Duality relation for the correlation length In this answer to my previous question, Yvan Velenik mentioned the equality for correlation lengths of dual Ising models on a square lattice
$$
\xi(T) = \xi(T^*)/2.
$$
I have the following questions about this equality:

*

*Is this true for all temperatures or only near the critical point?

*I saw, including in Yvan Velenik's comment, that this equality follows from the relationship between the correlation length and the surface tension in the Ising model. The correlation length is a function of direction. Does this mean that surface tension is also a function of direction?

*Is the same equality true for the correlation lengths of dual Ising models on triangular and hexagonal lattices?

 A: Before answering your three questions, let me copy here the comments I made in the other thread, as they are prone to disappear.

This relation is a consequence of the duality enjoyed by the planar
Ising model. Namely, the surface tension of the model at temperature
$T<T_{\rm c}$ is easily seen to be equal to the rate of exponential decay
of the 2-point function at the dual temperature $T^∗>T_{\rm c}$
(essentially by a direct comparison of the low and high-temperature
expansions of these two quantities). Now, at least heuristically, the
correlation between 2 distant spins at $T<T_{\rm c}$ is due to the presence
of a large Peierls contour surrounding both of them simultaneously (see
this answer). Note that such a contour creates two $+/−$ interfaces and
thus costs twice the surface tension.

Now, let us turn to your questions.

*

*Yes, the statement holds at all temperatures.

*Yes, the surface tension is direction dependent. It becomes isotropic only in the limit $T\uparrow T_c$. That's why the Wulff shape (that is, the shape of an equilibrium droplet of one phase immersed in the other phase) is not a disk in this model (the pictures are for decreasing values of the temperature, starting very close to $T_{\rm c}$; taken from Section 4.12.1 of our book):



*

*Yes. The graphs contributing to the high-temperature expansion of the 2-point function on the first lattice coincide with those contributing to the low-temperature expansion of the surface tension on the other lattice (provided that the two spins involved in the 2-point function are located on the dual vertices corresponding to the endpoints of the interface). The only other properties needed in the proof of this identity follow from correlation inequalities valid on all these graphs.

If you are interested, a proof of the equality between the surface tension and the rate of exponential decay of the 2-point function at the dual temperature can be found, for instance, in Lemma 6.3 of this paper. Alternatively, it can be deduced from exact computations of both quantities (although that might not be as enlightening).
