Spectral decimation of the magnetic Laplacian on the Sierpinski gasket: Hofstadter's butterfly, determinants, and loop soup entropy
The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by complex-valued weights. When properly interpreted, these complex weights give rise to magnetic fluxes through cycles in the graph. 
In this paper we establish the spectrum of the magnetic Laplacian on the Sierpinski gasket graph ($SG$) where the magnetic fluxes equal $\alpha$ through the smallest upright triangles, and $\beta$ through the smallest downright triangles. This is achieved upon showing the spectral self-similarity of the magnetic Laplacian via a 3-parameter map involving non-rational functions, which takes into account $\alpha$, $\beta$, and the spectral parameter $\lambda$. When $\alpha,\beta\in \{0,\frac{1}{2}\}$, we can list the eigenvalues according to their multiplicities. When not both $\alpha$ and $\beta$ are in $\{0,\frac{1}{2}\}$, we show that the spectrum is essentially determined by the Julia set of the 3-parameter map. 
We provide two applications of our results: 1) demonstrate the true Hofstadter's butterfly appearing in the energy spectrum on $SG$ under uniform magnetic field; and 2) compute the magnetic Laplacian determinant which corresponds to the partition function for random cycle-rooted spanning forests on $SG$, derive the corresponding asymptotic complexity, and give its probabilistic interpretation in terms of a "loop soup entropy."