Extremal distance and conformal radius of a CLE_4 loop

Conformal Skorokhod embeddings and related extremal problems

The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset\mathbb{C}$ whose exit time embeds a given probability distribution $μ$ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding $μ$ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb{U}$ …

随机共形雪花

Random conformal snowflakes

在经典分析的许多问题中，极值位形表现出复杂的分形结构。这使得描述极值和解决此类问题变得更加困难。其中许多问题都与调和测度的多重分形分析有关。我们认为，在这样的问题中寻找极值，应该与随机分形，而不是确定性的。我们引入了一类新的分形随机共形雪花，并研究其性质，开发工具，以估计光谱，并显示极值可以在这一类。作为一个应用，我们显着改善已知的估计，从下面的极值行为的调和测度，显示如何构建一个相当简单的雪花，它有一个频谱非常接近所述的极值。

关于共形不变的CLE探索

On conformally invariant CLE explorations

我们研究了一些共形不变的动态方法来构建共形循环合奏与简单的循环在早期的文件中介绍了谢菲尔德，谢菲尔德和沃纳。一个结果是当识别每个循环的所有点时，测量CLE（4）循环到CLE（4）“内”的边界的距离的共形不变方式。

Conformal symmetries in the extremal process of two-dimensional discrete Gaussian Free Field

We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions ($x$) and centered values ($h$) of the extreme local maxima in lattice versions of a bounded domain $D\subset\mathbb C$ converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure $Z^D(dx)\otimes e^{-αh}d h$, where $α$ is a constant and $Z^D$ is a random a.s.-finite measure on $D$. The random measures $\{Z^…

Conformal invariance of CLE$_κ$ on the Riemann sphere for $κ\in (4,8)$

The conformal loop ensemble (CLE) is the canonical conformally invariant probability measure on non-crossing loops in a simply connected domain in $\mathbb C$ and is indexed by a parameter $κ\in (8/3,8)$. We consider CLE$_κ$ on the whole-plane in the regime in which the loops are self-intersecting ($κ\in (4,8)$) and show that it is invariant under the inversion map $z \mapsto 1/z$. This shows that whole-plane CLE$_κ$ for $κ\in (4,8)$ defines a conformally invariant measure on loops on the Riemann sphere. The analogous statement in the regime in which the loops are simple ($κ\in (8/3,4]$) was p…

CLE$_4$上的共形不变度量 I：非简单CLE图度量的子序列极限

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

本文研究κ=4的共形环系综CLE₄，通过证明κ趋近于4⁺时CLE_κ环上图度量的重整化子序列极限存在，构造出CLE₄上共形不变的局部测地度量，其边界度量球增长与Werner和Wu的均匀探测一致。