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chapters/chap3.tex

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+% chapters/chap2.tex
+\documentclass[../main.tex]{subfiles}
+
+\ifSubfilesClassLoaded{
+  \addbibresource{chap3.bib}
+}
+
+\begin{document}
+
+\chapter{Seigel-Bargmann Space}
+
+In this chapter, we will collect ideas and other perspective we have understanding the concentration of measure phenomenon. Especially with symmetric product of $\C P^1$ and see how it relates to Riemman surfaces and Seigel-Bargmann spaces.
+
+\begin{figure}[h]
+    \centering
+    \begin{tikzpicture}[node distance=40mm, thick,
+        main/.style={draw, draw=white},
+        towards/.style={->},
+        towards_imp/.style={<->,red},
+        mutual/.style={<->}
+        ]
+        \node[main] (cp) {$\mathbb{C}P^{n}$};
+        \node[main] (c) [below of=cp] {$\mathbb{C}^{n+1}$};
+        \node[main] (p) [right of=cp] {$\mathbb{P}^n$};
+        \node[main] (sym) [below of=p] {$\operatorname{Sym}_n(\mathbb{C}P^1)$};
+        % draw edges
+        \draw[towards] (c) -- (cp) node[midway, left] {$z\sim \lambda z$};
+        \draw[towards] (c) -- (p) node[midway, fill=white] {$w(z)=\sum_{i=0}^n Z_i z^i$};
+        \draw[towards_imp] (cp) -- (p) node[midway, above] {$w(z)\sim w(\lambda z)$};
+        \draw[mutual] (p) -- (sym) node[midway, right] {root of $w(z)$};
+    \end{tikzpicture}
+    \caption{Majorana stellar representation}
+    \label{fig:majorana_stellar_representation}
+\end{figure}
+
+Basically, there is a bijection between the complex projective space $\mathbb{C}P^n$ and the set of roots of a polynomial of degree $n$.
+
+We can use a symmetric group of permutations of $n$ complex numbers (or $S^2$) to represent the $\mathbb{C}P^n$, that is, $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$.
+
+One might be interested in the random sampling over the $\operatorname{Sym}_n(\mathbb{C}P^1)$ and the concentration of measure phenomenon on that.
+
+\ifSubfilesClassLoaded{
+  \printbibliography[title={References for Chapter 2}]
+}
+\end{document}