

\subsection{\bf Choice of Points in Lattice Monte Carlo}

\begin{slide}
\begin{itemize}
\item Assume you have a particular choice $s_1^{(0)} \ldots
s_n^{(0)}$ with $H^{(0)} = H \left( s_1^{(0)} \ldots
s_n^{(0)}\right)$.  

\item Remember $\exp \lbrack - H/kT\rbrack$ is a tiny
number (as $H$ is energy of full system $H/kT$ is approximately
the number of objects in system) and more importantly, varies
rapidly.  

\item Thus, we cannot choose points ``uniformly''.  

\item Rather, we assume point $\underline{s}^{(1)}$ ``near''
$\underline{s}^{(0)}$.  So that $\exp \lbrack -H/kT\rbrack$ only
changes a little.  This can be achieved by only changing one of the
spins at a time, i.e., we move $s_k^{(0)} \rightarrow s_k^{(1)}$ and
$s_i^{(0)} i\not= k$ unchanged.

\item Typically, this produces a change $\Delta H \sim kT$ as $kT$ is
energy per site (up to numerical constants).

\item Remember classical result of energy of $1/2 kT$ per degree of
freedom.

\item So change in $\exp\lbrack -H/kT\rbrack$ is roughly $\exp\lbrack
I\,kT/kT\rbrack$ (i.e., modest).
\item Thus, we will not move off ``peak'' of integrand.
\end{itemize}
\end{slide}

