Is it possible to find a 2-way infinite (self-avoiding) path $\{x_i\}_{i\in \mathbb Z}$ in the standard Cayley graph of $\mathbb Z^2$, i.e. the square grid, such that the distance between $x_i$ and $x_{i+n}$ is of order $o(n)$? If yes, how small can this distance be? Here I'm asking for upper bounds $f(n)$ that are independent of $i$. Let me make this more precise:

**Is there a 2-way infinite (self-avoiding) path $\{x_i\}_{i\in \mathbb Z}$ in $\mathbb Z^2$, and a number M, such that for every i and every $n>M$, we have $d(x_i,x_{i+n}) < f(n)$ where $f(n)$ is $o(n)$?**

Here $d$ denotes the graph-distance on $\mathbb Z^2$.

If the answer is yes, I would like to know what is the smallest $f(n)$ for which this is possible. Easily, $f(n)= \Omega(\sqrt{n})$.