Exact scaling functions for one-dimensional stationary KPZ growth

J. Stat. Phys. 115 (1-2), 255-279 (2004) 

doi 10.1023/B:JOSS.0000019810.21828.fc

as preprint arXiv:cond-mat/0212519 

Michael Prähofer and Herbert Spohn

Date: March 29, 2003

Abstract:

We determine the stationary two-point correlation function of the one-dimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann-Hilbert problem related to the Painlevé II equation. We solve these equations numerically with very high precision and compare with the prediction of Colaiori and Moore  obtained from the mode coupling approximation. 

Tables:

double precision tables (16 digits):

• [Table] The Hastings-McLeod solution u(s) of Painlevé II and associated functions U(s), u'(s) in double precision (16 digits) for values s=-40 to 200, stepsize 1/16. The columns are space separated with the following entries:

  s

  U(s)

  u(s)

  u'(s)

  ln(U(s))

  ln(-u(s))

  ln(u'(s))

• [Table] Same for V(s), v(s), and u(s)² with entries

  s

  V(s)

  v(s)

  u(s)2

  ln(V(s))

  ln(-v(s))

  ln(u(s)2)

• [Table] The GUE Tracy-Widom distribution function F₂(s)=exp(-V(s)) in the form

  s

  F2(s)

  log(F'2(s))

  The index two is used by convention and has nothing to do with F_y(s), setting y=2.
  
  

• [Table] The GOE Tracy-Widom distribution function F₁(s)=exp(-½(V(s)+U(s))) in the form

  s

  F1(s)

  log(F'1(s))

  2-2/3s

  The index one is used by convention and has nothing to do with F_y(s), setting y=1.
  
  

• (updated July 31, 2014) [Table] The Baik-Rains distribution function F₀(s)=(1-(s+2u'(s)+2u(s)²)v(s))exp(-2U(s)-V(s)) in the form

  s

  F0(s)

  log(F'0(s))

  This is indeed F_y(s), setting y=0 from the paper.

Accuracy of the decimal numbers in the following tables is about 100 digits:

• The scaling function g(y) in an ASCII file, first column is y, second column is g(y), y=0, y=1/128, y=2/128, ... up to y=8.84375:
  [gy.txt]
• g(y) with derivatives in an ASCII file, first column is y, second column is g(y), and every sixteenth line column 3 to 6 contain g'(y), g''(y), g'''(y), and g''''(y), respectively. Values of y range from -8.625 to 8.625:
  [gy-deriv.txt].
  
  To properly import the file in Mathematica, use for example:

gg = Import["gy-deriv.txt","Table"];
g = Interpolation[Table[{Rationalize[gg[[i]][[1]]],Rest[gg[[i]]]},{i,Length[gg]}],InterpolationOrder -> 57];
f = Function[y,g''[y]/4];
Plot[f[y],{y,-3,3}]