Critical Value of the Quantum Ising Model on Star-Like Graphs

We present a rigorous determination of the critical value of the ground-state quantum Ising model in a transverse field, on a class of planar graphs which we call star-like. These include the star graph, which is a junction of several copies of Z at a single point. Our approach is to use the graphical, or FK-, representation of the model, and the probabilistic and geometric tools associated with it.