Correspondence between spanning trees and the Ising model on a square lattice

Abstract

An important problem in statistical physics concerns the fascinating connections between partition functions of lattice models studied in equilibrium statistical mechanics on the one hand and graph theoretical enumeration problems on the other hand. We investigate the nature of the relationship between the number of spanning trees and the partition function of the Ising model on the square lattice. The spanning tree generating function T (z) gives the spanning tree constant when evaluated at z = 1, while giving the lattice green function when differentiated. It is known that for the infinite square lattice the partition function Z(K) of the Ising model evaluated at the critical temperature K = Kc is related to T (1). Here we show that this idea in fact generalizes to all real temperatures. We prove that (Z(K) sech 2K ) 2 = k exp T (k) , where k = 2 tanh(2K) sech(2K). The identical Mahler measure connects the two seemingly disparate quantities T (z) and Z(K). In turn, the Mahler measure is determined by the random walk structure function. Finally, we show that the the above correspondence does not generalize in a straightforward manner to non-planar lattices.

The hypergeometric series for the partition function of the 2D Ising model

“…Guttmann and Rogers have defined a generating function that generalizes the spanning tree constant [12]. Subsequently, it was shown that this spanning tree generating function is related to the partition function of the isotropic Ising model in a precise way [13]. The connection is due to the fact that the same Mahler measure appears in both, which in turn is due to the random walk structure function for the square lattice.…”

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

“…As reviewed in [21], the spanning tree constant is known to bear a close relation with the critical temperature T c in the Ising model. Moreover, it has been shown in [21]-using the STGF T(z)-which for two simple vertex-transitive lattices (square and triangular), this type of relation is also valid at any temperature value. So, legitimate questions would be: are there other physical models in which the STGF can map into some relevant quantity characterizing the systems?…”

“…An illustrative example is the relation between the spanning tree constant z G of an infinite periodic graph G (for proper definitions, see section 2) and certain discrete spin-like models on G [19,20]. As explained in details in [21] (even with a brief historical account, refer to the references therein), for some problems like the Ising in the square lattice, the partition function at the critical temperature can be expressed as function of z sq .…”

“…Nonetheless, in [24] the framework has been put in very solid and general grounds, establishing T(z) as an important tool for calculating z G . Remarkably, by analyzing the corresponding STGFs it was shown [21] that T(z) and the partition function of the Ising model in square and triangular lattices are related not only at the T c , but also for any value of the temperature.…”

“…Our second goal is to embody more generally-e.g. than through only specific solvable instances as in [21]-the potential intrinsic connection between STGF's and some lattice models. We demonstrate that the free energy of the random walk loop soup (RWLS) model [31]-valuable in various branches of theoretical physics-can be given in terms of T e (z) for arbitrary periodic regular lattices L. As an application we unveil that for any L the system critical point (if existing) is directly related to z L .…”