C^1 fractals in statistical mechanics It is well-known--even famous--that the Schramm-Loewner curves appear as domain boundaries between phases at second-order critical points like the critical Ising model or percolation in two dimensions. These curves are fractals and their Hausdorff dimensions are related to the critical exponents of correlation functions in the underlying stat mech model.
I have recently learned a bit about Nash's embedding theorem and the wrinkled geometries that appear there. I saw them described as "$C^1$" fractals, because they are nearly self-similar but admit tangent planes (pdf link).
The study of generalized fractal structures is very important in scaling theories, eg. multifractality in turbulent flows, so I ask the following question:
Are there stat mech models where the domain boundaries are expected to be $C^1$ or better? Is there some understanding of the degree of smoothness in terms of the correlation functions? Perhaps it has to do with the order of the phase transition?
 A: I will start with an example and then discuss a potential example that turns out to be (most likely) a non-example.
An example is the fractional Gaussian field with sufficiently large Hurst parameter. Informally, the fractional Gaussian field $h$ on $\mathbb{R}^d$ with index $s \in \mathbb{R}$ is given by $h = (-\Delta)^{-s/2} W$, where $(-\Delta)^{s/2}$ is the fractional Laplacian on $\mathbb{R}^d$ and $W$ is a white noise on $\mathbb{R}^d$ (the usual Gaussian free field is the case $s = 1$). A rigorous definition, as well as a comprehensive survey of this class of processes is given in this paper by Lodhia, Sheffield, Sun, and Watson.
The Hurst parameter is defined by $H = s - d/2$. If $H > 0$, then $h$ lives in a quotient of the space $C^{\lceil H \rceil - 1}(\mathbb{R}^d)$ of $(\lceil H \rceil - 1)$-times continuously differentiable functions. In particular, an example of a model with $C^1$ level sets is the fractional Gaussian field with $H > 1$. Regarding the correlation functions, the regularity of a Gaussian field is closely related to the regularity of its covariance; in the case of the fractional Gaussian field, see Theorem 3.3 of the aforementioned paper.
Going a bit further, we might ask whether the fractional Gaussian field arises as the scaling limit of models that are not a priori Gaussian. A natural example to consider is the Ising model with interaction $J_{x,y} = (-\Delta_{\mathbb{Z}^d})^s_{x,y}$ for $x, y \in \mathbb{Z}^d$ with $\Delta_{\mathbb{Z}^d}$ the graph Laplacian of $\mathbb{Z}^d$ (when $s = 1$, $J$ is a nearest-neighbour interaction).
Whether or not the scaling limit of this model is (expected to be) Gaussian depends on the dimension $d$. Precisely, it is believed that there exist a number $d_c$ (not necessarily an integer) such that the scaling limit is Gaussian if and only if $d > d_c$. In this case, I would expect the scaling limit to be a fractional Gaussian field with index $s$.
When the interaction is given as above, all evidence points to the fact that $d_c = 4 s$. See, for instance, this paper of Heydenreich, van der Hofstad, and Sakai (in this paper, $\alpha = 2 s$ in the notation I am using). In particular, if $d > d_c$ then $H < 0$ and the predicted Gaussian scaling limit cannot be defined as a continuous function, so this turns out be a non-example.
