Target-Mass Borel–Cantelli Laws for IID Random LSV Maps

Abstract

We prove an annealed target-mass Borel–Cantelli theorem for the two-map IID random LSV model with a common neutral fixed point. Averaging first-return operators yields a deterministic renewal convolution and a physical-time limsup law for every deterministic interval sequence with divergent Lebesgue-length sum. A target-mass covariance estimate gives a quantitative strong law on the return base. Under a fibrewise cone-coupling hypothesis and a finite regularization-time condition, this transfers to physical-time quenched strong laws for regularizing targets, including complete environment-adapted neutral shells and deterministic neutral annuli. In the stationary two-map model, we compute the quenched endpoint-density asymptotic, the neutral residence-mark law, and the laminar persistence kernel; these identify the alpha=1/2 crossover and show why invariant-mass divergence alone cannot yield a neutral-endpoint strong law. We also prove an annealed Poisson law for expanding-branch entrances at scale epsilon^{-1} and a uniform atom-aligned quenched compensator-normalized power-law mark kernel. The full moving-endpoint marked or strong-law extension remains open because it requires a partial-boundary weighted-renewal estimate.