It’s where you talk about orthogonal projectors:

Not me! Tomi Johnson wrote this post!

]]>.. Each acts as identity only on vectors in the space spanned by $phi_k$ and as zero on all others,….

This is before the end diagram summary of the section on normalized laplacians. You can search for the word projector and see it.

]]>Small typo: phi_k -> above the diagram in the section on the normalized laplacian.

]]>The difference between the finite automata and what you are doing is that in the finite automata, the graph edges are labelled with a symbol (the monoid element), and thus a graph walk corresponds to a sequence of symbols (edges walked). A random walk on a graph induces a measure on the set of strings of symbols (aka ‘the language’). If the random walk is independent of the history of the path, then it is Markovian, and the measure factorizes. (if random walk is not independent of the history, then it must be generated by a push-down automaton (context free language) or even a full Turing machine).

If I take a random walk on a graph, and, after coming to a vertex, I assign equal probability to leaving by any edge, then I get your stochastic Laplacian, above. Or, as John Baez suggests, I could twiddle the edge weights, and preferentially leave on some edges. Or I could twiddle the exit probabilities *at each vertex* (i.e. each edge has two weights not one, depending on whether one is coming or going), and recover a classic Markov chain (minus diagonal) instead of the Laplacian.

And here is where I get confused. The word ‘Markov’ in my last paragraph is not really the same as in my first paragraph, and yet its closely related, and so mentally I circle back, and wonder “what other sets can the monoid act on?” … and perhaps I now can answer my question, like so:

In measure theory, measures must the real (and positive), and so the measure assigned to a language (the set of all random walks on graph) is real-valued, and thus “stochastic”. Perhaps(?) one can contemplate measures with an additional U(1) in them, and perhaps this is what the quantum walk is providing!? So when I ask “what other sets can the monoid act on, and what would the Laplacian, etc. generalize to in such cases?” then perhaps I am contemplating set-valued measures on languages? Hmmmm.

Sorry for the long post. Sometimes, mathematics is like a visit to a candy shop; each treat looks more delicious than the last, and picking out just one to enjoy is just too hard.

]]>ii. Even in the wikipedia definition of “random walk normalized Laplacian” it’s defined as mentioned.

I was meaning, as you mentioned. The books I have seen use the definition we use however.

Thanks for the link.

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