Exact scaling functions for onedimensional stationary KPZ growth
Date: March 29, 2003
Abstract:
We determine the stationary twopoint correlation function of the onedimensional 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 RiemannHilbert 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.Paper [PDF]
Tables:
double precision tables (16 digits): [Table] The HastingsMcLeod 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:
 [Table] Same for V(s), v(s), and u(s)^{2} with entries

[Table]
The GUE TracyWidom distribution function
F_{2}(s)=exp(V(s)) in the form
s F_{2}(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 TracyWidom distribution function
F_{1}(s)=exp(½(V(s)+U(s))) in the form
s F_{1}(s) log(F'_{1}(s)) 2^{2/3}s
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 BaikRains distribution function F_{0}(s)=(1(s+2u'(s)+2u(s)^{2})v(s))exp(2U(s)V(s)) in the forms F_{0}(s) log(F'_{0}(s))
This is indeed F_{y}(s), setting y=0 from the paper.
s  U(s)  u(s)  u'(s)  ln(U(s))  ln(u(s))  ln(u'(s)) 
s^{ }  V(s)^{ }  v(s)^{ }  u(s)^{2}  ln(V(s))^{ }  ln(v(s))^{ }  ln(u(s)^{2}) 
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 n 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: [gyderiv.txt].
To properly import the file in Mathematica, use:
gg = Import["gyderiv.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}]
Michael Prähofer
20030329