arXiv subjects

Yizheng Yuan

Publications and source records attributed to Yizheng Yuan.

13 recordsLinked to original sources

arXiv2026

Multiple SLE$_κ$ from CLE$_κ$

We introduce multichordal CLE$_κ$ which is a random collection of non-crossing loops together with additional chords connecting a set of marked boundary points. The chords have a random link pattern, and their law conditionally on the link pattern is a (global) multichordal SLE$_κ$. We show that multichordal CLE$_κ$ arises as the conditional law of the remainder of a partially explored CLE$_κ$. The multichordal CLE$_κ$ are the conjectural scaling limits of FK and loop $O(n)$ models with some wiring patterns of the boundary arcs. We further explain how CLE$_κ$ configurations can be locally resampled, and show that the partially explored strands can be relinked in any possible way with positive probability. Our results also establish a useful local independence property of CLE$_κ$. Altogether, the results and estimates in this paper serve to provide a toolbox for studying CLE$_κ$ and global multiple SLE$_κ$.

math.PR↗

arXiv2026

Minkowski content construction of the CLE gasket measure

We show that the canonical conformally covariant measure on the conformal loop ensemble (CLE$_κ$) gasket/carpet, 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 $δ$ 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 $κ= 6$ $\big(\mathrm{CLE}_6\big)$, 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

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

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$_κ$) in the regime $κ\in (4,8)$ 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$_κ$ 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 $κ=6$ is the continuum scaling limit of the chemical distance metric for critical percolation in two dimensions. We further conjecture that for $κ\in (4,8)$, the geodesic CLE$_κ$ metric is the scaling limit of the chemical distance metric associated with discrete models that converge to CLE$_κ$.

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$_κ$) for $κ\in (4,8)$. This is the range of parameter values so that the loops of a CLE$_κ$ 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$_κ$ 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$_κ$ for $κ\in (4,8)$ (e.g., critical percolation for $κ=6$).

math.PR↗

arXiv2025

Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity

We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the SLE with its natural parametrisation. More precisely, denoting by $d\in(0,2]$ the dimension of the curve, we show the following. 1. The optimal $ψ$-variation is $ψ(x)=x^d(\log\log x^{-1})^{-(d-1)}$ in the sense that $η$ is a.s. of finite $ψ$-variation for this $ψ$ and not for any function decaying more slowly as $x \downarrow 0$. 2. The optimal modulus of continuity is $ω(s) = c\,s^{1/d}(\log s^{-1})^{1-1/d}$, i.e. for some random $c>0$ we have $|η(t)-η(s)| \le ω(t-s)$ a.s., while this does not hold for any function $ω$ decaying faster as $s \downarrow 0$. 3. $\limsup_{t\downarrow 0} |η(t)|\,\big(t^{1/d}(\log\log t^{-1})^{1-1/d}\big)^{-1}$ is a.s. equal to a deterministic constant in $(0,\infty)$. We also show that the natural parametrisation of SLE is given by the fine mesh limit of the $ψ$-variation. As part of our proof, we show that every stochastic process whose increments satisfy a particular moment condition attains a certain variation regularity.

math.PR↗

arXiv2025

Refined regularity of SLE

We prove refined (variation and Hölder-type) regularity statements for the SLE trace (under capacity parametrisation). More precisely, we show that the trace has finite $ψ$-variation for $ψ(x) = x^d(\log 1/x)^{-d-\varepsilon}$ and Hölder-type modulus $φ(t) = t^α(\log 1/t)^β$ where $d$ and $α$ are the optimal $p$-variation and Hölder exponents of SLE$_κ$ which have been previously identified by Viklund, Lawler (2011) and Friz, Tran (2017). For SLE$_8$, we simplify a step in the proof by Kavvadias, Miller, and Schoug (2021), and get the modulus $φ(t) = (\log 1/t)^{-1/4}(\log\log 1/t)^{2+\varepsilon}$. Finally, for $κ\ge 8$, we prove regularity estimates for the uniformising maps that hold uniformly in time, namely $\sup_t |\hat f_t'(u+iv)| \lesssim v^{2α-1}(\log 1/v)^β$ in case $κ>8$ and $v^{-1}(\log 1/v)^{-1/4}(\log\log 1/v)^{1+\varepsilon}$ in case $κ=8$. Our results are obtained from analysing the forward Loewner differential equation (in contrast to the other mentioned works which analyse the backward equation).

math.PR↗

arXiv2023

On Loewner chains driven by semimartingales and complex Bessel-type SDEs

We prove existence (and simpleness) of the trace for both forward and backward Loewner chains under fairly general conditions on semimartingale drivers. As an application, we show that stochastic Komatu-Loewner evolutions SKLE$_{α,b}$ are generated by curves. As another application, motivated by a question of A. Sepúlveda, we show that for $α>3/2$ and Brownian motion $B$, the driving function $|B_t|^α$ generates a simple curve for small $t$. On a related note we also introduce a complex variant of Bessel-type SDEs and prove existence and uniqueness of strong solution. Such SDEs appear naturally while describing the trace of Loewner chains. In particular, we write SLE$_κ$, $κ<4$, in terms of stochastic flow of such SDEs.

math.PR↗

arXiv2023

Law of the SLE tip

We analyze the law of the SLE tip at a fixed time in capacity parametrization. We describe it as the stationary law of a suitable diffusion process, and show that it has a density which is a unique solution of a certain PDE. Moreover, we identify the phases in which the even negative moments of the imaginary value are finite. For the negative second and negative fourth moments we provide closed-form expressions.

math.PR↗

arXiv2022

Topological characterisations of Loewner traces

The (chordal) Loewner differential equation encodes certain curves in the half-plane (aka traces) by continuous real-valued driving functions. Not all curves are traces; the latter can be defined via a geometric condition called the local growth property. In this paper we give two other equivalent conditions that characterise traces: 1. A continuous curve is a trace if and only if mapping out any initial segment preserves its continuity (which can be seen as an analogue of the domain Markov property of SLE). 2. The (not necessarily simple) traces are exactly the uniform limits of simple traces. Moreover, using methods by Lind, Marshall, Rohde (2010), we infer that uniform convergence of traces imply uniform convergence of their driving functions.

math.CV↗

arXiv2021

Regularity of SLE in $(t,κ)$ and refined GRR estimates

Schramm-Loewner evolution (SLE$_κ$) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $\sqrtκ$ times Brownian motion. This yields a (half-plane) valued random field $γ= γ(t, κ; ω)$. (Hölder) regularity of in $γ(\cdot,κ;ω$), a.k.a. SLE trace, has been considered by many authors, starting with Rohde-Schramm (2005). Subsequently, Johansson Viklund, Rohde, and Wong (2014) showed a.s. Hölder continuity of this random field for $κ< 8(2-\sqrt{3})$. In this paper, we improve their result to joint Hölder continuity up to $κ< 8/3$. Moreover, we show that the SLE$_κ$ trace $γ(\cdot,κ)$ (as a continuous path) is stochastically continuous in $κ$ at all $κ\neq 8$. Our proofs rely on a novel variation of the Garsia-Rodemich-Rumsey (GRR) inequality, which is of independent interest.

math.PR↗

arXiv2020

A support theorem for SLE curves

For all $κ> 0$, we show that the support of SLE$_κ$ curves is the closure in the sup-norm of the set of Loewner curves driven by nice (e.g. smooth) functions. It follows that the support is the closure of the set of simple curves starting at $0$.

math.PR↗