arXiv subjects

Cillian Doherty

Publications and source records attributed to Cillian Doherty.

3 recordsLinked to original sources

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 The square-summability of diameters has been proved for $\kappa \in (8/3, 4]$ in unpublished work by Rohde and Werness using a different method.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↗

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↗

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↗