Consider the Ising model on an $L \times L$ lattice with periodic boundary conditions in the east/west directions and with spins on the north boundary fixed as $+1$ and the spins on the south boundary fixed as $-1$. Then, if you fix a cold temperature $T < T_c$, as $L \to \infty$ you expect that the domain wall between the $+1$ spin cluster on the top and the $-1$ spin cluster on the bottom approaches a flat line (dimension 1), and exactly at $T = T_c$ you expect there to be some nontrivial behaviour, where you end up with the domain wall being some fractal dimension between $1$ and $2$.
This is just intuition I've gleaned from a talk I attended ~2 years ago, but I can't find any references following this line of thought. Are there any places to look to see experiments in this direction? In particular, I'm mainly curious how to rigorously define what the fractal dimension we are measuring at $T = T_c$ is in the thermodynamic limit, and what this number turns out to be.
As an example of how to formalise what I'm talking about, we can consider the box-counting dimension, and simply define the fractal dimension of the domain wall to be: $$\text{dim}_\text{box}(\text{Domain Wall}) = \lim_{\epsilon = \frac{1}{L} \to 0} \frac{\langle\text{Number of sites that the domain wall passes through}\rangle_\text{Gibbs Distribution on L x L lattice}}{\log(1/\epsilon)}$$ There might be other formulations based on the actual thermodynamic limit measure, but this isn't something I'm too familiar with