The near critical random bond ising model via embedding deformation

Abstract

Using the formalism of differential equations, we introduce a new method to continuously deform the $s$-embeddings associated with a family of Ising models as their coupling constants vary. This provides a geometric interpretation of the critical scaling window $\asymp n^{-1}$ for the model on the $n \times n$ box. We then drive this deterministic deformation process by i.i.d.\ Brownian motions on each edge, centered at the critical model, thereby generating random $s$-embeddings as solutions to stochastic differential equations attached to near-critical random bond Ising models. In this setting, with high probability with respect to the random environment, the Ising model remains conformally invariant in the scaling limit, even when the standard deviation of the random variables (up to logarithmic corrections) is $n^{-\frac{1}{3}} \gg n^{-1}$, far exceeding the deterministic critical window. We also construct an Ising model with slightly correlated (in space) random coupling constants, whose critical window is $ \asymp \log(n)^{-1}$ on the $n \times n$ box. Our method, which can also be applied to the dimer context, naturally extends to a much broader class of graphs and opens a new approach to understanding the critical Ising model in random environments.