Single-graph inference for fractal Gaussian networks

Source abstract

We study inference on the strength of Gaussian multiplicative chaos from one geometric graph with unobserved vertex positions. In the planar model, the edge-count statistic has a piecewise deterministic limit with a transition at $γ=1$, whereas degree quantiles consistently estimate $ν=γ^2/2$ throughout the subcritical range. Fractional moments give uniform finite-sample risk bounds. At finite resolution, common graph envelopes calibrate tests over continuous parameter boxes with unknown Poisson intensity. They give simultaneous coverage under adaptive refinement for a fixed statistic, and conditional coverage for an independently piloted size--degree residual on a fixed partition. For the exact periodic FFT--pixel approximation, we prove explicit total-variation rates for the annealed spatial Cox law and the observed graph law, uniformly on every compact subcritical range $0\leγ\leΓ<2$. The rates allow intensity to grow with resolution and imply uniform asymptotic coverage after recalibration at each resolution. Numerical studies examine parameter refinement, residual calibration, and multiresolution stability.