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A random walk in the plane with step size $f(n)$ is cool if there is some constant $C$ such that it returns within a distance $C$ of the origin infinitely many times with probability $1$.

At the $n$-th step it moves $f(n)$ in one of the directions $x,y,-x,-y$ with equal probability.

$f(n)=1$ is cool. Is there a cool $f(n)$ such that $\lim\limits_{n\rightarrow\infty} f(n)=\infty$?

If $f(n)$ eventually dominates $n!$, then $f(n)$ is uncool; is there a sharpest upper bound on the growth rate for $f(n)$ to be cool?

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    If you take $f(n)$ growing sufficiently slowly, then the expected number of returns near $0$ is infinite, and (again making a bunch of unsubtantiated independence assumptions), you could hope that $f(n)$ is cool.2012-02-24

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