Resonance statistics, Fock-space branching, and long-range pair networks in slowly varying interacting chains

Abstract

In a slowly varying aperiodic potential $h_i=h\cos(2πβi^n+ϕ)$ 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 $x_{\rm pair}^{4-2n-α}\ln x_{\rm pair}\propto h^2/(Vt_0)$ 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 $N_0\propto L^{2-n}/h$, 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 $L\le18$ shows that the order-one forward-branching scale $h_{\rm FB}\propto L^{2-n}$ carries no giant component: one-step branching and connectivity are inequivalent. We separate the fixed-pair matching law from the shell-averaged $q\ln(1/q)$ 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.