Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.19204

The square summability of the CLE complementary component diameters

Cillian Doherty·Jason Miller
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 The square-summability of diameters has been proved for $κ\in (8/3, 4]$ in unpublished work by Rohde and Werness using a different method.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

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cillian Doherty, Jason Miller. 2025-09-23. The square summability of the CLE complementary component diameters. https://arxiv.org/abs/2509.19204

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Distribution-uniform strong laws of large numbers

We revisit the question of whether the strong law of large numbers (SLLN) holds uniformly in a rich family of distributions, culminating in a distribution-uniform generalization of the Marcinkiewicz-Zygmund SLLN. These results can be viewed as extensions of Chung's distribution-uniform SLLN to random variables with uniformly integrable $q^\text{th}$ absolute central moments for $0 < q < 2$. Furthermore, we show that uniform integrability of the $q^\text{th}$ moment is both sufficient and necessary for the SLLN to hold uniformly at the Marcinkiewicz-Zygmund rate of $n^{1/q - 1}$. These proofs centrally rely on novel distribution-uniform analogues of some familiar almost sure convergence results including the Khintchine-Kolmogorov convergence theorem, Kolmogorov's three-series theorem, a stochastic generalization of Kronecker's lemma, and the Borel-Cantelli lemmas. We also consider the non-identically distributed case.

math.PR↗
arXiv2026

Malliavin Calculus for rough stochastic differential equations

In this work we show that rough stochastic differential equations (RSDEs), as introduced by Friz, Hocquet, and Lê (2021), are Malliavin differentiable. We use this to prove existence of a density when the diffusion coefficients satisfies standard ellipticity assumptions. Moreover, when the coefficients are smooth and the diffusion coefficients satisfies a Hörmander condition, the density is shown to be smooth. The key ingredient is to develop a comprehensive theory of linear rough stochastic differential equations, which could be of independent interest.

math.PR↗
arXiv2026

Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation

We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly.

math.PR↗