Spanning tree generating functions for infinite periodic graphs L and connections with simple closed random walks on L

A spanning tree generating function T(z) for infinite periodic vertex-transitive (vt) lattices L _vt has been proposed by Guttmann and Rogers (2012 J. Phys. A: Math. Theor. 45 494001). Their spanning tree constants $z_{L_{vt}}$ are then given by T(z = 1). Here an extended T _e(z) is constructed, relaxing the previous vt condition to q-regular lattices L and likewise leading to z _L = T _e(1). Further, in the vt case the method to derive T _e yields a new integral formula for the lattice Green function. As examples, spanning tree generating functions for all the eleven vt Archimedean (surprisingly, easier to obtain from T _e than from T) and two relevant non-vt, martini and the (4, 8²) covering/medial, lattices are derived. The importance of T _e (and T)—beyond just being a tool to calculate z _L—is illustrated with the proof that the free energy of the random walk loop soup model (defined from closed random walks, the loops, over L) can be written in terms of T _e(z). From such result, it is shown that the system critical point—existing only for d = 1 and d = 2—is directly related to z _L. (paper)