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

Tomohiro Sasamoto; Herbert Spohn

doi:10.1016/j.nuclphysb.2010.03.026

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

Tomohiro Sasamoto
, Herbert Spohn

TUM Emeriti of Excellence

Chiba-U

Research output: Contribution to journal › Article › peer-review

154 Scopus citations

Abstract

We consider the KPZ equation in one space dimension with narrow wedge initial condition, h(x,t=0)=-|x|/δ, δ≪1, evolving into a parabolic profile with superimposed fluctuations. 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 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.

Original language	English
Pages (from-to)	523-542
Number of pages	20
Journal	Nuclear Physics, Section B
Volume	834
Issue number	3
DOIs	https://doi.org/10.1016/j.nuclphysb.2010.03.026
State	Published - Aug 2010

Access to Document

10.1016/j.nuclphysb.2010.03.026

Cite this

@article{c44eebb839e84fb29986f2eb2dd8f8a2,

title = "Exact height distributions for the KPZ equation with narrow wedge initial condition",

abstract = "We consider the KPZ equation in one space dimension with narrow wedge initial condition, h(x,t=0)=-|x|/δ, δ≪1, evolving into a parabolic profile with superimposed fluctuations. 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 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.",

author = "Tomohiro Sasamoto and Herbert Spohn",

note = "Funding Information: We are grateful to Michael Pr{\"a}hofer for many illuminating discussions. H.S. thanks Jeremy Quastel for emphasizing the importance of the crossover WASEP. This work is supported by a DFG grant. In addition T.S. acknowledges the support from KAKENHI ( 9740044 ) and H.S. from Math-for-Industry of Kyushu University . ",

year = "2010",

month = aug,

doi = "10.1016/j.nuclphysb.2010.03.026",

language = "English",

volume = "834",

pages = "523--542",

journal = "Nuclear Physics, Section B",

issn = "0550-3213",

publisher = "Elsevier B.V.",

number = "3",

}

TY - JOUR

T1 - Exact height distributions for the KPZ equation with narrow wedge initial condition

AU - Sasamoto, Tomohiro

AU - Spohn, Herbert

N1 - Funding Information: We are grateful to Michael Prähofer for many illuminating discussions. H.S. thanks Jeremy Quastel for emphasizing the importance of the crossover WASEP. This work is supported by a DFG grant. In addition T.S. acknowledges the support from KAKENHI ( 9740044 ) and H.S. from Math-for-Industry of Kyushu University .

PY - 2010/8

Y1 - 2010/8

N2 - We consider the KPZ equation in one space dimension with narrow wedge initial condition, h(x,t=0)=-|x|/δ, δ≪1, evolving into a parabolic profile with superimposed fluctuations. 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 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.

AB - We consider the KPZ equation in one space dimension with narrow wedge initial condition, h(x,t=0)=-|x|/δ, δ≪1, evolving into a parabolic profile with superimposed fluctuations. 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 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.

UR - https://www.scopus.com/pages/publications/77952885302

U2 - 10.1016/j.nuclphysb.2010.03.026

DO - 10.1016/j.nuclphysb.2010.03.026

M3 - Article

AN - SCOPUS:77952885302

SN - 0550-3213

VL - 834

SP - 523

EP - 542

JO - Nuclear Physics, Section B

JF - Nuclear Physics, Section B

IS - 3

ER -

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

Abstract

Access to Document

Other files and links

Fingerprint

Cite this