arXiv subjects

Jason Miller

Publications and source records attributed to Jason Miller.

At least 37 records · Page 2Linked to original sources

arXiv2024

SLE$_κ(ρ)$ processes in the light cone regime on Liouville quantum gravity

We study the relationship between certain SLE$_κ(ρ)$ processes, which are variants of the Schramm-Loewner evolution with parameter $κ$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG). These processes are defined whenever $ρ> -2-κ/2$ and in this work we will focus on the light cone regime, meaning that $κ\in (0,4)$ and $\max(κ/2-4,-2-κ/2) < ρ< -2$. Such processes are self-intersecting even though ordinary SLE$_κ$ curves are simple for $κ\in (0,4)$. We show that such a process drawn on top of an independent $\sqrtκ$-LQG surface called a weight $(ρ+4)$-quantum wedge can be represented as a gluing of a pair of trees which are described by the two coordinate functions of a correlated $α$-stable Lévy process with $α= 1-2(ρ+2)/κ$. Combined with another work, this shows that bipolar oriented random planar maps with large faces can be identified in the scaling limit with an SLE$_κ(κ-4)$ curve on an independent $\sqrtκ$-LQG surface for $κ\in (4/3,2)$.

math.PR↗

arXiv2024

Connectivity of the adjacency graph of complementary components of the SLE fan

Suppose that $h$ is an instance of the Gaussian free field (GFF) on a simply connected domain $D \subseteq {\mathbf C}$ and $x,y \in \partial D$ are distinct. Fix $κ\in (0,4)$ and for each $θ\in {\mathbf R}$ let $η_θ$ be the flow line of $h$ from $x$ to $y$. Recall that for $θ_1 < θ_2$ the fan ${\mathbf F}(θ_1,θ_2)$ of flow lines of $h$ from $x$ to $y$ is the closure of the union of $η_θ$ as $θ$ varies in any fixed countable dense subset of $[θ_1,θ_2]$. We show that the adjacency graph of components of $D \setminus {\mathbf F}(θ_1,θ_2)$ is a.s. connected, meaning it a.s. holds that for every pair $U,V$ of components there exist components $U_1,\ldots,U_n$ so that $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. We further show that ${\mathbf F}(θ_1,θ_2)$ a.s. determines the flow lines used in its construction. That is, for each $θ\in [θ_1,θ_2]$ we prove that $η_θ$ is a.s. determined by ${\mathbf F}(θ_1,θ_2)$ as a set.

math.PR↗

arXiv2024

Equivalence of metric gluing and conformal welding in $γ$-Liouville quantum gravity for $γ\in (0,2)$

We consider the $γ$-Liouville quantum gravity (LQG) model for $γ\in (0,2)$, formally described by $e^{γh}$ where $h$ is a Gaussian free field on a planar domain $D$. Sheffield showed that when a certain type of LQG surface, called a quantum wedge, is decorated by an appropriate independent SLE curve, the wedge is cut into two independent surfaces which are themselves quantum wedges, and that the original surface can be recovered as a unique conformal welding. We prove that the original surface can also be obtained as a metric space quotient of the two wedges, extending results of Gwynne and Miller in the special case $γ= \sqrt{8/3}$ to the whole subcritical regime $γ\in (0,2)$. Since the proof for $γ= \sqrt{8/3}$ used estimates for Brownian surfaces, which are equivalent to $γ$-LQG surfaces only when $γ=\sqrt{8/3}$, we instead use GFF techniques to establish estimates relating distances, areas and boundary lengths, as well as bi-Hölder continuity of the LQG metric w.r.t. the Euclidean metric at the boundary, which may be of independent interest.

math.PR↗

arXiv2023

On the Sobolev removability of the graph of one-dimensional Brownian motion

Suppose that $B$ is a one-dimensional Brownian motion and let $Γ= \{ (t, B_t) : t \in [0,1]\}$ be the graph of $B|_{[0,1]}$. We characterize the Sobolev removability properties of $Γ$ by showing that $Γ$ is almost surely not $W^{1,p}$--removable for all $p \in [1, \infty)$ but is almost surely $W^{1,\infty}$--removable.

math.PR↗

arXiv2023

Geodesics in the Brownian map: Strong confluence and geometric structure

We study geodesics in the Brownian map $(\mathcal{S},d,ν)$, the random metric measure space which arises as the Gromov-Hausdorff scaling limit of uniformly random planar maps. Our results apply to all geodesics including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most $5$, and the maximal number of geodesics which connect any pair of points is $9$. For each $1\le k \le 9$, we obtain the Hausdorff dimension of the pairs of points connected by exactly $k$ geodesics. For $k=7,8,9$, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points in $\mathcal{S}$, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting $ν$-typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame of $\mathcal{S}$, the union of all of the geodesics in $\mathcal{S}$ minus their endpoints, has dimension one, the dimension of a single geodesic.

