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A celebrated result due to Duminil-Copin and Smirnov states that the connectivity constant for the honeycomb lattice is equal to $\sqrt{2+\sqrt{2}}$.

My question is the following: apart from the honeycomb lattices, for which lattices is it known exact estimates for the connectivity constant (if any)?

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  • $\begingroup$ I think the answer is "no other lattices." Also, this is almost the same question as: mathoverflow.net/questions/408850/… $\endgroup$
    – Sam Hopkins
    Commented Jan 27, 2023 at 19:52
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    $\begingroup$ See also the Wiki page: en.wikipedia.org/wiki/Connective_constant. I guess there is a known answer for another lattice (the "$(3.12^2)$ lattice," whatever that means) basically because of a reduction to the hexagonal lattice. $\endgroup$
    – Sam Hopkins
    Commented Jan 27, 2023 at 20:08

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