$s$ uniformly at random from the interval $[a,b]$. Randomly generate a loop $sy$ of size $s$ and count $(1-n)/s^5$ if the loop hits $n\geq 2$ objects, and $0$ otherwise.

Here $[a,b]$ is an interval big enough so that integrating over that interval does not differ noticeably from integrating over all positive reals.

Looking at that random process more closely, one notes that the information about the random loop we use is the number of objects it hits. We have to pay particular attention to short loops that are just long enough to barely touch multiple objects, for they give the largest contribution to the sum. One should note that it is not possible to attribute a useful physical meaning to absolute differences in loop scaling factors $s$: for a loop that hits (at least) two objects, the effect of changing $s$ to $s+0.1$ very much depends on what the magnitude of $s$ is. As relative changes of the scaling factor hence are more important than absolute changes, we much prefer a distribution, when sampling loops, that handles all orders of magnitude equally. In other words, we prefer a distribution where the logarithm of $s$ is uniformly distributed on $[\ln(a),\ln(b)]$.

When 