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Posted online: 2020-10-04 04:57:12Z by Tomas Kojar10

Cite as: P-201004.1

The KPZ equation is given by

$$ \partial_{t}h(x,t)=\partial_{x}^{2}h(x,t)+\lambda(\partial_{x}h(x,t))^{2}+\xi, $$

where $\xi$ denotes space-time white noise and $\lambda\in \mathbb{R}$ is a parameter describing the strength of its "asymmetry". It has been conjectured (see [BPRS93, BG97, Cor 12 ,GJ14,HQ18] for a number of results in this direction) that the KPZ equation has a “universal” character in the sense that any one-dimensional model of surface growth should converge to it provided that it has the following features:

• There is a microscopic smoothing mechanism. Pictorially this means that large valleys are quickly filled.

• The system has microscopic fluctuations with short-range correlations. Pictorially this means that height function change depends only on neighboring heights.

• The system has some “lateral growth” mechanism in the sense that the growth speed depends in a nontrivial way on the slope. The vertical effective growth rate depends non-linearly on local slope.

• At the microscopic scale, the strengths of the growth and fluctuation mechanisms are well separated: either the growth mechanism dominates (intermediate disorder) or the fluctuations dominate (weak asymmetry). Growth is drive by noise which quickly decorrelates in space / time and is not heavy tailed.

Here is a concrete surface growth mathematical model to give a sense of the above features. The random deposition model is one of the simplest (and least realistic) models for a randomly growing one-dimensional interface. Unit blocks fall independently and in parallel from the sky above each site of $\mathbb{Z}$ according to exponentially distributed waiting times. Recall that a random variable X has exponential distribution of rate $\lambda>0$ (or mean $1/\lambda$) if $P(X > x) = e^{-\lambda x}$. Such random variables are characterized by the memoryless property – conditioned on the event that $X > x$, $X - x$ still has the exponential distribution of the same rate. Consequently, the random deposition model is Markov – its future evolution only depends on the present state (and not on its history). The ballistic deposition (or sticky block) model was introduced by Vold [V59] in 1959 and, as one expects in real growing interfaces, displays spatial correlation. As before, blocks fall according to iid exponential waiting times, however, now a block will stick to the first edge against which it becomes incident. This creates overhangs and we define the height function h(t, x) as the maximal height above x which is occupied by a box.

The question is to prove that the solution to the tuned KPZ $$ \partial_{t}h(x,t)=\partial_{x}^{2}h(x,t)+\delta(\partial_{x}h(x,t))^{2}+\delta^{1/2}\xi, $$

, after addition of the Itô factor t/12, converges in distribution, in the topology of uniform convergence on compact subsets, to the KPZ fixed point. (See details in [2] for the case of continuous initial data bounded above by an AIry function A(1+|x|) plus a finite collection of narrow wedges)

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Created at: 2020-10-04 04:57:12Z

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