KPZ fixed point

Question

It was conjectured, later proved that KPZ converges to a universal limit called the KPZ fixed point.

Are there analogue of this for other classes of SPDE like the stochastic quantization equation?

Answer 1

The stochastic quantization equation reads $$ \partial_t\Phi = \Delta\Phi + C \Phi - \Phi^3 + \xi $$ with the cube interpreted in a suitably renormalised way and $\xi$ denotes space-time white noise. It makes sense for $d < 4$. The natural conjecture is that, for $d \in \{2,3\}$ there exists a critical value of $C$ which separates a regime where the equation admits a unique invariant measure and a regime where uniqueness fails. For $d=2$, at the critical value of $C$, there should exist an exponent $z$ such that the rescaled field $$ \Phi_\varepsilon(t,x) = \varepsilon^{-1/8}\Phi(t/\varepsilon^z,x/\varepsilon) $$ converges to a distributional Markovian space-time random field whose distribution at any fixed time coincides with the random field described in the paper by Camia, Garban and Newmann.

Note though that unlike in the case of KPZ, $z$ isn't known (numerically it's about $2.1665$) and, a fortiori, the full limit isn't known either. In $d=3$ a similar result should hold, but even the analogue of the exponent $-1/8$ isn't known and not much is known about the limiting field. In dimensions $d\ge 4$, one should start with $\xi$ having smooth correlations and the limit should simply be the additive stochastic heat equation, but this isn't known either.

Answer 2

While convergence to the KPZ fixed for a large class of random systems remains open, much progress has been made in recent years. For a nice informal review of topics in KPZ universality, see the article cited below by Ivan Corwin, [1]. Moreover another nice paper by Duncan Dauvergne and Lingfu Zhang ''gives a framework for proving convergence to the directed landscape given convergence to the KPZ fixed point. They apply this framework to prove convergence for a range of models, some without exact solvability: asymmetric exclusion processes with potentially non-nearest neighbour interactions, exotic couplings of ASEP, the random walk and Brownian web distance, and directed polymer models.''

As for recent progress in universality phenomena not directly related to the KPZ fixed point, one can look at objects like the stochastic quantization of the Yang-Mills measure in dimensions 2 and 3, Liouville quantum gravity metrics and other objects connected to the Gaussian free field that appear in the study of statistical physics, stochastic partial differential equations, random geometries and random matrices.

References

Corwin, I., Kardar-Parisi-Zhang universality, Notices Am. Math. Soc. 63, No. 3, 230-239 (2016). ZBL1342.82098.