More about the Schrödinger equation on a finite graph

It looks like the finite graph model is not just a toy model! It’s called a continuous-time quantum random walk and is used in quantum computing in a way similar to how random walks on graphs are used in classical computing. The fact that quantum random walks mix sooner than classical random walks relates to [...]

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Test your intuition: consecutive tails

Something very unfortunate has happened: several things I have recently written that could have been blog entries are instead answers on math.SE! In the interest of exposition beyond the Q&A format I am going to “rescue” one of these answers. It is an answer to the following question, which I would like you to test [...]

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Walks on graphs and statistical mechanics

I finally learned the solution to a little puzzle that’s been bothering me for awhile. The setup of the puzzle is as follows. Let be a weighted undirected graph, e.g. to each edge is associated a non-negative real number , and let be the corresponding weighted adjacency matrix. If is stochastic, one can interpret the [...]

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The McKay correspondence I

Today we’re going to relate the representation graphs introduced in this blog post to something I blogged about in the very first and second posts in this blog! The result will be a beautiful connection between the finite subgroups of , the Platonic solids, and the ADE Dynkin diagrams. This connection has been written about [...]

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Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on [...]

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The Lindstrom-Gessel-Viennot lemma

One of my favorite results in algebraic combinatorics is a surprisingly useful lemma which allows a combinatorial interpretation of the determinant of certain integer matrices. One of its more popular uses is to prove an equivalence between three other definitions of the Schur functions (none of which I have given yet), but I find its [...]

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Newton’s sums, necklace congruences, and zeta functions II

In a previous post I gave essentially the following definition: given a discrete dynamical system, i.e. a space and a function , and under the assumption that has a finite number of fixed points for all , we define the dynamical zeta function to be the formal power series . What I didn’t do was [...]

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Newton’s sums, necklace congruences, and zeta functions

The goal of this post is to give a purely combinatorial proof of Newton’s sums which would have interrupted the flow of the previous post. Recall that, in the notation of the previous post, Newton’s sums (also known as the first Newton-Girard identity) state that . One way to motivate a combinatorial proof is to [...]

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The magic of the Frobenius map II

Once upon a time I discussed some interesting uses of the Frobenius map to solve some Putnam-style problems. Unfortunately, I wrote that post before becoming really interested in combinatorics, so I neglected to develop that particular side of the story, which I’d like to do now. The beginning of this story is the folklore combinatorial [...]

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The spectral radius of a simple graph

I’ve decided not to port over the posts from AoPS. Among other things, I’d prefer to preserve the comments on older posts as they are. So here’s something I’ve been thinking about. Given a simple connected graph , define to be its largest positive eigenvalue; in other words, if counts the number of walks (equivalently, [...]

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