math.PR↗

arXiv2023

Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles

We prove the existence and uniqueness of the canonical conformally covariant volume measure on the carpet/gasket of a conformal loop ensemble (CLE$_κ$, $κ\in (8/3,8)$) which respects the Markov property for CLE. The starting point for the construction is the existence of the canonical measure on CLE in the context of Liouville quantum gravity (LQG) previously constructed by the first author with Sheffield and Werner. As a warm-up, we construct the natural parameterization of SLE$_κ$ for $κ\in (4,8)$ using LQG which serves to complement earlier work of Benoist on the case $κ\in (0,4)$.

math.PR↗

arXiv2022

Conformal removability of SLE$_4$

We consider the Schramm-Loewner evolution (SLE$_κ$) with $κ=4$, the critical value of $κ> 0$ at or below which SLE$_κ$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable, answering a question posed by Sheffield. Such curves arise as the conformal welding of a pair of independent critical ($γ=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\mathbf C}$ to be conformally removable which applies in the case that $X$ is not necessarily the boundary of a simply connected domain.

math.PR↗

arXiv2022

A continuous proof of the existence of the SLE$_8$ curve

Suppose that $η$ is a whole-plane space-filling SLE$_κ$ for $κ\in (4,8)$ from $\infty$ to $\infty$ parameterized by Lebesgue measure and normalized so that $η(0) = 0$. For each $T > 0$ and $κ\in (4,8)$ we let $μ_{κ,T}$ denote the law of $η|_{[0,T]}$. We show for each $ν, T > 0$ that the family of laws $μ_{κ,T}$ for $κ\in [4+ν,8)$ is compact in the weak topology associated with the space of probability measures on continuous curves $[0,T] \to {\mathbf C}$ equipped with the uniform distance. As a direct byproduct of this tightness result (taking a limit as $κ\uparrow 8$), we obtain a new proof of the existence of the SLE$_8$ curve which does not build on the discrete uniform spanning tree scaling limit of Lawler-Schramm-Werner.

math.PR↗

arXiv2022

Bipolar oriented random planar maps with large faces and exotic SLE$_κ(ρ)$ processes

We consider bipolar oriented random planar maps with heavy-tailed face degrees. We show for each $α\in (1,2)$ that if the face degree is in the domain of attraction of an $α$-stable Lévy process, the corresponding random planar map has an infinite volume limit in the Benjamini-Schramm topology. We also show in the limit that the properly rescaled contour functions associated with the northwest and southeast trees converge in law to a certain correlated pair of $α$-stable Lévy processes. Combined with other work, this allows us to identify the scaling limit of the planar map with an SLE$_κ(ρ)$ process with $ρ= κ-4 < -2$ on $\sqrtκ$-Liouville quantum gravity for $κ\in (4/3,2)$ where $α, κ$ are related by $α= 4/κ-1$.

math.PR↗

arXiv2021

Tightness of approximations to the chemical distance metric for simple conformal loop ensembles

Suppose that $Γ$ is a conformal loop ensemble (CLE$_κ$) with simple loops ($κ\in (8/3,4)$) in a simply connected domain $D \subseteq {\mathbf C}$ whose boundary is itself a type of CLE$_κ$ loop. Let $Υ$ be the carpet of $Γ$, i.e., the set of points in $D$ not surrounded by a loop of $Γ$. We prove that certain approximations to the chemical distance metric in $Υ$ are tight. More precisely, for each path $ω\colon [0,1] \to Υ$ and $ε> 0$ we let ${\mathfrak N}_ε(ω)$ be the Lebesgue measure of the $ε$-neighborhood of $ω$. For $z,w \in Υ$ we let ${\mathfrak d}_ε(z,w;Γ) = \inf_ω{\mathfrak N}_ε(ω)$ where the infimum is over all paths $ω\colon [0,1] \to Υ$ with $ω(0) = z$, $ω(1) = w$ and let ${\mathfrak m}_ε$ be the median of $\sup_{z,w \in \partial D} {\mathfrak d}_ε(z,w;Γ)$. We prove that $(z,w) \mapsto {\mathfrak m}_ε^{-1} {\mathfrak d}_ε(z,w;Γ)$ is tight and that any subsequential limit defines a geodesic metric on $Υ$ which is Hölder continuous with respect to the Euclidean metric. We conjecture that the subsequential limit is unique, conformally covariant, and describes the scaling limit of the chemical distance metric for discrete loop models which converge to CLE$_κ$ for $κ\in (8/3,4)$ such as the critical Ising model.

