Cramer's random model leads us to expect that the
primes are distributed in a Poisson distribution around their
mean spacing. It is conjectured that, for any given positive
number $\lambda$ and nonnegative integer $m$, the proportion of
$n \le x$ for which $(n,n + \lambda \log n]$ contains exactly $m$
primes is asymptotically equal to $\lambda^me^{-\lambda}/m!$ as
$x \to \infty$. It is also conjectured that, for any given numbers
$b > a \ge 0$, the proportion of $n \le x$ for which
$d_n/\log n \in (a,b]$, where $d_n = p_{n+1} - p_n$ and $p_n$ is
the $n$th smallest prime, is asymptotically equal to
$\int_a^b e^{-t} dt$ as $x \to \infty$.
By combining an Erdos--Rankin type construction, which
produces large gaps between consecutive primes, with the
Maynard--Tao breakthrough on short gaps between primes, we are
able to show that the number of $n \le x$ for which
$\pi(n + \lambda\log n) - \pi(n) = m$ is at least $x^{1 - o(1)}$.
We are also able to show that at least $25\%$ of nonnegative
real numbers are limit points of the sequence $(d_n/\log n)$ of
normalized level spacings in the primes. We discuss an extension of
this limit point result, which utilizes the work of Ford, Green,
Konyagin, Maynard and Tao on long gaps between consecutive
primes.