In this paper, while generating a random data set (on origin spacing), it is written
"All we require is to define an average separation between adjacent origins and the existence of an associated standard deviation (i.e. we assume the distribution has no power law tail)"
What does this mean, exactly?
As mentioned by @Qiaochu Yuan, this is imprecise. However, as explained in the Wikipedia page on power law distributions, if random quantity $X$ has a "power law distribution", then its distribution (density or mass function) has the form
$$P(X>x) \propto L(x)x^{-(\alpha-1)}\, \text{ where } \;\;\alpha>1 \text { and }\;\lim_{x \to \infty} \frac{L(r\cdot x)}{L(x)} = 1\;\;\forall r> 0$$
The point of factoring out the $L(x)$ is that this is the "close-in" behavior (small $x$) and it is eventually dominated by the decay factor $x^{-(\alpha-1)}$ when $x$ is large (i.e., in the tails).
For example, in for a Pareto distribution (a type of power law distribution), if $\alpha \leq 1$ then $E[X]=V[X]=\infty$, which isn't a great model for genetic origin locations (it will tend to produce outlandishly large origin spacings every so often).
So they are just assuming the distribution of origin spacings is "well behaved" in terms of the mean and variance.