Poisson-Dirichlet statistics for the log-modulus of the Riemann zeta function on expanding intervals near the critical line

Source abstract

For $T > 0$ large, let $τ$ be uniformly distributed on $[T,2T]$ and let $σ= σ(T)$ be a deterministic real part approaching $1/2$ from the right. For the Riemann zeta function $ζ$, we study the Gibbs measure with density proportional to $|ζ(σ+ \mathrm{i}τ+ \mathrm{i} h)|^β$ on $I_θ := [ - (\log{T})^θ,(\log{T})^θ]$, for every $θ> 0$ and $β> 0$. As $β$ increases, the Gibbs measure increasingly favors positions where $|ζ|$ is large, providing a way to study the spatial organization of extreme values. Under explicit conditions on the rate at which $σ$ approaches the critical line, a phase transition is established at the critical inverse temperature $β_{c}(θ) := 2\sqrt{1 + θ}$. The limiting distribution of the correlation (or overlap) between the field values at two positions sampled from the Gibbs measure is concentrated at $0$ when $β\leq β_{c}(θ)$, and assigns masses $β_{c}(θ)/β$ and $1 - β_{c}(θ)/β$ to $0$ and $1$, respectively, when $β> β_{c}(θ)$. Joint convergence of the overlaps is also established, with the limiting arrays described by Poisson-Dirichlet weights in the low-temperature regime. The Ghirlanda-Guerra identities, which provide recursive relations for the joint overlap distributions, are established in the limit, uniformly for bounded deterministic observables of the sampled positions. All our results are established independently of the Riemann hypothesis.