Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains
Abstract
In a slowly varying aperiodic potential hi=hcos(2πβin+ϕ) with random power-law density interactions Vij/|i−j|α, the resonant-object ensemble changes across α=2n, wherever Hartree fragmentation is operative, from isolated two-level bonds to a mixture of fragmented multi-site clusters and isolated wing bonds that survive at finite density, while the pair-resonance scaling xpair4−2n−αln xpair∝ h2/(Vt0) is unchanged~\cite{letter}. Here we develop the microscopic resonance theory underlying these results, together with its domain of validity. We derive the exact phase-averaged resonance statistics --- the supply N0∝ L2−n/h, the correlated common-phase comb, and the closed-form Hartree variance --- proving that the interaction leaves the leading supply law unchanged. Exact construction of the resonant Fock-space graph at L≤18 shows that the order-one forward-branching scale hFB∝ L2−n carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged qln(1/q) law of the bond ensemble and develop the comb into a mesoscopic shell theory; we map the fragmentation domain, with its support threshold V*(α) and the boundary-healing recursion; and we treat the marginal case α=2, where shell and matching marginalities compound into a double logarithm. Three long-range thresholds emerge with distinct meanings --- α=1/2 (the exact variance threshold of the random Fock-space energy and the square-summability boundary of the leading LIOM-dressing estimate), α=2n (change of the local resonant objects), and α=2 (marginality of the long-range shell sum) --- none of which is, by itself, a localization transition.
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BibTeX
@misc{prasad2026resonance,
title = {{Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains}},
author = {Yogeshwar Prasad},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30761},
eprint = {2608.30761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30761},
}
RIS
TY - UNPB
TI - Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains
AU - Yogeshwar Prasad
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30761
UR - https://arxiv.org/abs/2608.30761
AB - In a slowly varying aperiodic potential h_i=hcos(2πβi^n+ϕ) with random power-law density interactions V_ij/|i−j|^α, the resonant-object ensemble changes across α=2n, wherever Hartree fragmentation is operative, from isolated two-level bonds to a mixture of fragmented multi-site clusters and isolated wing bonds that survive at finite density, while the pair-resonance scaling x_pair^4−2n−αln x_pair∝ h^2/(Vt_0) is unchanged~\cite{letter}. Here we develop the microscopic resonance theory underlying these results, together with its domain of validity. We derive the exact phase-averaged resonance statistics --- the supply N_0∝ L^2−n/h, the correlated common-phase comb, and the closed-form Hartree variance --- proving that the interaction leaves the leading supply law unchanged. Exact construction of the resonant Fock-space graph at L≤18 shows that the order-one forward-branching scale h_FB∝ L^2−n carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged qln(1/q) law of the bond ensemble and develop the comb into a mesoscopic shell theory; we map the fragmentation domain, with its support threshold V_*(α) and the boundary-healing recursion; and we treat the marginal case α=2, where shell and matching marginalities compound into a double logarithm. Three long-range thresholds emerge with distinct meanings --- α=1/2 (the exact variance threshold of the random Fock-space energy and the square-summability boundary of the leading LIOM-dressing estimate), α=2n (change of the local resonant objects), and α=2 (marginality of the long-range shell sum) --- none of which is, by itself, a localization transition.
ER -