Ewain Gwynne
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View article: Random walk on sphere packings and Delaunay triangulations in arbitrary dimension
Random walk on sphere packings and Delaunay triangulations in arbitrary dimension Open
We prove that random walks on a family of tilings of ‐dimensional Euclidean space, with a canonical choice of conductances, converge to Brownian motion modulo time parameterization. This class of tilings includes Delaunay triangulations (t…
View article: Permutons, meanders, and SLE-decorated Liouville quantum gravity
Permutons, meanders, and SLE-decorated Liouville quantum gravity Open
We study a class of random permutons which can be constructed from a pair of space-filling Schramm–Loewner evolution (SLE) curves on a Liouville quantum gravity (LQG) surface. This class includes the skew Brownian permutons introduced by B…
View article: Liouville Brownian motion and quantum cones in dimension $d > 2$
Liouville Brownian motion and quantum cones in dimension $d > 2$ Open
For $d > 2$ and $γ\in (0, \sqrt{2d})$, we study the Liouville Brownian motion associated with the whole-space log-correlated Gaussian field in $\mathbb{R}^d$. We compute its spectral dimension, i.e., the short-time asymptotics of the heat …
View article: Harmonic balls in Liouville quantum gravity
Harmonic balls in Liouville quantum gravity Open
Harmonic balls are domains that 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 Hele–…
View article: Percolation of thick points of the log-correlated Gaussian field in high dimensions
Percolation of thick points of the log-correlated Gaussian field in high dimensions Open
We prove that the set of thick points of the log-correlated Gaussian field contains an unbounded path in sufficiently high dimensions. This contrasts with the two-dimensional case, where Aru, Papon, and Powell (2023) showed that the set of…
View article: Approximation of length metrics by conformally flat Riemannian metrics
Approximation of length metrics by conformally flat Riemannian metrics Open
We present a proof of the folklore result that any length metric on $\mathbb R^d$ can be approximated by conformally flat Riemannian distance functions in the uniform distance. This result is used to study Liouville quantum gravity in anot…
View article: Area measures and branched polymers in supercritical Liouville quantum gravity
Area measures and branched polymers in supercritical Liouville quantum gravity Open
We study Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which has background charge $Q \in (0,2)$ and central charge $\mathbf{c}_{\mathrm{L}} = 1+6Q^2 \in (1,25)$. Recent works have shown how to defin…
View article: Reconstructing SLE-decorated Liouville quantum gravity surfaces from random permutons
Reconstructing SLE-decorated Liouville quantum gravity surfaces from random permutons Open
Permutons constructed from a Liouville quantum gravity surface and a pair of space-filling Schramm-Loewner evolutions (SLEs) have been shown -- or are conjectured -- to describe the scaling limit of various natural models of random constra…
View article: Gaussian curvature on random planar maps and Liouville quantum gravity
Gaussian curvature on random planar maps and Liouville quantum gravity Open
We investigate the notion of curvature in the context of Liouville quantum gravity (LQG) surfaces. We define the Gaussian curvature for LQG, which we conjecture is the scaling limit of discrete curvature on random planar maps. Motivated by…
View article: Random walk on sphere packings and Delaunay triangulations in arbitrary dimension
Random walk on sphere packings and Delaunay triangulations in arbitrary dimension Open
We prove that random walks on a family of tilings of d-dimensional Euclidean space, with a canonical choice of conductances, converge to Brownian motion modulo time parameterization. This class of tilings includes Delaunay triangulations (…
View article: Regularity and confluence of geodesics for the supercritical Liouville quantum gravity metric
Regularity and confluence of geodesics for the supercritical Liouville quantum gravity metric Open
We show that for each c M ∈ [1, 25), there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge c M .An earlier series of works by Ding-Dubédat-Dunlap-Falconet, Gwynne-Miller, and others showed that…
View article: Introduction to the Liouville quantum gravity metric
Introduction to the Liouville quantum gravity metric Open
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 gi…
View article: Cutting $γ$-Liouville quantum gravity by Schramm-Loewner evolution for $κ\not\in \{γ^2, 16/γ^2\}$
Cutting $γ$-Liouville quantum gravity by Schramm-Loewner evolution for $κ\not\in \{γ^2, 16/γ^2\}$ Open
There are many deep and useful theorems relating Schramm-Loewner evolution (SLE$_κ$) and Liouville quantum gravity ($γ$-LQG) in the case when the parameters satisfy $κ\in \{γ^2, 16/γ^2\}$. Roughly speaking, these theorems say that the SLE$…
View article: Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension
Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension Open
We prove the tightness of a natural approximation scheme for an analog of the Liouville quantum gravity metric on $\mathbb R^d$ for arbitrary $d\geq 2$. More precisely, let $\{h_n\}_{n\geq 1}$ be a suitable sequence of Gaussian random func…
