Asymptotic distance between $x^2+1$ primes?

As I recall, the most common difference between consecutive primes starts with $2$, moves on to $6$, then $30$, and is conjectured to progress like the primorials over a long time, without bound.

Is there any analogous prediction about $x^2+1$ or other polynomial prime gaps?

Those things are conjectured by the random model for the primes, the model on which every conjecture about primes is based. This model is saying that when $n$ is picked uniformly in $[1,N]$, for $p\le N^r$ then $n\bmod p$ is uniformly distributed in $1,\ldots, p$ and it is independent from one $p$ to the other. Of course this model is wrong for a fixed $N$, what we assume is that it gets less wrong as $N\to \infty$, ultimately giving correct predictions.

• What is $r$ ? It is the constant $r\in [1/2,1)$ such that the probability that $n$ (picked uniformly in $[1,N]$) is prime is $$\prod_{p \le N^r} Pr(p\nmid n)=\prod_{p \le N^r} (1-p^{-1})$$

  This $r$ can be evaluated from Mertens' theorem and the PNT : it is $r=e^{-\gamma}$, the only constant such that $$\lim_{N \to \infty}\frac{\prod_{p \le N^r} (1-p^{-1})}{\pi(N)/N}=\lim_{N \to \infty}\log N\prod_{p \le N^r} (1-p^{-1})=1$$

• The probability that $n,n+2$ are both primes is $$\prod_{p \le N^r} Pr(p\nmid n,p\nmid n+2)=(1-2^{-1})\prod_{3\le p \le N^r} (1-2 p^{-1})$$

  The conjectured number of twin primes $\le N$ is $$C_2\frac{N}{\log^2 N}\sim N(1-2^{-1})\prod_{3\le p \le N^r} (1-2 p^{-1})$$ where $C_2$ is the twin prime constant $$C_2=\lim_{N\to \infty}(\log^2 N) (1-2^{-1})\prod_{3\le p \le N^r} (1-2 p^{-1})=\lim_{N\to \infty} \frac{(1-2^{-1})\prod_{3\le p \le N^r} (1-2 p^{-1})}{\prod_{ p \le N^r} (1- p^{-1})^2}$$

• The probability that $n^2+1$ is prime is $$\prod_{p\le N^{2r}} Pr(p\nmid n^2+1)= (1-2^{-1})\prod_{p\le N^{2r},p\equiv 1\bmod 4} (1-2 p^{-1})$$

  Thus the conjectured number of primes $n^2+1$ with $n\le N$ is $$A \frac{N}{\log N}$$ where the obtained constant is $$A = \lim_{N\to \infty}(\log N) (1-2^{-1})\prod_{p\le N^{2r},p\equiv 1\bmod 4} (1-2 p^{-1})$$ $$= \lim_{N\to \infty}\frac{(1-2^{-1})\prod_{p\le N^{2r},p\equiv 1\bmod 4} (1-2 p^{-1})}{\frac12 \prod_{p\le N^{2r}}(1-p^{-1})}$$

  And the conjectured mean distance between two consecutive such primes $n^2+1,n_2^2+1$ is $$A\log n$$