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March 25, 2026

For next Thursday I’m supposed to present Lemma 10.2 and Lemma 13.1. Also, I’m wondering if the characterization of the lattice Maxwell action as a difference of Dirichlet forms gives an alternative proof of Lemma 13.1. But for that I’d need an upper bound on the 2nd term, which would not be a bottom of the spectrum thing.

February 20, 2026

The matrix of the form in section 2 of Chatterjee has a determinant which is invariant under a change of the ordering of the elements of the edge set. If you permute the edge set, this permutation corresponds to a sequence of transpositions. Each transposition corresponds to swapping two rows then two columns. When you swap two rows, the determinant changes by a factor or negative one. Swapping the corresponding columns changes the determinant back to what it was originally. This shows the the determinant is well defined regardless of the choice of ordering of the edges.

February 3, 2026

Today in my lecture we learned about the equivalence between Green’s function G(x,y) on a graph being finite and a graph being transient, meaning that a random walk won’t come back infinitely often to the same point. I also thought about how G(x,y) can be thought of as the total heat seen by the point x, as you propogate a heat flow with initial heat data defined by a value of 1 at y and a value of 0 everywhere else.

Note that for dimension greater than 2, the infinite volume lattice is transient, meaning that Green’s function on the infinite volume lattice is finite.

There was also a result about transience for tree graphs. I wonder if I can use that result in the context of the axial graph in lattice gauge theory.

Balaban wrote a paper about Green’s functions, and this paper led to his ultraviolet stability proof. Green’s function in the sense of distributions is formally speaking the two point quantum field expecation, where you formally insert a dirac delta into the quantum field distribution.

Brennecke saw that something in the Chatterjee paper is related to a differential operator. Also Balaban defined a 2nd order differential operator on the lattice. Also there is the Poincare inequality in Section 10 of the Chatterjee paper. I wonder if there is a bridge between Green’s functions and these operators. And between Dirichlet forms on graphs and these differential operators. Recalling the graph Laplacian, you just take local differences of the function argument.

Also, there is the so called Yang Mills heat flow in the continuum. What is the Yang Mill’s heat flow on a lattice? The heat kernel for a Laplacian on a graph is just e to the negative t times L. It is the heat semigroup.

Oh another interesting thing from today was that the Laplace transform provided a bridge between the land of resolvents and the land of heat semigroups or heat equation or Green’s functions. As long as alpha is in the resolvent set, meaning that alpha cannot be an eigenvalue of the Laplace operator, or that alpha lies below the smallest eigenvalue, then you can consider the inverse map (L+alpha)^{-1} and do spectral stuff with it.

January 6, 2025

When the coupling constant of U(N) lattice gauge theory is small, then beta is big. When beta is big, the format of the Yang mills probability density forces the Wilson action to be very small if there is to be any probability mass associated to a given configuration. The Wilson action being small means that most unitary matrices are equal to the identity matrix. Matrices mostly being equal to identity matrices implies that parallel transporting along plaquettes does not do anything. So the gauge field has no effect. I think that the gauge field having little effect and the coupling constant being small are related concepts. By analogy, smallness of the coupling constant for gravity means that gravity has little role.

I was previously confused because I noticed that when the coupling constant is large, then the Yang Mills gauge measure is just a product Haar measure. I figured product Haar measure means that there is no interaction, a contradiction to the coupling constant being large. Perhaps the reason that this is not a contradiction is that unitary matrices under Haar measure are not concentrated near the identity matrix, so during parallel transport, the gauge field plays a bigger role. And the gauge field taking stronger role fits with the coupling constant being larger.

December 24, 2025

Conditional probabilities are where you fix some prior knowledge/variables and then you ask about the probability for the remaining variables.

That seems like the right framework for axial gauge fixing. You fix the variables for the tree edges, then figure out the rest of the variables.

December 25, 2025

The log beta term in the formula for the free energy is related to the 1/beta smallness of the Wilson action.

Also, even though the math of cqft is very eclectic and broad, it also feels like a niche field. At least I get that feeling when looking at one of the early papers by Fröhlich and Brydges.