arXiv subjects

Jason Miller

Publications and source records attributed to Jason Miller.

At least 19 recordsLinked to original sources

arXiv2026

The conformally invariant metric on CLE$_4$ I: subsequential limits of the non-simple CLE graph metric

We consider the conformal loop ensemble (CLE) with the parameter $\kappa=4$, the critical value at or below which the loops are simple and do not intersect each other or the domain boundary. We show that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric so that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. This metric was previously constructed in unpublished work of Sheffield, Watson, and Wu. Our approach differs in that we show that the metric arises as the renormalized limit of the graph metric on CLE$_\kappa$ loops as $\kappa \downarrow 4$. In this first paper in a series of three, we prove that the subsequential limits exist and define a non-trivial conformally invariant metric on CLE$_4$ which is local and such that the metric ball growth from the boundary is given by the uniform exploration of Werner and Wu. In subsequent work, we will show that the subsequential limit exists as a true limit.

math.PR↗

arXiv2026

The conformally invariant metric on CLE$_4$ II: existence of geodesics

We continue our study of the conformal loop ensemble (CLE) with parameter $\kappa=4$, the critical threshold at or below which the loops are simple and disjoint, touching neither each other nor the domain boundary. This paper is the second in a series of three establishing that the loops of a CLE$_4$ uniquely determine a conformally invariant, local, and geodesic metric such that the metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this second paper, we prove the existence of geodesics, showing that any geodesic between two loops is supported on the CLE$_4$ loops (off a set of Hausdorff dimension zero) and does not intersect the domain boundary. Along the way, we establish sharp quantitative estimates for the CLE$_4$ metric geometry, including exponential tail bounds for rectangle distances and multi-scale four-arm SLE$_4$ non-intersection bounds for metric balls.

math.PR↗

arXiv2026

The conformally invariant metric on CLE$_4$ III: uniqueness

This paper is the third and final article in a series of papers constructing the canonical conformally invariant metric on the set of loops of the conformal loop ensemble (CLE) with critical parameter $\kappa=4$. The previous two articles construct, as a subsequential limit of the renormalized graph metric on the loops of CLE$_\kappa$ as $\kappa \downarrow 4$, a conformally invariant, local metric on the loops of a CLE$_4$ whose metric ball growth from the domain boundary coincides with the uniform exploration of Werner and Wu. In this paper, we establish that this metric is uniquely characterized by its properties, as are its geodesics, and that it is a measurable function of the CLE$_4$. In particular, we show that the renormalized CLE$_\kappa$ graph metric converges as $\kappa \downarrow 4$ without passing to a subsequence. A key step in the proof is to show that the metric is determined by the geodesics from each loop to the domain boundary, which are in turn determined by the uniform exploration; this representation will have important applications in future work.

math.PR↗

arXiv2026

Quasisymmetric rigidity of the Brownian sphere

The Brownian sphere, also known as the Brownian map, is a canonical random metric measure space homeomorphic to the two-dimensional sphere $\mathbf S^2$. It can be interpreted as the uniform measure on surfaces homeomorphic to $\mathbf S^2$, in the sense that it arises as the scaling limit of many natural models of random planar maps chosen uniformly from a given class. It is also equivalent to the $\sqrt{8/3}$-Liouville quantum gravity sphere. We prove that the Brownian sphere is quasisymmetrically rigid, meaning that, almost surely, it has no nontrivial quasisymmetric automorphisms. We also show that two independent Brownian spheres are almost surely not quasisymmetrically equivalent. Our argument also gives a new proof that the conformal structure of the Brownian sphere is almost surely determined by its metric structure.

math.PR↗

arXiv2026

The conformal dimension of the Brownian tree is one

The Brownian tree, also known as the continuum random tree, is a canonical random compact, geodesic $\mathbf R$-tree that arises as the universal scaling limit for numerous models of discrete random trees. A key quasisymmetric invariant of a metric space is its conformal dimension, defined as the infimum of the Hausdorff dimensions over all quasisymmetrically equivalent spaces. This value is always bounded below by the space's topological dimension and above by its Hausdorff dimension. In the present paper, we prove that the conformal dimension of the Brownian tree is $1$, matching its topological dimension.

