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

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. Writin