Why are large scale structures isotropic in the Ising model? I have at least a qualitative understanding of why the critical state of the Ising model is scale invariant, by arguments to do with renormalisation, which I understand only very roughly.
However, in addition to being invariant to changes in scale, the large-scale patterns at the critical point are also invariant with respect to rotations, even though the use of a (square) lattice makes the model anisotropic at the microscopic level. For that matter, the phase separated 'blob' pattern that forms when the Ising model is quenched also seems to be rotationally invariant.
Can someone offer me an explanation (intuitive or technical, or preferably both) for why one should expect large-scale structures in this type of model to be rotationally invariant?
 A: I don't know that this qualifies as a full answer, but the “intuitive” answer would be that on large enough scales, the details of the lattice won't matter: If you look from far enough away (conversely, the lattice parameter is sufficiently small), it will look like a continuum.
In this picture, you would expect large-scale structures to be fully rotationally invariant, but not smaller ones.  On the other hand, a similar statement would hold for the scale invariance.
Incidentally, when you say “seems to be rotationally invariant”, do you have any analytical or numerical evidence for that?  Once I looked at the correlation function, and I did see (numerically) that it was more or less circular.  
UPDATE: Here are those correlation functions.  I have been a bit reluctant to post them because (a) it does not really add anything as to why the correlations behave this way, and (b) I made them years ago for a course project, so the data (produced with the Swendsen-Wang algorithm) are maybe not as “safe”, and the plots not as nice as you would expect in a publication.  With this caveat, I will go ahead and post them anyway.
What to look for: At $T \gtrless T_c$, the (connected!) correlation function decays quickly, at $T=T_c$ it is far-ranged.  The isolines show how circularly symmetric the correlation function becomes at longer distance, while at short distance (in the high-$T$ plot) you can still see the lattice.  (Note the different scales on the plots.)








