determine…
Liouville quantum gravity (LQG) is, heuristically, a theory of random Riemannian geometry with Riemannian metric tensor $e^{\gamma h} (\mathrm{d} x^2 + \mathrm{d} y^2)$, where $h$ is a variant of the Gaussian free field and $\gamma > 0$ is…
We introduce the concept of a local metric of the Gaussian free field (GFF) $h$, which is a random metric coupled with $h$ in such a way that it depends locally on $h$ in a certain sense. This definition is a metric analog of the concept of…
There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge ${\mathbf{c}}_{\mathrm M}\in(-\infty,1]$. Via the DDK ansatz, LQG can equivalently be described as…
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 $x$ lying on the boundary. When the…
Harmonic balls are domains which satisfy the mean-value property for harmonic functions. We establish the existence and uniqueness of harmonic balls on Liouville quantum gravity (LQG) surfaces using the obstacle problem formulation of…
For $\gamma \in (0,2)$, $U\subset \mathbb C$, and an instance $h$ of the Gaussian free field (GFF) on $U$, the $\gamma$-Liouville quantum gravity (LQG) surface associated with $(U,h)$ is formally described by the Riemannian metric tensor…