# Final Project: Random Walks and Electric Networks Stuff I'm Omitting: * Energy dissipation stuff * Other proofs to Polya Outline: * Introduction * Example 1: Simple 1D example of rephrasing random walk to electric network. Here we talk about the properties of p(x) and v(x) harmonic function, uniqueness, all that stuff. * Example 2: Simple 2D example. Maybe talk about the different ways to solve Dirichlet although might not have to go too in depth. * Probabilitistic Interpetation of Circuits * Example 3: More general example (Still finite). How to get a random walk out of a general circuit. Voltage interpretation. * Example 4: Effective Resistence. Talk about peff = i/Ca. Also mention Raleigh monotonicity law. * Application on Infinite Networks * Example 5: d-dimensional lattices. Prove Polya's Theorem using cutting and shorting method * Example 6: k-fuzz graphs. Discuss the comparing two networks. * Example 7: BCC graph.