TITLE: Particularities about the honeycomb lattice for the computation of connectivity constant
QUESTION [8 upvotes]: After reading the paper The connective constant of the honeycomb lattice equals $\sqrt{2+\sqrt{2}}$ by Hugo Duminil-Copin and Stanislav Smirnov (arXiv:1007.0575) published some time ago in Annals Math. it comes to my mind the following question/doubt.
We have the result for the honeycomb lattice but there is not an analogue result for the quadrangular lattice (as far as I know), or the triangular lattice. Maybe the answer would be 'it is not possible to adapt in any way the techinque used in the honeycomb lattice to other environments', but if this is the case, my question is what is so special/particular (not special computations, but the geometry of the lattice, deeper considerations, etc) of the honeycomb lattice that make this computation feasable there but not possible in other very regular lattices.

REPLY [6 votes]: What fails for other lattices is that there seems to be no parafermionic observable with properties as nice as for hexagonal lattice; specifically, there is no analog of Lemma 1.
In the definition of the parafermionic observable, one has two parameters to play with: $\sigma$ and $x$. The condition that the two configurations on the left in the picture on page 4, combined, contribute zero to (1), fixes the value of $\sigma$. The condition that the three pictures on the right contribute nothing fixes the value of $x$. This clearly uses the trivalence, but also the symmetry under the rotation by $e^{2\pi i/3}$. There seems to be no other lattice where you have as many parameters as conditions.
Curiously, it is conjectured based on numerics that the connective constant on the square lattice is also a root of a biquadratic polynomial, this time
$$
581x^4+7x^2-13.
$$
(see Conway, A. R.; Enting, I. G.; Guttmann, A. J., Algebraic techniques for enumerating self-avoiding walks on the square lattice, J. Phys. A, Math. Gen. 26, No. 7, 1519-1534 (1993). ZBL0772.60094.). As far as I know, there is no heuristic explanation for this, but the conjecture is claimed to have been checked to 12 digits of accuracy (Clisby, Nathan; Jensen, Iwan, A new transfer-matrix algorithm for exact enumerations: self-avoiding polygons on the square lattice, J. Phys. A, Math. Theor. 45, No. 11, Article ID 115202, 15 p. (2012). ZBL1241.82040.)
Another famous example of when a result is only know for the honeycomb lattice is the conformal invariance of critical percolation (on faces), see this MO answer and the reference therein.