The Poisson Tail Conjecture for Primes in Short Intervals

In 1976, Gallagher showed that, conditional on the Hardy--Littlewood conjectures, the number of primes below $x$ in a randomly chosen short interval of length $\lambda \log x$ asymptotically follows a Poisson distribution with mean $\lambda$. Correspondingly, the normalized gaps between consecutive primes follow an exponential distribution, provided that the scaling parameter $\lambda$ is fixed. We investigate the validity and limitations of the associated folklore Poisson Tail Conjecture as $\lambda$ is allowed to grow. For \edit{slowly growing} $\lambda$, and conditional on a strong variant of the Hardy--Littlewood conjectures, we establish asymptotics demonstrating that the local counting statistics rigorously align with these predictions. Furthermore, we identify a phase transition and explore the breakdown of these distributions for larger $\lambda$, capturing the precise deviations when $\lambda$ grows slower than any fixed power of $\log x$. The proof relies on a novel combination of extremal interval sieve estimates and concentration inequalities from probability.