Numerical experiments show the distribution of prime gaps conforms to some quite firm constraints.
The following plot visualises these constraints - it shows the log of the count of prime gaps against gaps.
Question: I can't find much information on how this distribution is explained by properties of primes. Is this because it is genuinely an area of mathematics that hasn't been explored, or yielded results?
I have looked through many of the standard texts (Stopple, Edwards, Apostol, Derbyshire,...) and done internet searches for university course content (indicating the study of prime gaps is well established) - but I have found very little, or nothing. The content I can find is a few blogs or YouTube videos which touch on this question but don't explain it.
I am not university trained in mathematics so I may be unaware of the state of the art, so apologies if the answer is (1) obvious, or (2) well known as an open question in mathematics.
Note: Is this question more difficult because it is essentially about the additive properties of primes rather than the multiplicative properties.
(As usual for open problems about primes) the first answer is the random model for the primes.
The simplest model for the gap $g(n)=p_{n+1}-p_n$ is that it follows approximately an exponential distribution of parameter $1-\frac2{\log n}$, following from that the probability that $p_n+2j$ is not prime is about $1-\frac2{\log (p_n+2j)}\approx 1-\frac2{\log n}$ so that
$$Pr(g(n)=2k)\approx (1-\frac2{\log n})^{k-1} \frac2{\log n}$$ And thus your curve is not surprising at all:
when looking at the gaps for $n\le N$ then most $\log n$ are $\approx \log N$ so you expect about $(1-\frac2{\log N})^{k-1} \frac{2N}{\log N} $ gaps of size $2k$ ie. a $\log$ proportion $\approx (k-1) \log (1-\frac2{\log N})-\log\log N$