Probability, Random Walk, Distance from Origin after N steps in 2 and 3 dimensions I am looking for a formula of the distance from Origin after N equal steps in random directions in a 2 or 3 dimensional spaces. Can someone help me with a reference to a book, article or any publication dealing with this subject? Thanks!
 A: note the question asks for a formula, simulating the thing is trivial..
As a first step, since you only ask for the distance, work out the incremental change in distance with each step assuming a unit step in a random direction. You can then develop a simple 1D numerical approach to find the distance after n steps: (mathematica code, the question was orignally posted to the mathematica site)
md[n_Integer] := (
       Mean@Table[
        dissquared = 0;
        Do[ 
          dissquared += 
               1 + 2 Cos[RandomReal[{0, 2 Pi}]] Sqrt[dissquared ];
         , {n}];
         Sqrt[dissquared], {2000}])


 Show[Plot[ Sqrt[Pi i/4 ], {i, 1, 500}, PlotStyle -> Red], 
      ListPlot[Table[{i, md[i]}, {i, 1, 500, 10}]]]

oops I dont seem to be allowd to post a graphic.  Anyway we can emperically work out an expression for the average distance as 
d[n] = Sqrt[ n Pi /4 ]

Note obviously d[1] must be 1, but the simulation converges to the above for n larger than 4..
If someone can finish up and show that analytically I'd like to see..
A: 2 apprioriate models for your task could be:
1) Markovian matrix with probabilities between each state.
or 2) Brownian motion
A: $\newcommand{\+}{^{\dagger}}%
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$\large{\mbox{Random steps of length}\ a\ \mbox{in}\ 3D}$:

\begin{align}
\pp_{N}\pars{\vec{r}}
&\equiv
\int\dd^{3}\vec{r}_{1}\,{\delta\pars{r_{1} - a} \over 4\pi a^{2}}\ldots
\int\dd^{3}\vec{r}_{N}\,{\delta\pars{r_{N} - a} \over 4\pi a^{2}}
\delta\pars{\vec{r} - \vec{r}_{1} - \cdots - \vec{r}_{N}}
\\[3mm]&=
\int\dd^{3}\vec{r}_{1}\,{\delta\pars{r_{1} - a} \over 4\pi a^{2}}\ldots
\int\dd^{3}\vec{r}_{N}\,{\delta\pars{r_{N} - a} \over 4\pi a^{2}}
\int\dd^{3}\vec{k}\,
\expo{\ic\vec{k}\cdot\pars{\vec{r} - \vec{r}_{1} - \cdots - \vec{r}_{N}}}
\\[3mm]&=
\int\dd^{3}\vec{k}\,\expo{\ic\vec{k}\cdot\vec{r}}
\bracks{\int\dd^{3}\vec{\rho}\,{\delta\pars{\rho - a} \over 4\pi a^{2}}
\expo{-\ic\vec{k}\cdot\vec{\rho}}}^{N}
\\[3mm]&=
\int\dd^{3}\vec{k}\,\expo{\ic\vec{k}\cdot\vec{r}}
\bracks{\int_{0}^{\infty}\dd\rho\,4\pi\rho^{2}\,
{\delta\pars{\rho - a} \over 4\pi a^{2}}
\int\expo{-\ic\vec{k}\cdot\vec{\rho}}{\dd\Omega_{\vec{\rho}} \over 4\pi}}^{N}
\\[3mm]&=
\int\dd^{3}\vec{k}\,\expo{\ic\vec{k}\cdot\vec{r}}
\bracks{\int_{0}^{\infty}\dd\rho\,\delta\pars{\rho - a}
\,{\sin\pars{k\rho} \over k\rho}}^{N}
=
\int\dd^{3}\vec{k}\,\expo{\ic\vec{k}\cdot\vec{r}}
\bracks{\sin\pars{ka} \over ka}^{N}
\\[3mm]&=
\int_{0}^{\infty}\dd k\,4\pi k^{2}\,{\sin\pars{kr} \over kr}\,
\bracks{\sin\pars{ka} \over ka}^{N}
=
{4\pi \over a^{3}}\int_{0}^{\infty}x^{2}{\sin\pars{x\tilde{r}} \over x\tilde{r}}\,
\bracks{\sin{x} \over x}^{N}\,\dd x
\end{align}
where $\tilde{r} \equiv r/a$.

