Log Gamma Planar Random Walks

The Log Gamma pdf for    0< N  ≤ 1 arises from gamma pdf for    x = Log(1/N) > 0
The case α=1 coincides with a uniform distribution. 
Cases α >> 1  yield approximations to Gaussians truncated to [0,1].
Here we generate some unit step random walks in the plane 
with angle   0≤θ ≤ 2π   distribution given by   θ = 2π N

CDF Log Gamma

(∫_0^n)((((((1/N)))^(1-(α/μ))) ((((α/μ)))^α) ((Log[(1/N)])^(-1+α)))/(Gamma[α]))\[DifferentialD]N//PowerExpand//FullSimplify

CDFLG[α_,μ_,n_]:=((Gamma[α,-((α Log[n])/μ)])/(Gamma[α]))

CDFLG[1,0.5,n]

Solve[CDFLG[1,0.5,n]==0.752440668329,n]

Plot[{CDFLG[1,0.9,n],CDFLG[1,1,n],CDFLG[1,1.1,n]},{n,0,1},PlotRange->{0,1},Frame->True,GridLines->Automatic]

Plot[Cos[2π CDFLG[.4,0.5,n]],{n,0,1},PlotRange->All,Frame->True,GridLines->Automatic]

Uniform Angle Unit Step Random Walk

This is the standard usual planar random walk with unit step length and uniform direction distribution

R=Table[Random[],{k,1,10000}];
R[[1]]
ran=Table[{Cos[2π R[[k]]],Sin[2π R[[k]]]}, {k,1,10000}];

{ran[[1]],(∑_(j=1)^10000)ran[[j]]}
ListLinePlot[Table[(∑_(j=1)^J)ran[[j]],{J,1,10000}],Frame->True,GridLines->Automatic]

Log Gamma Angle Unit Step Random Walk α=0.5

sol=Table[Solve[CDFLG[1,0.5,n]==R[[k]],n][[2]], {k,1,10000}];
ranlg2=n/.sol;
ranlg2[[1]]
t2=Table[{Cos[2π ranlg2[[k]]],Sin[2π ranlg2[[k]]]}, {k,1,10000}];

{t1[[1]],(∑_(j=1)^10000)t2[[j]]}
ListLinePlot[Table[(∑_(j=1)^J)t2[[j]],{J,1,10000}],Frame->True,GridLines->Automatic]

Log Gamma Angle Unit Step Random Walk α=0.9

sol2=Table[Solve[CDFLG[1,0.9,n]==R[[k]],n][[1]], {k,1,10000}];
ranlg2=n/.sol2;
ranlg2[[1]]
t2=Table[{Cos[2π ranlg2[[k]]],Sin[2π ranlg2[[k]]]}, {k,1,10000}];

{t2[[1]],(∑_(j=1)^10000)t2[[j]]}
g09=ListLinePlot[Table[(∑_(j=1)^J)t2[[j]],{J,1,10000}],Frame->True,GridLines->Automatic,AspectRatio->Automatic]

{g09,g10,g11}

Show[{g09,g10,g11},PlotRange->All,AspectRatio->Automatic,GridLines->Automatic]

Log Gamma Angle Unit Step Random Walk α=1

This should be the same as the standard uniform direction distribution case

sol2=Table[Solve[CDFLG[1,1,n]==R[[k]],n][[1]], {k,1,10000}];
ranlg2=n/.sol2;
ranlg2[[1]]
t2=Table[{Cos[2π ranlg2[[k]]],Sin[2π ranlg2[[k]]]}, {k,1,10000}];

{t2[[1]],(∑_(j=1)^10000)t2[[j]]}
g10=ListLinePlot[Table[(∑_(j=1)^J)t2[[j]],{J,1,10000}],Frame->True,GridLines->Automatic,AspectRatio->Automatic]

Log Gamma Angle Unit Step Random Walk α=1.1

sol2=Table[Solve[CDFLG[1,1.1,n]==R[[k]],n][[1]], {k,1,10000}];
ranlg2=n/.sol2;
ranlg2[[1]]
t2=Table[{Cos[2π ranlg2[[k]]],Sin[2π ranlg2[[k]]]}, {k,1,10000}];

{t2[[1]],(∑_(j=1)^10000)t2[[j]]}
g11=ListLinePlot[Table[(∑_(j=1)^J)t2[[j]],{J,1,10000}],Frame->True,GridLines->Automatic]

Log Gamma Angle Unit Step Random Walk α=2

sol2=Table[Solve[CDFLG[1,2,n]==R[[k]],n][[1]], {k,1,10000}];
ranlg2=n/.sol2;
ranlg2[[1]]
t2=Table[{Cos[2π ranlg2[[k]]],Sin[2π ranlg2[[k]]]}, {k,1,10000}];

{t2[[1]],(∑_(j=1)^10000)t2[[j]]}
ListLinePlot[Table[(∑_(j=1)^J)t2[[j]],{J,1,10000}],Frame->True,GridLines->Automatic]

Log Gamma Angle Unit Step Random Walk α=5

sol2=Table[Solve[CDFLG[1,5,n]==R[[k]],n][[1]], {k,1,10000}];
ranlg2=n/.sol2;
ranlg2[[1]]
t2=Table[{Cos[2π ranlg2[[k]]],Sin[2π ranlg2[[k]]]}, {k,1,10000}];

{t2[[1]],(∑_(j=1)^10000)t2[[j]]}
ListLinePlot[Table[(∑_(j=1)^J)t2[[j]],{J,1,10000}],Frame->True,GridLines->Automatic]