The purpose of this post is to present the very basics of potential theory for finite Markov chains. This post is by no means a complete presentation but rather aims to show that there are intuitive finite analogs of the potential kernels that arise when studying Markov chains on general state spaces. By presenting a piece of potential theory for Markov chains without the complications of measure theory I hope the reader will be able to appreciate the big picture of the general theory.
I would like to briefly introduce disconnectivity graphs — striking visualizations of multidimensional energy landscapes that I had never seen before. While it’s not immediately obvious how useful they are, it should be straightforward to adapt them for visualizing probability distributions. A quick Google search for ‘disconnectivity graph’ will turn up lots of examples. These things look like chandeliers and are meant to summarize the potential energy surface of a molecule, potentially with many degrees of freedom and many local optima
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