View article: Supercritical Liouville quantum gravity and CLE$_4$
Supercritical Liouville quantum gravity and CLE$_4$ Open
We establish the first relationship between Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) in the supercritical (a.k.a. strongly coupled) phase, which corresponds to central charge values $\mathbf c_{\mathrm L} \in (1,…
View article: Critical Liouville quantum gravity and CLE$_4$
Critical Liouville quantum gravity and CLE$_4$ Open
Consider a critical ($γ=2$) Liouville quantum gravity (LQG) disk together with an independent conformal loop ensemble (CLE) with parameter $κ=4$. We show that the critical LQG surfaces parametrized by the regions enclosed by the CLE$_4$ lo…
View article: Scaling limits of planar maps under the Smith embedding
Scaling limits of planar maps under the Smith embedding Open
The Smith embedding of a finite planar map with two marked vertices, possibly with conductances on the edges, is a way of representing the map as a tiling of a finite cylinder by rectangles. In this embedding, each edge of the planar map c…
View article: A support theorem for exponential metrics of log-correlated Gaussian fields in arbitrary dimension
A support theorem for exponential metrics of log-correlated Gaussian fields in arbitrary dimension Open
Let $h$ be a log-correlated Gaussian field on $\R^d$, let $γ\in (0,\sqrt{2d}),$ let $μ_h$ be the $γ$-Gaussian multiplicative chaos measure, and let $D_h$ be an exponential metric associated with $h$ satisfying certain natural axioms. In th…
View article: Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons
Power-law bounds for increasing subsequences in Brownian separable permutons and homogeneous sets in Brownian cographons Open
The Brownian separable permutons are a one-parameter family -- indexed by $p\in(0,1)$ -- of universal limits of random constrained permutations. We show that for each $p\in (0,1)$, there are explicit constants $1/2 < α_*(p) \leq β^*(p) < 1…
View article: On the geometry of uniform meandric systems
On the geometry of uniform meandric systems Open
A meandric system of size $n$ is the set of loops formed from two arc diagrams (non-crossing perfect matchings) on $\{1,\dots,2n\}$, one drawn above the real line and the other below the real line. A uniform random meandric system can be v…
View article: Uniqueness of the critical and supercritical Liouville quantum gravity metrics
Uniqueness of the critical and supercritical Liouville quantum gravity metrics Open
We show that for each cM ∈ [ 1 , 25 ) ${\\mathbf {c}}_{\\mathrm{M}} \\in [1,25)$ , there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge cM ${\\mathbf {c}}_{\\mathrm{M}}$ . An earlier series of…
View article: Internal DLA on mated-CRT maps
Internal DLA on mated-CRT maps Open
We prove a shape theorem for internal diffusion limited aggregation on mated-CRT maps, a family of random planar maps which approximate Liouville quantum gravity (LQG) surfaces. The limit is an LQG harmonic ball, which we constructed in a …
View article: The Minkowski content measure for the Liouville quantum gravity metric
The Minkowski content measure for the Liouville quantum gravity metric Open
A Liouville quantum gravity (LQG) surface is a natural random two-dimensional surface, initially formulated as a random measure space and later as a random metric space. We show that the LQG measure can be recovered as the Minkowski measur…
View article: Tightness of supercritical Liouville first passage percolation
Tightness of supercritical Liouville first passage percolation Open
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 \epsilon >…
View article: Harmonic balls in Liouville quantum gravity
Harmonic balls in Liouville quantum gravity Open
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 Hele…
View article: Permutons, meanders, and SLE-decorated Liouville quantum gravity
Permutons, meanders, and SLE-decorated Liouville quantum gravity Open
We study a class of random permutons which can be constructed from a pair of space-filling Schramm-Loewner evolution (SLE) curves on a Liouville quantum gravity (LQG) surface. This class includes the skew Brownian permutons introduced by B…
View article: Loewner evolution driven by complex Brownian motion (with simulations by\n Minjae Park)
Loewner evolution driven by complex Brownian motion (with simulations by\n Minjae Park) Open
We study the Loewner evolution whose driving function is $W_t = B_t^1 + i\nB_t^2$, where $(B^1,B^2)$ is a pair of Brownian motions with a given covariance\nmatrix. This model can be thought of as a generalization of Schramm-Loewner\nevolut…
View article: Loewner evolution driven by complex Brownian motion (with simulations by Minjae Park)
Loewner evolution driven by complex Brownian motion (with simulations by Minjae Park) Open
We study the Loewner evolution whose driving function is $W_t = B_t^1 + i B_t^2$, where $(B^1,B^2)$ is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm-Loewner evolution…
View article: An invariance principle for ergodic scale-free random environments
An invariance principle for ergodic scale-free random environments Open
does the same with M and M ′ reversed.Note that the integrand is equal to e -r for each r>0 such an invariance principle for ergodic scale-free random environments 305 that no such homeomorphism exists.When we speak of random embedded latt…