What does Cramer's model say?

Question

I know that Cramer's model is stated as follows:

"With a probability =1, the relation

$$\displaystyle \limsup_{n\to\infty}\frac{p_{n+1}-p_n}{(\log p_n)^2}=1$$ is satisfied"

Could someone explain to me in "softer" words what this means? I have a clear notation $\limsup$ and $\liminf$

For example, if I write this $\liminf_{n\to\infty} (p_{n+1}-p_n)=2$, I know you mean that there are infinitely many primes such that if difference is 2.

In the case of the cramer model, what does it say? I think I'm confused by the fact of the quotient or the text "With a probability =1"

I would appreciate an explanation, it is the first time that I approach this result and I would like it to be clear to me.

Answer 1

First, ignoring the "probability 1" comment, the equation:

$$\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{(\log p_n)^2} = 1$$

can (non-rigorously) be rewritten as $\limsup_{n\to\infty} p_{n+1}-p_n = (\log p_n)^2$. Strictly speaking, this makes no sense --- $n$ is not a free variable on the left-hand side (we are taking a limit over it), but on the right-hand side it is. Still, it can be useful to translate your intuition (which I will rewrite as)

for arbitrarily large $n$, there exists prime gaps of size at least 2.

to

for arbitrarily large $n$, all prime gaps (asymptotically) are of size of at most $(\log p_n)^2$.

So instead of claiming that there are some prime gaps that are incredibly small (an "existence" result that is a "lower bound"), it is stating that (in an asymptotic sense) prime gaps cannot get "too large" (or a "universal result" that is an "upper bound"). Note that the heuristic form $\limsup_{n\infty} (p_{n+1}-p_n) = (\log p_n)^2$ is stronger than having written $O((\log p_n)^2)$ on the RHS --- it is claiming to pin down a precise constant.

Now, what does the probability 1 comment mean? The primes $\mathcal{P} \subseteq \mathbb{N}$ are a fixed (deterministic) subset. Cramer's random model replaces $\mathcal{P}$ with a certain random subset of $\mathbb{N}$. Formally (this is from Cramer vs. Cramer. On Cramer's Probabilistic Model For Primes by Pintz) this can be defined as follows.

For $n\geq 3$, define the independent set of random variables $\xi(n)$, supported on $\{0,1\}$ where

$$\Pr[\xi(n) = 1]= \frac{1}{\log n},\qquad \Pr[\xi(n) = 0] = 1 - \frac{1}{\log n}$$

The collection of $(\xi(n))_{n\geq 3}$ can be seen as defining a random subset of $\mathbb{N}_{\geq 3}$, e.g. a "random model of the primes" (we view this as being a random model of the primes as each $\xi(n)$ has the "right density" according to the prime number theorem). The statement

$$\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{(\log p_n)^2} = 1$$

then holds with probability $1$ over the choice of random model of the primes $(\xi(n))_{n\geq 3}$, and more properly can be written as

$$\Pr_{(\xi(n))_{n\geq 3}}\left[\limsup_{n\to\infty}\frac{\xi(n+1)-\xi(n)}{(\log \xi(n))^2}=1\right]=1$$