Covariance-dependent participation scaling of Comtet disorder on open hyperbolic lattices

Abstract

Negative curvature makes the electrostatic Green function decay exponentially, producing a finite disorder-variance budget even as radial volume grows exponentially. We investigate whether this near-field concentration also determines wave-function participation on open hyperbolic lattices. A Gaussian control preserves the full one-site distribution of Comtet disorder at every vertex while removing its off-diagonal covariance. Complete-window participation ratios on $\{7,3\}$ graphs with $N=847,2240,5887$ yield an effective-exponent difference of $-0.028$ at $W/t=0.5$ and $+0.116$ at $W/t=3$; both signs persist upon extending to $N=15435$. Size-resolved squared-kernel radii lie near $2.45\ell$ on the larger graphs of this sequence, below the cutoff-matched continuum value $3.12\ell$. Distance-shell weights resolve the finite discrete measure associated with this shift, while higher participation moments and spacing ratios characterize the accompanying wave-function and spectral structure. A marginal-matched exponential-kernel family gives positive exponent contrasts at all five sampled kernel lengths at $W/t=0.5,3$, separating the Comtet response from a description based on the decay length alone. These results distinguish a concentrated variance budget from the covariance-dependent participation response of finite open hyperbolic systems.