Overlap distributions under the Gibbs measure of a Haar model of $\log|ζ|$ on and near the critical line

Abstract

We study the Gibbs measure of $H(h) = \Re\sum_{p^{j} \leq X}U_{p}^{j}/(j p^{j(σ+ i h)})$, a random field modeling large values of the log-modulus of the Riemann zeta function on and near the critical line, where the $U_{p}$ are independent and uniformly distributed on the complex unit circle. Writing $t = \log \log T$, the cutoff $X$ and real part $σ$ may vary with $T$ subject to $T^{1/t^Φ} \leq X \leq T^{1/t^ϕ}$ and $0 \leq σ- 1/2 \leq C_σt^Φ/\log T$, for fixed $Φ\geq ϕ> 1/2$ and $C_σ > 0$. On $I_θ = [ - (\log T)^θ,(\log T)^θ]$, define the Gibbs measure $G_{β,σ,T}^{\mathrm{H}}(\mathrm{d}h) := e^{βH(h)}\mathrm{d}h/\int_{I_θ}e^{βH(h')}\mathrm{d}h'$. For every fixed $θ> - 1$ and inverse temperature $β> 0$, the limiting joint distributions of correlations (overlaps) among points sampled from this measure are determined after averaging over the Haar phases. At and below $β_{c}(θ) = 2\sqrt{1 + (θ\vee 0)}$, the limiting two-overlap distribution is concentrated at $q_{0} = |θ\wedge 0|$. Above this threshold, it assigns masses $β_{c}(θ)/β$ and $1 - β_{c}(θ)/β$ to $q_{0}$ and $1$, respectively, and the limiting joint overlap distribution is described by a Poisson-Dirichlet process of parameter $β_{c}(θ)/β$. Approximate Ghirlanda-Guerra identities are shown to hold uniformly for bounded deterministic functions of the sampled positions. This paper is the first in a series devoted to extending these results to $\log|ζ|$ near the critical line.