math.PR↗

arXiv2026

Minkowski content construction of the CLE gasket measure

We show for $\kappa \in (4,8)$ that the canonical conformally covariant measure on the conformal loop ensemble (CLE$_\kappa$) gasket, previously constructed indirectly by the first co-author and Schoug, can be realized as the limit of several natural approximation schemes. These include the Euclidean Minkowski content and its box-count variants, the properly renormalized number of dyadic squares that intersect the gasket, and the properly renormalized minimal number of balls of radius $\delta$ necessary to cover the gasket with respect to both its canonical geodesic and resistance metrics. This in particular allows us to identify the CLE$_6$ gasket measure with the conformally covariant measure constructed by Garban-Pete-Schramm as a scaling limit of the number of vertices in a macroscopic critical percolation cluster on the triangular lattice. Along the way, we show that the CLE gasket measure of every fixed compact set has finite moments of all orders; previously this was only known for first moments.

math.PR↗

arXiv2026

The scaling limit of random walk and the intrinsic metric on planar critical percolation

We consider critical site percolation ($p=p_c=1/2$) on the triangular lattice $\mathbf{T}$ in two dimensions. We show that the simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion which lives in the gasket of a conformal loop ensemble with parameter $\kappa = 6$ $\big(\mathrm{CLE}_6\big)$, the so-called $\mathrm{CLE}_6$ Brownian motion. We also show that the intrinsic (i.e., chemical distance) metric converges in the scaling limit to the geodesic $\mathrm{CLE}_6$ metric. As a consequence, we deduce the existence of the chemical distance exponent, the resistance exponent, and the spectral dimension of the critical percolation clusters. Moreover, we show that the exponents satisfy the Einstein relations.

math.PR↗

arXiv2026

The conformal dimension of the Brownian sphere is two

The conformal dimension of a metric space $(X, d)$ is equal to the infimum of the Hausdorff dimensions among all metric spaces quasisymmetric to $(X, d)$. It is an important quasisymmetric invariant which lies non-strictly between the topological and Hausdorff dimensions of $(X, d)$. We consider the conformal dimension of the Brownian sphere (a.k.a. the Brownian map), whose law can be thought of as the uniform measure on metric measure spaces homeomorphic to the standard sphere $\mathbf S^2$ with unit area. Since the Hausdorff dimension of the Brownian sphere is $4$, its conformal dimension lies in $[2, 4]$. Our main result is that its conformal dimension is equal to $2$, its topological dimension.

math.PR↗

arXiv2026

MOOSEnger -- a Domain-Specific AI Agent for the MOOSE Ecosystem

MOOSEnger is a tool-enabled AI agent tailored to the Multiphysics Object-Oriented Simulation Environment (MOOSE). MOOSE cases are specified in HIT ".i" input files; the large object catalog and strict syntax make initial setup and debugging slow. MOOSEnger offers a conversational workflow that turns natural-language intent into runnable inputs by combining retrieval-augmented generation over curated docs/examples with deterministic, MOOSE-aware parsing, validation, and execution tools. A core-plus-domain architecture separates reusable agent infrastructure (configuration, registries, tool dispatch, retrieval services, persistence, and evaluation) from a MOOSE plugin that adds HIT-based parsing, syntax-preserving ingestion of input files, and domain-specific utilities for input repair and checking. An input precheck pipeline removes hidden formatting artifacts, fixes malformed HIT structure with a bounded grammar-constrained loop, and resolves invalid object types via similarity search over an application syntax registry. Inputs are then validated and optionally smoke-tested with the MOOSE runtime in the loop via an MCP-backed execution backend (with local fallback), translating solver diagnostics into iterative verify-and-correct updates. Built-in evaluation reports RAG metrics (faithfulness, relevancy, context precision/recall) and end-to-end success by actual execution. On a 125-prompt benchmark spanning diffusion, transient heat conduction, solid mechanics, porous flow, incompressible Navier--Stokes, phase field and plasticity, MOOSEnger achieves a 0.90 execution pass rate versus 0.06 for an LLM-only baseline.

cs.AI↗

arXiv2025

Existence and uniqueness of the canonical Brownian motion in non-simple conformal loop ensemble gaskets

