The free-fermionic C2(1) loop model, double dimers and Kashaev's recurrence

Abstract

We study a two-color loop model known as the C (1) 2 loop model. We define a free-fermionic regime for this model, and show that under this assumption it can be transformed into a double dimer model. We then compute its free energy on periodic planar graphs. We also study the star-triangle relation or Yang-Baxter equations of this model, and show that after a proper parametrization they can be summed up into a single relation known as Kashaev's relation. This is enough to identify the solution of Kashaev's relation as the partition function of a C (1) 2 loop model with some boundary conditions, thus solving an open question of Kenyon and Pemantle [29] about the combinatorics of Kashaev's relation. 1 Kashaev's initial equation contained a +4 instead of a −4 coefficient for the last terms, but one can easily get from one to another, for instance by multiplying g by −1 at a vertex of the cube and its three neighbors.

The Schwarzian octahedron recurrence (dSKP equation) I: explicit solutions

“…As mentioned, such combinatorial solutions of discrete evolution equations are often related to limit shapes phenomena for the associated statistical mechanics models, which generalize the celebrated arctic circle phenomenon for tilings of the Aztec diamond; on this classical theory, see [CEP96,CKP01,Gor21] and references therein, and for approaches similar to ours, [DFSG14,PS05,Mel18,Geo21]. In our case, it is unclear if one can hope for probabilistic interpretations of this sort for oriented dimers or complementary trees and forests, first because the solution is not a partition function but a ratio of partition functions, and second because configurations come with signs.…”

“…One of the main purposes of this paper is to prove a combinatorial expression of this rational function. The corresponding problem has been solved for various similar recurrences [CS04,Spe07,KP16,Mel18], and has led to fruitful developments such as limit shapes results [PS05,DFSG14,Geo21].…”