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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.

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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.

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    $\begingroup$ I'll just add a brief comment: the best numerical evidence now suggests that the conjecture for the connective constant on the square lattice is wrong - see "On the growth constant for square-lattice self-avoiding walks" published by Jacobsen, Scullard, and Guttmann in J. Phys. A in 2016. Links: iopscience.iop.org/article/10.1088/1751-8113/49/49/494004 and arxiv.org/abs/1607.02984 $\endgroup$ Feb 17 at 11:55
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    $\begingroup$ @NathanClisby, thanks a lot! I remember hearing about this new evidence, but for some reason did not find the reference when writing my answer. $\endgroup$
    – Kostya_I
    Feb 17 at 12:42
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It's worth mentioning that there is another lattice for which the precise value of the connective constant has been established. In the paper "Self-avoiding walks and trails on the $3.12^2$ lattice" by Guttmann, Parviainen and Rechnitzer (J. Phys. A: Math. Gen. 38 (2005), 4309--4325), it is shown that the connective constant of the $3.12^2$ lattice is equal to the largest positive real root of the equation $$ x^{12} -4x^8 - 8 x^7 - 4 x^6 + 2 x^4 + 8 x^3 + 12x^2 + 8x + 2 = 0, $$ approximately equal to $1.71104$. This is done through a reduction to the case of the honeycomb lattice (which was conjectural at the time this paper was written, but is now established rigorously).

While the dependence on Duminil-Copin and Smirnov's result means we are still relying on their particular, and rather miraculous, "parafermionic observable" (mentioned in @Kostya_I's answer), this result at least demonstrates that it is perhaps not unreasonable to expect to find additional lattices for which the connective constant can be shown to be an explicitly identifiable algebraic number. Whether it is reasonable to expect such a thing for particularly nice lattices such as the square and triangular lattice seems rather more difficult to say.

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