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\author{Jeremy Morris}
\title{Self-avoiding Walks\\Extra Credit}

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The code I have written will find self-avoiding walks using the Metropolis algorithm.  It does this for the general case, in other words, for any choice of $d$ (dimension of the lattice) and $n$ (path length).

In order to check to see if the code is working, I have run two sets of tests.  The results of which can be seen in Figures (\ref{N}) and (\ref{beta}).

The first question is how big should the value $N$ be in order to assure convergence.  $N$ is the number of times we carry out the product $\nu = \nu P$.  I have chosen the values $N_{i} = 10,10^{2},10^{4},10^{6}$ and kept the values $n=3,d=2,\beta=\infty$.  The results can be seen in Figure (\ref{N}) shows the results.  It turns out that we get exactly what is expected for very low values of $N$.  For example, when $N = 100$ we already get the uniform distribution on the self-avoiding walks.  This may be due to the choice for $n,d,\beta$.  Unfortunately, it takes a lot of time to compute the stationary distribution for larger values of $n$ and $d$.  At a later time I may try to increase these values to see what happens.


\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.6]{expN.eps}
\caption{Plots Varying $N$}\label{N}
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\end{figure}

The next question is how does the value of $\beta$ affect these calulations.  I have run some experiments setting $\beta = 1,10^{4},10^{16},10^{32}$.  It is interesting that there is only a difference when $\beta = 1$.  For all other values of $\beta$ we have the uniform distribution.  Again, this may have something to do with the choice of $n$ and $d$ which remained the same for these experiments.

The next step would be to run some similar experiments when $n$ and $d$ are altered to see how $N$ and $\beta$ affect the results.  This could happen during a long weekend or something when we have more computing time at our disposal.

\begin{figure}[ht]
\begin{center}
\includegraphics[scale=0.6]{expbeta.eps}
\caption{Plots Varying $\beta$}\label{beta}
\end{center}
\end{figure}

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