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) = ℜ∑pj ≤ XUpj/(j pj(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 Up 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 T1/tΦ ≤ X ≤ T1/tϕ and 0 ≤ σ− 1/2 ≤ CσtΦ/log T, for fixed Φ≥ ϕ> 1/2 and Cσ > 0. On Iθ = [ − (log T)θ,(log T)θ], define the Gibbs measure Gβ,σ,TH(dh) := eβH(h)dh/∫IθeβH(h')dh'. 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 βc(θ) = 2√(1 + (θ∨ 0)), the limiting two-overlap distribution is concentrated at q0 = |θ∧ 0|. Above this threshold, it assigns masses βc(θ)/β and 1 − βc(θ)/β to q0 and $1$, respectively, and the limiting joint overlap distribution is described by a Poisson-Dirichlet process of parameter βc(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.
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BibTeX
@misc{ouimet2026overlap,
title = {{Overlap distributions under the Gibbs measure of a Haar model of $\log|ζ|$ on and near the critical line}},
author = {Frédéric Ouimet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14901},
eprint = {2610.14901},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14901},
}
RIS
TY - UNPB
TI - Overlap distributions under the Gibbs measure of a Haar model of log|ζ| on and near the critical line
AU - Frédéric Ouimet
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14901
UR - https://arxiv.org/abs/2610.14901
AB - We study the Gibbs measure of H(h) = ℜ∑_p^j ≤ XU_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 ≤ X ≤ T^1/t^ϕ and 0 ≤ σ− 1/2 ≤ C_σt^Φ/log T, for fixed Φ≥ ϕ> 1/2 and C_σ > 0. On I_θ = [ − (log T)^θ,(log T)^θ], define the Gibbs measure G_β,σ,T^H(dh) := e^βH(h)dh/∫_I_θe^βH(h')dh'. 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√(1 + (θ∨ 0)), the limiting two-overlap distribution is concentrated at q_0 = |θ∧ 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.
ER -