The \(Z\)-invariant massive Laplacian on isoradial graphs
A planar graph \(G=(V,E)\) (where \(V\) is the set of vertices and \(E\) is the set of edges) is called isoradial if it can be embedded in the plane in such a way that all internal faces are inscribable in a circle, with all circles having the same radius, and such that all circumcenters are in the interior of the faces. Assume that for each \(l=(xy)\in E\), a positive number (conductance) \(\rho(l)=\rho(xy)\) is given and that for each \(x\in V\), a mass \(m(x)\) is given. Then define the massive Laplacian operator \(\Delta^m:\mathbb C^V\to \mathbb C^V\) by \[ (\Delta^m)f(x)=d(x)f(x)-\sum_{y\sim x}\rho(xy)f(y), \] where \(d(x)=m^2(x)+ \sum_{y\sim x}\rho(xy)\). A function \(f\in \mathbb C^V\) is massive harmonic on \(G\) if \(\Delta^mf=0\). In the paper the authors introduce a one-parameter family of massive Laplacian operators \((\Delta^{m(k)})_{k\in [0,1)}\) in which \(\rho(xy)=\rho(\theta_{xy}|k)\) and \(m(x)=m(x|k)\) are given by \[ \rho(\theta_{xy}|k)=\text{sc}(\theta_{xy}|k), \;\;m^2(x|k)=\sum_{y\sim x}[A(\theta_{xy}|k)-\rho(\theta_{xy}|k)], \] where \(\text{sc}\) is one of Jacobi's twelve elliptic functions and the function \(A\) is related to integrals of squared Jacobi elliptic functions. An explicit formula for the inverse of \(\Delta^{m(k)}\), the massive Green function, is proved. It is shown that the Green function only depends on the local geometry of the graph and its asymptotics is computed. An explicit local formula for an infinite volume Boltzmann measure and for the free energy of the model are given. It is shown that the model has a second-order phase transition at \(k=2\). Moreover, it is proved that the massive Laplacian operator provides a one-parameter family of \(Z\)-invariant rooted spanning forest models. An explicit parametrization of the spectral curve of the massive Laplacian (when the graph is \(Z^2\)-periodic) is given. It is shown that the spectral curve is Harnack and has geometric genus 1.
scientific article