The Loewner Equation and the derivative of its solution
Author
carrling
Last Updated
10 years ago
License
Creative Commons CC BY 4.0
Abstract
The Loewner Equation and the derivative of its solution
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\title{The Loewner Equation and the derivative of its solution}
\subtitle{}
\author{Carl Ringqvist}
\institute{Master's Thesis presentation, KTH \& SU}
\begin{document}
\maketitle
\begin{frame}[fragile]
\frametitle{Introduction}
Charles Loewner and the introduction of the Loewner Equation.
\begin{figure}
\centering
\includegraphics[width=6cm,height=6cm]{firstex.png}
\end{figure}
\end{frame}
\begin{frame}[fragile]
\frametitle{Illustrations}
Some examples of driving functions $U_{t}$
\begin{figure}
\centering
\includegraphics[width=11cm,height=4cm]{intro2.png}
\end{figure}
\tiny
Source: "Spacefilling Curves and Phases of the Loewner Equation", J.Lind, S.Rohde
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{Illustrations}
And the sets they generate
\begin{figure}
\centering
\includegraphics[width=8cm,height=4cm]{intro1.png}
\caption{van Koch curve, the half-Sierpinski gasket, and the Hilbert space-filling curve}
\end{figure}
\tiny
Source: "Spacefilling Curves and Phases of the Loewner Equation", J.Lind, S.Rohde
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{the SLE-curve}
Let $U_{t}=\sqrt{\kappa}B_{t}$, where $\kappa \in \mathbb{R}$ and $B_{t}$ is a standard Brownian motion. Then, with probability one, the set $H_{t}$ is generated by a curve, i.e $H_{t}=\mathbb{H}\backslash \gamma(0,t]$ for some continuous curve $\gamma$. This $\gamma$ is called an SLE-curve. Two interesting facts about the SLE-curve:
\begin{itemize}
\item The curve spirals at every point
\item The trace is extremely sensitive to the value of $\kappa$. In fact
\begin{itemize}
\item For $0 \leq \kappa \leq 4$ the trace $\gamma$ is simple with probability one.
\item For $4 \leq \kappa < 8$ the trace $\gamma$ intersects itself and every point is contained in a loop but the curve is not space-filling (with probability 1).
\item For $\kappa \geq 8$ the trace $\gamma$ is space-filling (with probability 1).
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{Illustrations}
$\kappa=2$, yielding a simple trace:
\begin{figure}
\centering
\includegraphics[width=8cm,height=4cm]{simple.png}
\end{figure}
\small
Source: http://iopscience.iop.org
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{Illustrations}
$\kappa=6$, yielding a non-simple trace:
\begin{figure}
\centering
\includegraphics[width=8cm,height=4cm]{nonsimple.png}
\end{figure}
\small
Source: http://iopscience.iop.org
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{Discrete Gaussian free fields}
Rougher grid left, finer grid right
\begin{figure}
\centering
\includegraphics[width=8cm,height=4cm]{gaussian.png}
\end{figure}
\small
Source: "Finding SLE paths in the Gaussian free field ", S.L. Watson
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{Purpose of Third part}
Aim: Investigate upper bounds of the quantity $|\arg g^{-1 \prime}_{t}(z)|$ as $z$ approaches the boundary, for driving functions of H\"older-1/2 continuous driving functions\\
(Recall H\"older-1/2 continuity: $|U_{t+s}-U_{t}| \leq \sigma \sqrt{s}$ for some $\sigma \in \mathbb{R}^{+}$)
\end{frame}
\begin{frame}[fragile]
\frametitle{Apriori expectations}
\begin{itemize}
\item $\boldsymbol{\sigma<2\sqrt{2}}$: Nontrivial bound for argument should exist and this should be possible to prove with methods from RTZ\\
\item $\boldsymbol{2\sqrt{2} \leq \sigma < 4}$. Non-trivial bound should exist for the argument since it does exist for the absolute value for $\sigma <4$; but this cannot be proved with methods outlined in RTZ
\end{itemize}
\end{frame}
\begin{frame}[fragile]
\frametitle{An illuminating example}
\small
In the article "Collisions and spirals of Loewner traces" by Lind, Marshall and Rohde, the Loewner trace and conformal map of the driving function $U_{t} = \sigma\sqrt{1-t}$, $0 < \sigma < 4$ are calculated. In fact here the following is stated:\\
Given $0 < \sigma < 4$, set $\theta := -\sin^{-1}(\sigma/4)$, $\beta = 2ie^{i\theta}$, and set
\begin{center}
$k(z) = \frac{(z-\beta)(z-\bar{\beta})^{e^{2i\theta}}}{(\sigma - \beta)(\sigma - \bar{\beta})^{e^{2i\theta}}}$ \\
$g_{t}(z) = (1-t)^{1/2}k^{-1}((1-t)^{-\cos\theta e^{i\theta}}k(z))$
\end{center}
Then $k$ is a conformal map of $\mathbb{H}$ onto $\mathbb{C}\backslash G$ where $ G := \{e^{te^{i\theta}} ; t \geq 0 \}$, is a logarithmic spiral in $\mathbb{C}$ beginning at $1$ and tending to $\infty$, and where $g_{t}$ satisfies the Loewner equation
\begin{center}
$\displaystyle{\dot g_{t} = \frac{2}{g_{t} - \sigma\sqrt{1-t}} \; g_{0} \equiv z} $
\end{center}
The trace $\gamma = k^{-1}(\{e^{-te^{i\theta}} ; t > 0\})$ is a curve in $\mathbb{H}$ beginning at $\sigma \in \mathbb{R}$ spiraling around $\beta \in \mathbb{H}$
\normalsize
\end{frame}
\begin{frame}[fragile]
\frametitle{}
\begin{figure}
\includegraphics[width=10cm,height=8cm]{spiral.png}
\end{figure}
\end{frame}
\begin{frame}[fragile]
\frametitle{Results}
\begin{itemize}
\item $\boldsymbol{\sigma<\sqrt{2}}$: A non-trivial bound is obtained in line with expectation. \\
\item $\boldsymbol{\sqrt{2} \leq \sigma < 2\sqrt{2}}$: We show the methods of RTZ are insufficient for obtaining a non-trivial bound in this interval. However, we still suspect such a nontrivial bound to exist\\
\item $\boldsymbol{2\sqrt{2} \leq \sigma < 4}$: We have shown no non-trivial bound can exist in this interval
\end{itemize}
\end{frame}
\plain{Questions?}
\end{document}