'Null distribution' is short for the sampling distribution of a statistic under the null hypothesis. 'Sampling distribution' you have to understand from the context: in the context you describe it also means the sampling distribution of a statistic under the null hypothesis, but in another context it could refer to the sampling distribution of a statistic under an alternative hypothesis.
A Dirichlet distribution is often used to probabilistically categorize events among several categories. Suppose that weather events take a Dirichlet distribution. We might then think that tomorrow's weather has probability of being sunny equal to 0.25, probability of rain equal to 0.5, and probability of snow equal to 0.25. Collecting these values in a vector creates a vector of probabilities.
Another way to think about a Dirichlet distribution is the process of breaking a stick. Imagine a stick of unit length. Break that stick anywhere and retain one of the two pieces. Then break the remaining piece into two pieces and continue this as long as you desire. All of the pieces together must sum to unit length, and allocating pieces of different lengths to different events represents the probability of that event.
If you're familiar with the beta distribution, the Dirichlet distribution might become even more clear. A beta distribution is often used to describe a distribution of probabilities of dichotomous events, so its restricted to the unit interval. For example, for a Bernoulli trial, there is only a parameter $\theta$ describing the probability of a "success." Often we think of $\theta$ as being fixed, but if we are uncertain about the "true" value of $\theta$, we could think about a distribution of all possible $\theta$s, with a larger likelihood for those we consider more plausible, so perhaps $\theta \sim \text{B}(\alpha, \beta)$, where $\alpha>\beta$ concentrates more of the mass near 1 and $\beta > \alpha$ concentrates more of the mass near 0.
One might object that the beta distribution only describes the probability of a single probability, that is, for example, that $P(\theta<0.25)=0.5$, which is a scalar number. But keep in mind that the beta distribution is describing dichotomous outcomes. So by applying Kolmogorov's second axiom, we also know that $P(\theta \ge 0.25)=0.5$ as well. Collecting these results in a vector gives us a vector of probabilities.
Extending the beta distribution into three or more categories gives us the Dirichlet distribution; indeed, the PDF of the Dirichlet for two groups is the exact same as the beta distribution.
Best Answer
I believe it means that the log is uniformly distributed, and the variable takes values in the range $[128, 4000]$.
From a footnote of the paper:
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