Simulating a random walk on a filament in Sage Math.

Consider a protein that moves along an actin filament starting at position $x=0.$ At every

timestep, it has an equal probability to go right or left, incrementing or decrementing $x$ by

one unit. When it returns to $x=0,$ it falls off and the process stops.

$($a$)$ Implement a function that will simulate this walker from its initial state to when it falls

off. The function should output two values: the maximal absolute displacement $|x{|}_{\text{m}\text{a}\text{x}}$

that the walker attained, and the time $t$ at which it fell off.

$($b$)$ Run the above simulation $100$ times to generate a distribution of $|x{|}_{\text{m}\text{a}\text{x}},t$ pairs. $($Note:

this may take a while, depending on the "luck" of your random number generator!$).$

Construct a plot of $|x{|}_{\text{m}\text{a}\text{x}}$ vs $t.$ Does it look linear? What about $|x{|}_{\text{m}\text{a}\text{x}}$ vs $\sqrt[\phantom{2}]{t}?$ What

does this mean about how long we would need to wait for it to go twice the greatest

$|x{|}_{\text{m}\text{a}\text{x}}$ that you observed?

$($c$)$ Plot the histogram of $t$ on a regular scale. What do you observe? Does this make sense?

Why or why not?

$($d$)$ Now plot the histogram on a log$-$log scale $($i$.$e$.,$ the histogram of $log(t)$ with a logarithmic

$y$ axis$).$

Hint: the following code will put the counts for each bin into the variable $n,$ and the

edge for each bin in variable bin:

You can modify the above so that plt$.$hist computes the histogram of $log(x),$ and

plt$.$scatter plots $log(n)$ rather than $n.$

$($e$)$ The "time of first return" for a random walk in $1-$D $($which is what your value $t$ is

measuring$)$ follows a power law distribution. Either by eye or using regression, use your

results from part $($d$)$ to estimate the scale parameter $.$