math.PR↗

arXiv2021

Simple Conformal Loop Ensembles on Liouville Quantum Gravity

We show that when one draws a simple conformal loop ensemble (CLE$_κ$ for $κ\in (8/3,4)$) on an independent $\sqrtκ$-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum surfaces (e.g., corresponding to the interior of the CLE loops) that are cut out form a Poisson point process of quantum disks. This construction allows us to make direct links between CLE on LQG, asymmetric $(4/κ)$-stable processes, and labeled branching trees. The ratio between positive and negative jump intensities of these processes turns out to be $-\cos (4 π/ κ)$, which can be interpreted as a "density" of CLE loops in the CLE on LQG setting. Positive jumps correspond to the discovery of a CLE loop (where the LQG length of the loop is given by the jump size) and negative jumps correspond to the moments where the discovery process splits the remaining to be discovered domain into two pieces. Some consequences are the following: (i) It provides a construction of a CLE on LQG as a patchwork/welding of quantum disks. (ii) It allows to construct the "natural quantum measure" that lives in a CLE carpet. (iii) It enables us to derive some new properties and formulas for SLE processes and CLE themselves (without LQG) such as the exact distribution of the trunk of the general asymmetric SLE$_κ(κ-6)$ processes. The present work deals directly with structures in the continuum and makes no reference to discrete models, but our calculations match those for scaling limits of O(N) models on planar maps with large faces and CLE on LQG. Indeed, our Lévy-tree descriptions are exactly the ones that appear in the study of the large-scale limit of peeling of discrete decorated planar maps such as in recent work of Bertoin, Budd, Curien and Kortchemski. The case of non-simple CLEs on LQG is the topic of another paper.

math.PR↗

arXiv2021

Machine Learning for Discovering Effective Interaction Kernels between Celestial Bodies from Ephemerides

Building accurate and predictive models of the underlying mechanisms of celestial motion has inspired fundamental developments in theoretical physics. Candidate theories seek to explain observations and predict future positions of planets, stars, and other astronomical bodies as faithfully as possible. We use a data-driven learning approach, extending that developed in Lu et al. ($2019$) and extended in Zhong et al. ($2020$), to a derive stable and accurate model for the motion of celestial bodies in our Solar System. Our model is based on a collective dynamics framework, and is learned from the NASA Jet Propulsion Lab's development ephemerides. By modeling the major astronomical bodies in the Solar System as pairwise interacting agents, our learned model generate extremely accurate dynamics that preserve not only intrinsic geometric properties of the orbits, but also highly sensitive features of the dynamics, such as perihelion precession rates. Our learned model can provide a unified explanation to the observation data, especially in terms of reproducing the perihelion precession of Mars, Mercury, and the Moon. Moreover, Our model outperforms Newton's Law of Universal Gravitation in all cases and performs similarly to, and exceeds on the Moon, the Einstein-Infeld-Hoffman equations derived from Einstein's theory of general relativity.

astro-ph.EP↗

arXiv2021

An invariance principle for ergodic scale-free random environments

There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant {\em modulo scaling}. For our purposes, an ``environment'' consists of an infinite random planar map embedded in $\mathbb C$, each of whose edges comes with a positive real conductance. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense. Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program. One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an {\em infinite} energy embedding, it suffices to show that one can perturb it to make the energy finite.

math.PR↗

arXiv2021

Non-simple conformal loop ensembles on Liouville quantum gravity and the law of CLE percolation interfaces

We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble CLE$_{κ'}$ for $κ'$ in $(4,8)$ that is drawn on an independent $γ$-LQG surface for $γ^2=16/κ'$. The results are similar in flavor to the ones from our paper dealing with CLE$_κ$ for $κ$ in $(8/3,4)$, where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the CLE$_{κ'}$ in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: Our previous paper "CLE percolations" described the law of interfaces obtained when coloring the loops of a CLE$_{κ'}$ independently into two colors with respective probabilities $p$ and $1-p$. This description was complete up to one missing parameter $ρ$. The results of the present paper about CLE on LQG allow us to determine its value in terms of $p$ and $κ'$. It shows in particular that CLE$_{κ'}$ and CLE$_{16/κ'}$ are related via a continuum analog of the Edwards-Sokal coupling between FK$_q$ percolation and the $q$-state Potts model (which makes sense even for non-integer $q$ between $1$ and $4$) if and only if $q=4\cos^2(4π/κ')$. This provides further evidence for the long-standing belief that CLE$_{κ'}$ and CLE$_{16/κ'}$ represent the scaling limits of FK$_q$ percolation and the $q$-Potts model when $q$ and $κ'$ are related in this way. Another consequence of the formula for $ρ(p,κ')$ is the value of half-plane arm exponents for such divide-and-color models (a.k.a. fuzzy Potts models) that turn out to take a somewhat different form than the usual critical exponents for two-dimensional models.

