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\homework{3: Current and Random Walks}{Due Friday, Week 2}

Do \textbf{one} of the following \textbf{three} problems!

\begin{enumerate}
\item Consider a graph $G$ corresponding to a tiling of $\mathbb{R}^2$ with equilateral triangles with side length 1.   Pick some vertex $A$ to denote a starting location, and suppose we model a random walk on this graph starting from $A$.  

What is $p_{\textrm{esc}}$?  In particular, is it 0 or nonzero?
\item Prove Rayleigh's Monotonicity Theorem:
\begin{theorem}
If any of the individual resistances in a circuit increase, then the overall effective resistance of the circuit can only increase or stay constant; conversely, if any of the individual resistances in a circuit decrease, the overall effective resistance of the circuit can only decrease or stay constant. 

In specific, cutting wires (setting certain resistances to infinity) only increases the effective resistance, while fusing vertices together (setting certain resistances to 0) only decreases the effective resistance.
\end{theorem}
\item Suppose that $V$ is a vector space with a subfield $W$, such that $V$ and $W$ are both $n$-dimensional.  Prove that $V = W$.
\end{enumerate}



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