

\subsection{\bf Example~3:\ \ Lattice Monte Carlo}
\subsection{\bf Theoretical Physics}

\begin{slide}
\begin{itemize}
\item A typical ``theoretical'' calculation in condensed
matter or high energy physics involves integrals of form
$$
I = \int \exp \left\lbrack {-H\over{kT}}\right\rbrack\,
\Theta\, ds_1, \ldots ds_n
$$
where $s_i$ is spin (gluon field) at site $i$.  

\item Typically, the sites are in a regular lattice in two, three, or
four dimensions.

\item This integral would involve solve $10^4 \rightarrow 10^6$
dimensions and so must be evaluated by Monte Carlo.

\item $\Theta$ is not a very rapidly varying function (typically a
polynomial in the $s_i$).

\item Also, we want to do integral for several different choices for
$\Theta$.

\item The Hamiltonian $H$ is proportional to size of system and so
the weighting function $\exp\lbrack - H/kT \rbrack$ is very
small and one must be very careful to choose one's integration
points appropriately.
\end{itemize}
\end{slide}

