1-dimensional Random Walk

Question

I am studying random walk and I would like to create the image of one in 1-dimension. Here is a pretty good one for 2-dimensional walks: Drawing random paths in TikZ (although for some reason it doesn't recalculate each time - which doesn't really matter). But I can't figure out how to create one for 1 dimension.

I'd like it to look something like this:

What I've tried so far:

\pgfmathsetseed{1}

\begin{tikzpicture}[x=6pt,y=6pt,baseline,scale=1,rotate=45]
 \coordinate (current point) at (0,0);
 \coordinate (old velocity) at (0,0);
 \coordinate (new velocity) at (0.5,rand);
 \foreach \i in {0,1,...,20}
 {
   \draw (current point) 
   .. controls ++([scale=0.2]old velocity) and ++(new velocity) .. ++(0.5,rand)
      coordinate (current point);
   \coordinate (old velocity) at (new velocity);
   \coordinate (new velocity) at (0.5,rand);
 }
\end{tikzpicture}

Answer 1

Something like this perhaps?

\documentclass[tikz,margin=5]{standalone}
\begin{document}
\tikz[x=1pt,y=10pt]
  \foreach \c in {red, green, blue}
    \draw [\c] (0,0) \foreach ~ in {0,...,1000}{-- ++(1,{random(0,1)*2-1})};
\end{document}

Answer 2

Since you are slightly concerned about speed, here is the Mark Wilbrow's solution ported to Metapost + LuaTeX (I use ConTeXt, but you could use something similar with LuaLaTeX + mplib as well).

\setupMPpage[instance=doublefun]
\starttext
\dorecurse{3}{
\startMPpage
  ux := 0.5pt; 
  uy := 2pt;
  vardef jump = 2*(uniformdeviate 1) - 1 enddef;
  path p;
  z[0] = origin;
  p := z[0] for i = 1 upto 1000 :
    hide(z[i] = z[i-1] + (ux, jump*uy);) -- z[i]
  endfor;

  draw p ;
\stopMPpage}
\stoptext

The instance=doublefun line tells metapost to use double precision (instead of the usual scaled precision. I draw the line three times to get a fair speed comparison with TikZ.

On my 2013 Macbook, I get the following times:

• For the Mark Wibrow's Tikz solution:

  $time pdflatex --interaction=batchmode rand-ltx.tex 2.45s user 0.02s system 99% cpu 2.491 total

• For the above luatex + metapost (ConTeXT) solution:

  $time contextjit rand-c --once --noconsole 0.77s user 0.08s system 97% cpu 0.867 total

So metapost is about 3 times faster (Perhaps more, because the load time of luatex + opentype fonts is much larger than the load time of pdftex).

You can also play around with different type of line connections to see which result looks better (I am trying with --, .. tension 2 .. and `.. tension 3 ``.

\setupMPpage[instance=doublefun]
\starttext
\setvalue{line:1}{--}
\setvalue{line:2}{.. tension 2 ..}
\setvalue{line:3}{.. tension 3 ..}
\startTEXpage
\dorecurse{3}
{\startMPcode
  randomseed := 42;
  ux := 1pt; 
  uy := 4pt;
  vardef jump = 2*(uniformdeviate 1) - 1 enddef;
  path p;
  z[0] = origin;
  p := z[0] for i = 1 upto 100 :
    hide(z[i] = z[i-1] + (ux, jump*uy);) \getvalue{line:\recurselevel} z[i]
  endfor;

  draw p ;
\stopMPcode
\endgraf}
\stopTEXpage
\stoptext

which gives

There is a subtle difference between the second and the third curve.