Single-graph inference for fractal Gaussian networks

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 $\gamma =1$, whereas degree quantiles consistently estimate $\nu ={\gamma }^2\mathrm{/}2$ 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 $0\le \gamma \le \mathrm{\Gamma }<2$. 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.