Related papers: Roughness of geodesics in Liouville quantum gravit…
For $\xi \geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\varepsilon \mathbb Z^2$ obtained by weighting each vertex by $\varepsilon e^{\xi h_\varepsilon(z)}$, where $h_\varepsilon(z)$ is the average of the…
Liouville first passage percolation (LFPP) with parameter $\xi >0$ is the family of random distance functions $\{D_h^\epsilon\}_{\epsilon >0}$ on the plane obtained by integrating $e^{\xi h_\epsilon}$ along paths, where $h_\epsilon$ for…
First passage percolation (FPP) on $\mathbb{Z}^d$ or $\mathbb{R}^d$ is a canonical model of a random metric space where the standard Euclidean geometry is distorted by random noise. Of central interest is the length and the geometry of 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…
Recent works have shown that there is a canonical way to to assign a metric (distance function) to a Liouville quantum gravity (LQG) surface for any parameter $\gamma \in (0,2)$. We establish a strong confluence property for LQG geodesics,…
For $\xi \geq 0$ and $d \geq 3$, the higher-dimensional Liouville first passage percolation (LFPP) is a random metric on $\epsilon \mathbb{Z}^d$ obtained by reweighting each vertex by $e^{\xi h_\epsilon(x)}$, where $h_\epsilon(x)$ is 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…
Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural $\gamma$-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the…
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 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 $z$,…
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…
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 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…
Liouville first passage percolation (LFPP) with parameter $\xi > 0$ is the family of random distance functions $\{D_h^\epsilon\}_{\epsilon >0}$ on the plane obtained by integrating $e^{\xi h_\epsilon}$ along paths, where…
We show that every possible metric associated with critical ($\gamma=2$) Liouville quantum gravity (LQG) induces the same topology on the plane as the Euclidean metric. More precisely, we show that the optimal modulus of continuity of the…
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…
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…
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…
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…
Liouville first passage percolation (LFPP) with parameter $\xi > 0$ is the family of random distance functions (metrics) $(D_h^{\epsilon})_{\epsilon > 0}$ on $\mathbb{C}$ obtained heuristically by integrating $e^{\xi h}$ along paths, where…