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Given the standard two-dimensional random walk (up, down, left, or right 1 unit with equal probability), what is the expected number of crossings of the origin after $x$ steps?

It strikes me as slightly unnatural to count each crossing separately (since there will generally be long periods without any crossings followed by short bursts of activity as the location moves near the origin). I'd be happy to accept an answer with a better measurement, whatever that might be.

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up vote 2 down vote accepted

Not very many.

The probability of the walk being at the origin after $2n$ steps is $\dfrac{{2n \choose n}^2}{16^n}$ so the expected number of returns after $x$ steps is $$\sum_{n=1}^{\lfloor x/2 \rfloor} \dfrac{{2n \choose n}^2}{16^n}.$$

This sum grows slowly: it almost reaches $1$ after $36$ steps, and for large $x$ I would expect it to be roughly proportional to $\log(x)$.

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