Large-scale limits of 3D continuous phase coexistence models beyond Gaussian noise and polynomial potentials: the unresolved cases of the Hairer-Xu classification

Hairer and Xu (2018) classified the large-scale behaviour of three-dimensional continuous phase coexistence models of Allen-Cahn type, solving $\partial_t u = \Delta u - \varepsilon V'_\theta(u) + \delta\hat\xi$ on a large torus, and showed that under diffusive rescaling the solutions converge to one of three universal limits - the dynamical $\Phi^4_3$ or $\Phi^3_3$ equation, or an Ornstein-Uhlenbeck process - according to whether two computable obstructions, the constants $A$ and $B$, vanish. The classification was proved for spatially Gaussian driving noise and, in the weakly nonlinear regime, for polynomial potentials. Several of the cases left open in the source paper have since been resolved: symmetric non-Gaussian noise (Shen-Xu, 2018), non-polynomial smooth nonlinearities under Gaussian noise (Furlan-Gubinelli, 2019), and the whole-space setting for the $\Phi^4_3$ branch (Zhu-Zhu, 2023). A precise open core nevertheless survives: (i) non-polynomial nonlinearities driven by non-Gaussian, short-range correlated noise with a space-time white-noise limit, which Corwin-Shen (2020) list as still open; (ii) the asymmetric-potential branches ($A \ne 0$ and $B \ne 0$, with $\Phi^3_3$ and Ornstein-Uhlenbeck limits) under non-Gaussian noise, together with the Shen-Xu conjecture that asymmetric noise with a symmetric potential likewise produces $\Phi^3_3$ or Ornstein-Uhlenbeck limits rather than $\Phi^4_3$ - both unproved; (iii) the non-$\Phi^4_3$ branches on the whole space $\mathbb{R}^3$ and for non-polynomial potentials; and (iv) a probabilistic interpretation of the constant $B$ analogous to the known integral formula for $A$.

Background

A continuous phase coexistence model is a stochastic partial differential equation (SPDE) of Allen-Cahn type, $\partial_t u = \Delta u - \varepsilon V'_\theta(u) + \delta\hat\xi,$ describing the evolution of an order parameter $u(t,x)$ in a two-phase medium: $V_\theta$ is a potential whose minima encode the coexisting phases, $\theta$ is a bifurcation parameter, $\varepsilon$ and $\delta$ are small scales controlling the strength of the nonlinearity and of the noise, and $\hat\xi$ is a random field modelling microscopic fluctuations, with spatial covariance of compact support and unit integral, so that under diffusive rescaling it approximates space-time white noise. Hairer and Xu studied the large-scale (i.e. long time, large distance) behaviour of such models in space dimension three, where the rescaled equation is singular and its limits - the dynamical $\Phi^4_3$ and $\Phi^3_3$ equations, the formal stochastic quantization equations of Euclidean quantum field theories with quartic and cubic self-interaction - are only defined after renormalization.

Two regimes are distinguished. In the weakly nonlinear regime one fixes $\delta = 1$ and lets $\varepsilon \to 0$, so the field is small but the noise is fully active; the effective nonlinearity is governed by the averaged potential $\langle V_\theta\rangle$ obtained by averaging $V_\theta$ against the stationary field of the linearized equation. In the weak noise regime one fixes $\varepsilon \approx 1$ and lets $\delta \to 0$, so the noise is weak and the dynamics track the deterministic bifurcation diagram of $V_\theta$. The averaged potential is assumed to undergo a pitchfork bifurcation at the origin (a symmetric loss of stability where a single minimum splits into two); generic asymmetric potentials instead undergo a saddle-node bifurcation (two equilibria colliding and annihilating), and this asymmetry is precisely what alters the classification.

