We study the 2-dimensional anisotropic KPZ equation (AKPZ), which is formally given by

$${\partial }_{t}h=\frac{1}{2}\mathrm{\Delta }h+\mathit{\lambda }({({\partial }_{1}h)}^2-{({\partial }_{2}h)}^2)+\mathrm{\xi },$$

where ξ denotes a space-time white noise and $\mathit{\lambda }>0$ is the so-called coupling constant. The AKPZ equation is a critical SPDE, meaning that not only is it analytically ill posed but also the breakthrough pathwise techniques for singular SPDEs in earlier works by Hairer, Gubinelli, Imkeller, and Perkowski are not applicable. As shown in recent work by the authors, the equation regularized at scale N has a diffusion coefficient that diverges logarithmically as the regularization is removed in the limit $N\to \mathrm{\infty }$. Here, we study the weak coupling limit where $\mathit{\lambda }={\mathit{\lambda }}_N=\hat{\mathit{\lambda }}∕\sqrt{logN}$: this is the correct scaling that guarantees that the nonlinearity has a still nontrivial but nondivergent effect. In fact, as $N\to \mathrm{\infty }$ the sequence of equations converges to the linear stochastic heat equation

$${\partial }_{t}h=\frac{{\mathrm{\nu }}_{\mathrm{eff}}}{2}\mathrm{\Delta }h+\sqrt{{\mathrm{\nu }}_{\mathrm{eff}}}\mathrm{\xi },$$

where ${\mathrm{\nu }}_{\mathrm{eff}}>1$ is explicit and depends nontrivially on $\hat{\mathit{\lambda }}$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearized via Cole–Hopf or any other transformation.

Let $f:X\to \mathrm{Spec}R$ be a 3-fold flopping contraction, where X has at worst Gorenstein terminal singularities and R is complete local. We describe the space of Bridgeland stability conditions on the null subcategory 𝒞 of $D^b(\mathrm{coh}X)$, which consists of those complexes that derive pushforward to zero, and also on the affine subcategory 𝒟, which consists of complexes supported on the exceptional locus. We show that a connected component ${\mathrm{Stab}}^{\circ }𝒞$ of $\mathrm{Stab}𝒞$ is the universal cover of the complexified complement of the real hyperplane arrangement associated to X via the homological MMP, and more generally that ${\mathrm{Stab}}_n^{\circ }𝒟$ is a regular covering space of the infinite hyperplane arrangement constructed in an earlier preprint by Iyama and the second named author. Neither arrangement is Coxeter in general. As a consequence, we give the first description of the stringy Kähler moduli space (SKMS) for all smooth irreducible 3-fold flops. The answer is surprising: we prove that the SKMS is always a sphere, minus either 3, 4, 6, 8, 12, or 14 points, depending on the length of the curve.

A long-standing open problem in the theory of hyperfinite equivalence relations asks whether the orbit equivalence relation generated by a Borel action of a countable amenable group is hyperfinite. In this paper, we prove that this question always has a positive answer when the acting group is polycyclic, and we obtain a positive answer for all free actions of a large class of groups including the lamplighter group ${\mathbb{Z}}_2\wr \mathbb{Z}$ and all virtually solvable groups having finite Prüfer rank. This marks the first time that a group of exponential volume-growth has been verified to have this property. In obtaining this result we introduce a new tool for studying Borel equivalence relations by extending Gromov’s notion of asymptotic dimension to the Borel setting. We show that countable Borel equivalence relations of finite Borel asymptotic dimension are hyperfinite, and more generally we prove under a mild compatibility assumption that increasing unions of such equivalence relations are hyperfinite. As part of our main theorem, we prove for a large class of solvable groups that all of their free Borel actions have finite Borel asymptotic dimension (and finite dynamic asymptotic dimension in the case of a continuous action on a 0-dimensional space). We also provide applications to Borel chromatic numbers, Borel and continuous Følner tilings, topological dynamics, and $C^{\ast }$-algebras.