Moments of the Riemann zeta function on short intervals of the critical line

We show that as $Tto infty$, for all $tin [T,2T]$ outside of a set of measure $mathrm{o}(T)$, $$ int_{-(log T)^{theta}}^{(log T)^{theta}} |zeta(tfrac 12 + mathrm{i} t + mathrm{i} h)|^{beta} mathrm{d} h = (log T)^{f_{theta}(beta) + mathrm{o}(1)}, $$ for some explicit exponent $f_{theta}(beta)$, where $theta > -1$ and $beta > 0$. This proves an extended version of a conjecture of Fyodorov and Keating (2014). In particular, it shows that, for all $theta > -1$, the moments exhibit a phase transition at a critical exponent $beta_c(theta)$, below which $f_theta(beta)$ is quadratic and above which $f_theta(beta)$ is linear. The form of the exponent $f_theta$ also differs between mesoscopic intervals ($-1<theta<0$) and macroscopic intervals ($theta>0$), a phenomenon that stems from an approximate tree structure for the correlations of zeta. We also prove that, for all $tin [T,2T]$ outside a set of measure $mathrm{o}(T)$, $$ max_{|h| leq (log T)^{theta}} |zeta(tfrac{1}{2} + mathrm{i} t + mathrm{i} h)| = (log T)^{m(theta) + mathrm{o}(1)}, $$ for some explicit $m(theta)$. This generalizes earlier results of Najnudel (2018) and Arguin et al. (2019) for $theta = 0$. The proofs are unconditional, except for the upper bounds when $theta > 3$, where the Riemann hypothesis is assumed.