The main theorem of Hairer and Xu is a trichotomy. In the weakly nonlinear regime, whether the rescaled field converges to the dynamical $\Phi^4_3$ equation at scale $\alpha = 1$ is decided by an obstruction constant $A = \int P(z)\, \mathbf{E}\big(V'_0(\Psi(0))\, V''_0(\Psi(z))\big)\,dz,$ where $P$ is the heat kernel, $z$ is a space-time variable, and the expectation is over the stationary Gaussian field $\Psi$ solving the linearized equation; $A$ vanishes exactly when the $\Phi^4_3$ limit is available, and when $A \ne 0$ the field instead converges, along suitable branches and at scales $\alpha = (1+\beta)/2$ or $5/6$ (where $\beta$ is the exponent in the tuning $\theta \sim \varepsilon^{\beta}$ of the bifurcation parameter along these branches), to Ornstein-Uhlenbeck (OU) processes - linear SPDEs of the form $\partial_t v = \Delta v - cv + \xi$ - while at the critical point one branch converges to the dynamical $\Phi^3_3$ equation at scale $\alpha = 8/9$. In the weak noise regime an analogous constant $B$, a distinguishable combination of Taylor coefficients of $V$ around the bifurcation point whose vanishing is equivalent to the potential being tunable to the symmetric ($\Phi^4_3$-producing) shape, plays the same role: if $B = 0$ one obtains the $\Phi^4_3$ family at scale $\alpha = 1/2$, while if $B \ne 0$ one obtains $\Phi^3_3$ at scale $\alpha = 7/9$ and OU limits at scale $\alpha = 2/3$, again along appropriately chosen branches of the bifurcation diagram.

The proofs rest on the solution theory for singular SPDEs developed in the 2010s, principally the theory of regularity structures and the companion theory of paracontrolled distributions, which give a renormalized, coordinate-free meaning to equations such as the dynamical $\Phi^4_3$ in which the ill-defined products are replaced by Wick-ordered (renormalized) ones and diverging constants are absorbed into counterterms. The classification itself is a weak-universality statement: it identifies which features of the microscopic model (the symmetry of $V$, the statistics of $\hat\xi$) survive at large scales.

The source paper states its results for Gaussian $\hat\xi$, and in the weakly nonlinear regime for polynomial $V$ (the weak noise regime already allows general $C^8$ potentials). It explicitly leaves open the extension to non-Gaussian noise, to non-polynomial potentials in the weakly nonlinear regime, to the whole space $\mathbb{R}^3$ in place of a torus, and to a probabilistic interpretation of $B$ of the kind available for $A$. A companion paper of Hairer and Xu on interface fluctuation models develops the parallel two-dimensional theory.

Problem Statement

Complete the Hairer-Xu classification of large-scale limits of three-dimensional continuous phase coexistence models in the settings its original proof does not reach. Specifically: (1) prove the trichotomy - convergence, under the diffusive rescalings of the source paper, to the renormalized dynamical $\Phi^4_3$ equation when the obstruction constant vanishes, and to the $\Phi^3_3$ or Ornstein-Uhlenbeck limits on the asymmetric branches - for equations $\partial_t u = \Delta u - \varepsilon V'_\theta(u) + \delta\hat\xi$ in which the nonlinearity $V$ is a general smooth non-polynomial function and the driving field $\hat\xi$ is non-Gaussian, spatially translation-invariant, short-range correlated noise whose rescaling approximates space-time white noise, thereby resolving the case listed as still open by Corwin-Shen; (2) settle, for the same non-Gaussian, spatially translation-invariant, short-range correlated noise class, the polynomial-potential case, namely the asymmetric branches of the trichotomy for polynomial potentials together with the conjecture of Shen-Xu Remark 1.5 (that asymmetric noise with a symmetric polynomial potential produces the same limits as an asymmetric potential, namely $\Phi^3_3$ or OU rather than $\Phi^4_3$); (3) establish the non-$\Phi^4_3$ branches of the trichotomy on the whole space $\mathbb{R}^3$ in place of the torus, in the Gaussian-noise setting of the source classification; and (4) determine whether the obstruction constant $B$ admits a probabilistic representation in terms of the driving field, analogous to the heat-kernel integral formula (1.11) for the constant $A$ in the source paper. The four parts are independent sub-problems, and together they cover the cases the original proof does not reach; each part is posed in the setting described for it above, and the relaxations named in the different parts are not combined.

References

1. Martin Hairer and Weijun Xu, "Large-Scale Behavior of Three-Dimensional Continuous Phase Coexistence Models", Communications on Pure and Applied Mathematics 71 (2018), no. 4, 688-746. DOI: 10.1002/cpa.21738, https://doi.org/10.1002/cpa.21738 (preprint: arXiv:1601.05138, https://arxiv.org/abs/1601.05138, titled "Large scale behaviour of 3D continuous phase coexistence models").
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