Exact height distributions for the KPZ equation with narrow wedge initial condition

Herbert Spohn; Tomohiro Sasamoto

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Exact height distributions for the KPZ equation with narrow wedge initial condition

Tomohiro Sasamoto, Herbert Spohn

Statistical Mechanics · Submitted 9 Feb 2010

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The authors' main claim Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we obtain a determinantal formula for the one-point distribution of the solution $h(x,t)$ valid for any $x$ and $t>0$.

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arXiv ID1002.1879 (version 2)

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CitationHerbert Spohn, Tomohiro Sasamoto. "Exact height distributions for the KPZ equation with narrow wedge initial condition." arXiv:1002.1879 (2010). https://pith.science/paper/J4MF2LDK

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.16 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.81 · compares with prior results 0.78 · mentions code or data availability 0.04.

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We consider the KPZ equation in one space dimension with narrow wedge initial condition, $h(x,t=0)=- |x|/\delta$, $\delta\ll 1$. Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we obtain a determinantal formula for the one-point distribution of the solution $h(x,t)$ valid for any $x$ and $t>0$. The corresponding distribution function converges in the long time limit, $t\to\infty$, to the Tracy-Widom distribution. The first order correction is a shift of order $t^{-1/3}$. We provide numerical computations based on the exact formula.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider the KPZ equation in one space dimension with narrow wedge initial condition, $h(x,t=0)=- \|x\|/\delta$, $\delta\ll 1$.	background 0.82	0.40	0.03	asserted 0.88
s1	Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we obtain a determinantal formula for the one-point distribution of the solution $h(x,t)$ valid for any $x$ and $t>0$.	contribution 0.73	0.91	0.40	asserted 1.00
s2	The corresponding distribution function converges in the long time limit, $t\to\infty$, to the Tracy-Widom distribution.	result 0.99	0.62	0.38	asserted 1.00
s3	The first order correction is a shift of order $t^{-1/3}$.	result 1.00	0.78	0.11	asserted 1.00
s4	We provide numerical computations based on the exact formula.	contribution 0.53	0.94	0.04	asserted 0.99

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Original abstract

We consider the KPZ equation in one space dimension with narrow wedge initial condition, $h(x,t=0)=- |x|/\delta$, $\delta\ll 1$. Based on previous results for the weakly asymmetric simple exclusion process with step initial conditions, we obtain a determinantal formula for the one-point distribution of the solution $h(x,t)$ valid for any $x$ and $t>0$. The corresponding distribution function converges in the long time limit, $t\to\infty$, to the Tracy-Widom distribution. The first order correction is a shift of order $t^{-1/3}$. We provide numerical computations based on the exact formula.

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12 arXiv papers cite this in 19 sentences

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Papers citing it · 11 sentences without a confident label

Builds on it 1 paper

To extend the results of [ref] and show that the $Airy_2$ process is the large time limit of KPZ with narrow wedge initial condition we would first try to extend the results of [ref] to joint distributions of particle positions in ASEP.

Formulas for Joint Probabilities for the Asymmetric Simple Exclusion Process · Introduction · 0.82

Cites it as support 1 paper

This last statement is substantiated by the developments in [ref] .

Painlev\'e Functions in Statistical Physics · Introduction · 0.99

Cites it as background 6 papers

Recently exact solutions have also been obtained for the KPZ equation at all times for various initial conditions [ref] .

Delta-Bose gas on a half-line and the KPZ equation: boundary bound states and unbinding transitions · Overview: KPZ in a half-space. · 0.83
A more detailed exposition can be found in [ref] .

The one-dimensional KPZ equation: an exact solution and its universality · Introduction · 0.89
Exact results have also been obtained for the KPZ equation itself, and its equivalent system, the continuous directed polymer [ref] .

Half-space stationary Kardar-Parisi-Zhang equation · Introduction · 0.83

What it cites · 2 sentences without a confident label

Cites as background 1 paper

Very recently, Balaźs, Quastel, and Seppäläinen [ref] studied the KPZ equation with initial condition $Z(x,0)=exp[(λ/2ν)b̃(x)]$, where $b̃(x)$ is two-sided Brownian motion.

Scaling exponent for the Hopf-Cole solution of KPZ/Stochastic Burgers · The Cole-Hopf solution · 0.93

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