Exact scaling functions for one-dimensional stationary KPZ growth

Abstract:

Paper [PDF]

Tables:

• [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.
  
  
• [Table] The Baik-Rains distribution function F_y(s) with y=0 in the form

  s

  F0(s)

  log(F'0(s))

• 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: [gy-deriv.txt].

To properly import the file in Mathematica, use:
 

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}]