interacting particle systems and KPZ

In one spatial dimension, the KPZ equation is formally

$$\partial_t H=\tfrac12\partial_x^2H+\tfrac12(\partial_xH)^2+\dot W,$$

where $\dot W$ is space-time white noise. Its Hopf–Cole solution is defined through the stochastic heat equation (SHE), $\partial_t Z=\tfrac12\partial_x^2Z+Z\dot W$, by $H=\log Z$. Write $\mathcal Z(s,x;t,y)$ for the SHE fundamental solution with delta initial data at $x$ at time $s$, and $\mathcal H(s,x;t,y)=\log\mathcal Z(s,x;t,y)$ for its logarithm. The $3:2:1$ scaling means time, space, and height scale respectively as $T$, $T^{2/3}$, and $T^{1/3}.$ The KPZ equation and the directed landscape (arXiv:2301.00547) Convergence of the KPZ line ensemble (arXiv:2106.08051)

A precise broad-initial-profile version of the universality question uses the scaled narrow-wedge kernel

$$\mathfrak H^T(s,x;t,y)=2^{1/3}T^{-1/3}\mathcal H(Ts,2^{1/3}T^{2/3}x;Tt,2^{1/3}T^{2/3}y)+(t-s)2^{1/3}T^{2/3}/24.$$

For a continuous profile $f:\mathbb R\to\mathbb R$ such that $f(x)\le C(1+|x|)$ for some finite $C$, define

$$(f\otimes_T\mathfrak H^T)(t,y)=2^{1/3}T^{-1/3}\log\int_{\mathbb R}\exp\!\left(2^{-1/3}T^{1/3}\big(f(x)+\mathfrak H^T(0,x;t,y)\big)\right)\,dx.$$

The question is whether, as $T\to\infty$, these scaled solutions converge in distribution to the KPZ fixed point with initial profile $f$. In its variational description that limit is $(f\,\overline\otimes\,\mathcal L)(t,y)=\sup_{x\in\mathbb R}\{f(x)+\mathcal L(0,x;t,y)\}$, where $\mathcal L$ is the directed landscape: a continuous random field with independent Airy-sheet increments on disjoint time intervals and the composition rule $\mathcal L(s,x;t,y)=\max_z[\mathcal L(s,x;r,z)+\mathcal L(r,z;t,y)]$ for $s<r<t$. The KPZ equation and the directed landscape (arXiv:2301.00547)

Background

Introduced as a model of random interface growth, the KPZ equation is a canonical object for the universality class in which height fluctuations grow on the $t^{1/3}$ scale and correlations extend over distances of order $t^{2/3}$. The formal equation is not interpreted by ordinary multiplication of distributions; the Hopf–Cole transformation instead defines its solution through the multiplicative-noise SHE. This connection also links interface fluctuations to directed polymers in random media. Convergence of the KPZ line ensemble (arXiv:2106.08051) The one-dimensional KPZ equation and its universality class (arXiv:1503.06185)

Exact formulas from integrable interacting particle systems supplied an early route to rigorous KPZ asymptotics. In particular, Tracy and Widom’s ASEP formulas for step initial data could be taken to the narrow-wedge KPZ equation through an appropriate scaling limit. A 2017 account emphasized that the exact methods then gave limited access to spatial correlations and that proving full scaling limits for broad classes of KPZ initial data remained a central challenge. Later work on the KPZ line ensemble and the directed landscape provides stronger process-level results than that account could report. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017) The one-dimensional KPZ equation and its universality class (arXiv:1503.06185) Convergence of the KPZ line ensemble (arXiv:2106.08051) The KPZ equation and the directed landscape (arXiv:2301.00547)

Known results

For narrow-wedge initial data, the 2010 work of Amir, Corwin, and Quastel, using ASEP results, gave an explicit formula for the Hopf–Cole solution and established the $t^{1/3}$ fluctuation scale and one-point asymptotics. The related ASEP formulas provided a route from a particle system to the continuum equation, but the one-point formula alone did not identify spatial correlations. Deift, Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems II (2017) The one-dimensional KPZ equation and its universality class (arXiv:1503.06185)

A stronger spatial result is convergence, as time grows, of the suitably scaled KPZ line ensemble to the parabolic Airy line ensemble. In particular, this gives a process limit across spatial locations at a fixed scaled time, rather than only a one-point fluctuation law. The proof uses local fluctuation estimates and tightness for the line ensemble. Convergence of the KPZ line ensemble (arXiv:2106.08051)

The preprint on the directed landscape reports convergence of the scaled narrow-wedge KPZ kernels to that landscape and, for every continuous $f$ bounded above by a linear-growth function, convergence of $f\otimes_T\mathfrak H^T$ in distribution in $C((0,\infty)\times\mathbb R,\mathbb R)$ to $f\,\overline\otimes\,\mathcal L$. It also records independent results establishing KPZ-equation convergence to the KPZ fixed point for general initial functions. Thus the precise profile-class question above has a positive answer in the stated topology; it should not be confused with an unrestricted assertion for every possible initial condition or with claims about all multi-time observables. The KPZ equation and the directed landscape (arXiv:2301.00547)