We construct the canonical Brownian motion on the gasket of conformal loop ensembles (CLE$_\kappa$) for $\kappa \in (4,8)$ (which is the range of parameter values in which loops of the CLE$_\kappa$ can intersect themselves, each other, and the domain boundary). More precisely, we show that there is a unique diffusion process on the CLE$_\kappa$ gasket whose law depends locally on the CLE$_\kappa$ and satisfies certain natural properties such as translation-invariance and scale-invariance (modulo time change). We characterize the diffusion process by its resistance form and show in particular that there is a unique resistance form on the CLE$_\kappa$ gasket that is locally determined by the CLE$_\kappa$ and satisfies certain natural properties such as translation-invariance and scale-covariance. We conjecture that the CLE$_\kappa$ Brownian motion describes the scaling limit of simple random walk on statistical mechanics models in two dimensions that converge to CLE$_\kappa$. In future work the results of this paper will be used to show that this is the case with $\kappa=6$ for critical percolation on the triangular lattice.

math.PR↗

arXiv2025

Existence and uniqueness of the conformally covariant geodesic metric on simple conformal loop ensemble carpets

We prove that for each $\kappa \in (8/3, 4)$ there exists a geodesic metric on the carpet of a CLE$_\kappa$ which is canonical in the sense that it is characterized by a certain list of axioms. Our metric can be constructed explicitly as the scaling limit of Minkowski first passage percolation (MFPP), i.e., the metric obtained by taking the infimum of the Lebesgue measure of the $\varepsilon$-neighborhood of all paths connecting each pair of points. Earlier work by the first co-author showed that MFPP admits nontrivial subsequential limits. The present paper shows that this subsequential limit is unique and is characterized by our list of axioms. We conjecture that our metric describes the scaling limit of the chemical distance metric for discrete loop models that converge to CLE$_\kappa$ for $\kappa \in (8/3, 4)$ in the scaling limit, e.g., the critical Ising model for $\kappa=3$. Our argument is inspired by recent works of Gwynne and Miller and Ding and Gwynne on the uniqueness of Liouville quantum gravity metrics.

math.PR↗

arXiv2025

The square summability of the CLE complementary component diameters

We show that the sum of the squares of the diameters of the complementary connected components of the CLE$_\kappa$ carpet/gasket is almost surely finite for $\kappa \in (8/3, 4) \cup (4, 8)$. This is a prerequisite for the application of a result of Ntalampekos which allows the CLE$_\kappa$ carpet/gasket to be uniformized to a round Sierpi\'nski packing, in analogy with the classical Koebe uniformization theorem for finitely connected domains. Our result is new in the case that $\kappa \in (4,8)$ and we provide a new proof for $\kappa \in (8/3, 4)$. In both cases we use the link between CLE and space-filling SLE. The square-summability of diameters has been proved for $\kappa \in (8/3, 4]$ in unpublished work by Rohde and Werness using a different method. Our work completes the proof that this property holds for all $\kappa$ for which CLE$_\kappa$ is defined.

math.PR↗

arXiv2025

Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets

We construct the canonical geodesic metric on the gasket of conformal loop ensembles (CLE$_\kappa$) in the regime $\kappa \in (4,8)$ where the loops intersect themselves, each other, and the domain boundary. Previous work of the authors and V. Ambrosio showed that the subsequential limits associated with certain approximation procedures for such a metric exist and are non-trivial. In this work, we show that the limit exists by proving that there is at most one geodesic metric on the CLE$_\kappa$ gasket which satisfies certain properties. Further, we obtain that the limit is conformally covariant. This paper is the foundation of future work which show that the metric for $\kappa=6$ is the continuum scaling limit of the chemical distance metric for critical percolation in two dimensions. We further conjecture that for $\kappa \in (4,8)$, the geodesic CLE$_\kappa$ metric is the scaling limit of the chemical distance metric associated with discrete models that converge to CLE$_\kappa$.

math.PR↗

arXiv2025

Tightness of approximations to metrics on non-simple conformal loop ensemble gaskets

