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If $P$ is the Markov transition matrix of a graph, the matrix $A=P\pi^{-1}$ is called the affinity matrix and $R=\pi^{1/2}A\pi^{1/2}=\pi^{1/2}P\pi^{-1/2}$ is called the normalized affinity matrix, see …

From the physics point of view, the answer to your question is an immediate "yes": a nonunique Gibbs measure arises if there is a phase transition into a phase with multiple ground states (say, a phas …

I could go back to Markov himself, who in 1913 applied the concept of a Markov chain to sequences of vowels and consonants in Alexander Pushkin's poem Eugene Onegin. In good approximation, the probabi …

I found George Lowther's blog on Feller processes quite instructive, with several helpful examples. The comments in particular give some simple constructions for Markov processes that are almost, but …

Let me denote the probability that that there is an event "first $A$ then $B$" by $P_{AB}$ and let me consider $1-P_{AB}$. One contribution to $1-P_{AB}$ is that the event $A$ does not happen at al …

Mark Jerrum has rigorous results (bounds) on mixing times for Markov Chain Monte Carlo algorithms, as summarized in this presentation. In the context of deep learning, such bounds have been used in La …

This may not be a viable route, but if your friend is familiar with Python, a hands-on course might be an effective way to explore Markov chains. The course designed by Sargent and Stachurski guides y …