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Let a particle in the plane $R^2$ executes random jumps at discrete times $t= 1, 2, ...$. At each step, the particle jumps from the point it is a distance of lenght one. The angle of any new jump (say, with the $x$ axis) is uniformly distributed in $[0,2\pi]$.

Question: If initially ($t=0$) the particle is at the origin, what is the probability that it gets back to the unit disk of the plane for each time $t= 2, 3, 4, ...$? In particular, what is the value of this probability for $t=3$?

Thanks for any help!

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$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ The probability density for the step $n$ is given by $\ds{{\rm p}\pars{\vec{r}_{n}} \equiv {\delta\pars{r_{n} - 1} \over 2\pi}}$. The probability density $\pp_{N}\pars{\vec{r}}$ of arriving at $\vec{r}$ after $N$ steps is given by:

\begin{align} \pp_{N}\pars{\vec{r}} &\equiv \int\dd^{2}\vec{r}_{1}\,{\rm p}\pars{\vec{r}_{1}}\ldots \int\dd^{2}\vec{r}_{N}{\rm p}\pars{\vec{r}_{N}} \delta\pars{\vec{r} - \sum_{\ell = 1}^{N}\vec{r}_{\ell}} \\[3mm]&= \int\dd^{2}\vec{r}_{1}\,{\rm p}\pars{\vec{r}_{1}}\ldots \int\dd^{2}\vec{r}_{N}{\rm p}\pars{\vec{r}_{N}} \int\exp\pars{\ic\vec{k}\cdot\bracks{\vec{r} - \sum_{\ell = 1}^{N}\vec{r}_{\ell}}} \,{\dd^{2}\vec{k} \over \pars{2\pi}^{2}} \\[3mm]&= \int\expo{\ic\vec{k}\cdot\vec{r}} \bracks{\int\dd^{2}\vec{R}\,{\rm p}\pars{\vec{R}}\expo{-\ic\vec{k}\cdot\vec{R}}}^{N} \,{\dd^{2}\vec{k} \over \pars{2\pi}^{2}} = \int\expo{\ic\vec{k}\cdot\vec{r}} \bracks{\int_{0}^{2\pi}\expo{-\ic k\cos\pars{\theta}}\,{\dd\theta \over 2\pi}}^{N} \,{\dd^{2}\vec{k} \over \pars{2\pi}^{2}} \end{align}

However $$ \int_{0}^{2\pi}\expo{-\ic k\cos\pars{\theta}}\,{\dd\theta \over 2\pi} = \int_{-\pi}^{\pi}\expo{\ic k\cos\pars{\theta}}\,{\dd\theta \over 2\pi} = {1 \over \pi}\int_{0}^{\pi}\expo{\ic k\cos\pars{\theta}}\,\dd\theta = {\rm J}_{0}\pars{k} $$ where ${\rm J}_{\nu}\pars{k}$ is the $\nu$-$\it\mbox{order Bessel Function of the First Kind}$.

\begin{align} \pp_{N}\pars{\vec{r}} &= \int\expo{\ic\vec{k}\cdot\vec{r}}{\rm J}_{0}^{N}\pars{k} \,{\dd^{2}\vec{k} \over \pars{2\pi}^{2}} = {1 \over 2\pi}\int_{0}^{\infty}\dd k\,k\,{\rm J}_{0}^{N}\pars{k}\int_{0}^{2\pi} \expo{\ic kr\cos\pars{\theta}}\,{\dd\theta \over 2\pi} \end{align} $$ \color{#ff0000}{\pp_{N}\pars{\vec{r}} = {1 \over 2\pi}\int_{0}^{\infty}{\rm J}_{0}\pars{kr}{\rm J}_{0}^{N}\pars{k}k \,\dd k} $$

The probability ${\rm P}_{N{\Huge\circ}}$ that it returns to the unit circle after $N$ steps is given by: \begin{align} \color{#0000ff}{\large{\rm P}_{N{\Huge\circ}}} &= \int_{r\ <\ 1}\pp_{N}\pars{\vec{r}}\,\dd^{2}\vec{r} = \int_{0}^{\infty}\overbrace{\bracks{\int_{0}^{1}{\rm J}_{0}\pars{kr}r\,\dd r}} ^{{\rm J}_{1}\pars{k}/k} {\rm J}_{0}^{N}\pars{k}k\,\dd k \\[3mm]&= \color{#0000ff}{\large\int_{0}^{\infty}{\rm J}_{1}\pars{k} {\rm J}_{0}^{N}\pars{k}\,\dd k} \end{align} We compute a few values with Wolfram Alpha: $$ \begin{array}{rclrcl} {\rm P}_{0{\Huge\circ}} =\int_{0}^{\infty}{\rm J}_{1}\pars{k} \,\dd k & = & 1\,,\quad {\rm P}_{1{\Huge\circ}} =\int_{0}^{\infty}{\rm J}_{1}\pars{k} {\rm J}_{0}\pars{k}\,\dd k & = & \half \\ {\rm P}_{2{\Huge\circ}} =\int_{0}^{\infty}{\rm J}_{1}\pars{k}{\rm J}_{0}^{2}\pars{k} \,\dd k & = & {1 \over 3}\,,\quad {\rm P}_{3{\Huge\circ}} =\int_{0}^{\infty}{\rm J}_{1}\pars{k} {\rm J}_{0}^{3}\pars{k}\,\dd k & = & {1 \over 4} \\ {\rm P}_{4{\Huge\circ}} =\int_{0}^{\infty}{\rm J}_{1}\pars{k}{\rm J}_{0}^{4}\pars{k} \,\dd k & = & {1 \over 5}\,,\quad {\rm P}_{5{\Huge\circ}} =\int_{0}^{\infty}{\rm J}_{1}\pars{k} {\rm J}_{0}^{5}\pars{k}\,\dd k & = & {1 \over 6} \end{array} $$

It $\tt\large seems$ the exact result is $\ds{% {\rm P}_{N{\Huge\circ}} = \int_{0}^{\infty}{\rm J}_{1}\pars{k}{\rm J}_{0}^{N}\pars{k} \,\dd k = {1 \over N + 1}}$

Mathematica can solve this integral and it yields the $\ul{\mbox{exact result}}$ $\color{#0000ff}{\Large{1 \over N + 1}}$ when $\color{#ff0000}{\large\Re N > -1}$.

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You can create a probility dristribution for the location after one step (the unit circle with a certain constant prbability). After two steps, I think that should be doable too. You can also carculate for each point the probability of entering de unit disk in the next step. Then, by combining the probability distribution ($P_1(x,y)$) and the probability of ending in the disk $P_2(x,y)$, you get $$ \int P_1(x,y)P_2(x,y) \,dx\,dy $$

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This is not a complete answer, but is kind of long for a comment.

For $t=2$, draw a picture. It's pretty easy to see the answer is $1/3$.

For the $t=3$ case, let the angles be $\theta_1$, $\theta_2$, and $\theta_3$. By symmetry, you can assume without loss of generality that $\theta_1 = 0$. So you need to find the probability that $\|\langle 1, 0 \rangle + \langle \cos\theta_2, \sin\theta_2 \rangle +\langle \cos\theta_3,\sin\theta_3\rangle \|<1$. A little computation shows that this is equals $P(1+\cos\theta_2+\cos\theta_3+\cos(\theta_2-\theta_3)<0)$, where $\theta_2$ and $\theta_3$ are chosen independently from the uniform distribution on $[0,2\pi]$. That's as far as I got.

This is the probability that the particle is in the unit disc at time $t=3$ (not $t$ equals 2 or 3).

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