We study a class of approximation schemes aimed at constructing conformally covariant metrics defined in the gasket of a conformal loop ensemble (CLE$_\kappa$) for $\kappa \in (4,8)$. This is the range of parameter values so that the loops of a CLE$_\kappa$ intersect themselves, each other, and the domain boundary. Its gasket is the closure of the union of the set of points not surrounded by a loop. The class of approximation schemes includes approximations to the geodesic metric and to the resistance metric. We show that the laws of these approximations are tight, and that every subsequential limit is a non-trivial metric on the CLE$_\kappa$ gasket satisfying a natural list of properties. Subsequent work of the second two authors will show that the limits exist and are conformally covariant both in the setting of the geodesic and resistance metrics. We conjecture that the geodesic (resp. resistance) metric describes the scaling limit of the chemical distance (resp. resistance) metric associated with discrete models that converge in the limit to CLE$_\kappa$ for $\kappa \in (4,8)$ (e.g., critical percolation for $\kappa=6$).

math.PR↗

arXiv2025

Two-sided heat kernel bounds for $\sqrt{8/3}$-Liouville Brownian motion

Liouville Brownian motion (LBM) is the canonical diffusion process on a Liouville quantum gravity (LQG) surface. In this work, we establish upper and lower bounds for the heat kernel for LBM when $\gamma=\sqrt{8/3}$ in terms of the $\sqrt{8/3}$-LQG metric which are sharp up to a polylogarithmic factor in the exponential.

math.PR↗

arXiv2025

Multiple SLE$_\kappa$ from CLE$_\kappa$ for $\kappa \in (4,8)$

We define multichordal CLE$_\kappa$ for $\kappa \in (4,8)$ as the conditional law of the remainder of a partially explored CLE$_\kappa$. The strands of a multichordal CLE$_\kappa$ have a random link pattern, and their law conditionally on the linking pattern is a (global) multichordal SLE$_\kappa$. The multichordal CLE$_\kappa$ are the conjectural scaling limits of FK and loop $O(n)$ models with some wiring patterns of the boundary arcs. We also explain how CLE$_\kappa$ configurations can be locally resampled, and show that the partially explored strands can be relinked in any possible way with positive probability. We will also establish several other estimates for partially explored CLE$_\kappa$. Altogether, these relationships and results serve to provide a toolbox for studying CLE$_\kappa$ and global multiple SLE$_\kappa$.

math.PR↗

arXiv2024

SLE$_{\kappa}(\rho)$ processes in the light cone regime on Liouville quantum gravity

We study the relationship between certain SLE$_\kappa(\rho)$ processes, which are variants of the Schramm-Loewner evolution with parameter $\kappa$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG). These processes are defined whenever $\rho > -2-\kappa/2$ and in this work we will focus on the light cone regime, meaning that $\kappa \in (0,4)$ and $\max(\kappa/2-4,-2-\kappa/2) < \rho < -2$. Such processes are self-intersecting even though ordinary SLE$_\kappa$ curves are simple for $\kappa \in (0,4)$. We show that such a process drawn on top of an independent $\sqrt{\kappa}$-LQG surface called a weight $(\rho+4)$-quantum wedge can be represented as a gluing of a pair of trees which are described by the two coordinate functions of a correlated $\alpha$-stable L\'evy process with $\alpha = 1-2(\rho+2)/\kappa$. 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$_\kappa(\kappa-4)$ curve on an independent $\sqrt{\kappa}$-LQG surface for $\kappa \in (4/3,2)$.

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 $\kappa \in (0,4)$ and for each $\theta \in {\mathbf R}$ let $\eta_\theta$ be the flow line of $h$ from $x$ to $y$. Recall that for $\theta_1 < \theta_2$ the fan ${\mathbf F}(\theta_1,\theta_2)$ of flow lines of $h$ from $x$ to $y$ is the closure of the union of $\eta_\theta$ as $\theta$ varies in any fixed countable dense subset of $[\theta_1,\theta_2]$. We show that the adjacency graph of components of $D \setminus {\mathbf F}(\theta_1,\theta_2)$ 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 ${\mathbf F}(\theta_1,\theta_2)$ a.s. determines the flow lines used in its construction. That is, for each $\theta \in [\theta_1,\theta_2]$ we prove that $\eta_\theta$ is a.s. determined by ${\mathbf F}(\theta_1,\theta_2)$ as a set.

math.PR↗