math.PR↗

arXiv2021

Liouville quantum gravity and the Brownian map II: geodesics and continuity of the embedding

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 naturally parameterized by the Euclidean sphere ${\mathbf S}^2$. Previous work in this series used quantum Loewner evolution (QLE) to construct a metric $d_{\mathcal Q}$ on a countable dense subset of ${\mathbf S}^2$. Here we show that $d_{\mathcal Q}$ a.s. extends uniquely and continuously to a metric $\bar{d}_{\mathcal Q}$ on all of ${\mathbf S}^2$. Letting $d$ denote the Euclidean metric on ${\mathbf S}^2$, we show that the identity map between $({\mathbf S}^2, d)$ and $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ is a.s. Hölder continuous in both directions. We establish several other properties of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$, culminating in the fact that (as a random metric measure space) it agrees in law with the Brownian map. We establish analogous results for the Brownian disk and plane. Our proofs involve new estimates on the size and shape of QLE balls and related quantum surfaces, as well as a careful analysis of $({\mathbf S}^2, \bar{d}_{\mathcal Q})$ geodesics.

math.PR↗

arXiv2021

Learning Interaction Kernels for Agent Systems on Riemannian Manifolds

Interacting agent and particle systems are extensively used to model complex phenomena in science and engineering. We consider the problem of learning interaction kernels in these dynamical systems constrained to evolve on Riemannian manifolds from given trajectory data. The models we consider are based on interaction kernels depending on pairwise Riemannian distances between agents, with agents interacting locally along the direction of the shortest geodesic connecting them. We show that our estimators converge at a rate that is independent of the dimension of the state space, and derive bounds on the trajectory estimation error, on the manifold, between the observed and estimated dynamics. We demonstrate the performance of our estimator on two classical first order interacting systems: Opinion Dynamics and a Predator-Swarm system, with each system constrained on two prototypical manifolds, the $2$-dimensional sphere and the Poincaré disk model of hyperbolic space.

cs.LG↗

arXiv2021

The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings) of certain random planar maps converge to $γ$-Liouville quantum gravity ($γ$-LQG). Specifically, we treat mated-CRT maps, which are discretized matings of correlated continuum random trees, and $γ$ ranges from $0$ to $2$ as one varies the correlation parameter. We also show that the associated space-filling path on the embedded map converges to space-filling SLE$_κ$ for $κ=16/γ^2$ (in the annealed sense) and that simple random walk on the embedded map converges to Brownian motion (in the quenched sense). This work constitutes the first proof that a discrete conformal embedding of a random planar map converges to LQG. Many more such statements have been conjectured. Since the mated-CRT map can be viewed as a coarse-grained approximation to other random planar maps (the UIPT, tree-weighted maps, bipolar-oriented maps, etc.), our results indicate a potential approach for proving that embeddings of these maps converge to LQG as well. To prove the main result, we establish several (independently interesting) theorems about LQG surfaces decorated by space-filling SLE. There is a natural way to use the SLE curve to divide the plane into "cells" corresponding to vertices of the mated-CRT map. We study the law of the shape of the origin-containing cell, in particular proving moments for the ratio of its squared diameter to its area. We also give bounds on the degree of the origin-containing cell and establish a form of ergodicity for the entire configuration. Ultimately, we use these properties to show (with the help of a general theorem proved in a separate paper) that random walk on these cells converges to a time change of Brownian motion, which in turn leads to the Tutte embedding result.

math.PR↗

arXiv2021

Characterizations of SLE$_κ$ for $κ\in (4,8)$ on Liouville quantum gravity

We prove that SLE$_κ$ for $κ\in (4,8)$ on an independent $γ=4/\sqrtκ$-Liouville quantum gravity (LQG) surface is uniquely characterized by the form of its LQG boundary length process and the form of the conditional law of the unexplored quantum surface given the explored curve-decorated quantum surface up to each time $t$. We prove variants of this characterization for both whole-plane space-filling SLE$_κ$ on an infinite-volume LQG surface and for chordal SLE$_κ$ on a finite-volume LQG surface with boundary. Using the equivalence of Brownian and $\sqrt{8/3}$-LQG surfaces, we deduce that SLE$_6$ on the Brownian disk is uniquely characterized by the form of its boundary length process and that the complementary connected components of the curve up to each time $t$ are themselves conditionally independent Brownian disks given this boundary length process. The results of this paper are used in another paper by the same authors to show that the scaling limit of percolation on random quadrangulations is given by SLE$_6$ on $\sqrt{8/3}$-LQG with respect to the Gromov-Hausdorff-Prokhorov-uniform topology, the natural analog of the Gromov-Hausdorff topology for curve-decorated metric measure spaces.

math.PR↗