math.PR — Probability

Records, scale flows, and Gaussian limits from lattice Poisson fields

Contributed by Brice Pouly

How does a rare-event field determine the history of its extremes? Starting from the complete signed run-field comparison in Long runs and rare patterns, we develop records, censoring, occupation and Gaussian limits for the resulting lattice Poisson target. The arithmetic input retains every site, full exact length and actual sign, at growing intensity under a one-factor information-intensity budget. Strict integer records, their signed imbalance and logarithmic leadership time have a joint phase-uniform Brownian limit with explicit covariance, independent of the terminal critical field and its extremal paths. Right censoring gives scalar $M_1$ convergence for the visible maximum and $J_1$ convergence for the first-holder sign. Physical occupation instead has a nondegenerate phase-indexed limit. Canonical interpolation connects threshold thinning and a stationary scale flow without identifying their record rules or their Gaussian constants. The signed spatial theory includes weighted exact levels, fixed-window Ornstein-Uhlenbeck paths, two correlated threshold-compression noises, and rectangle fields with explicit grid centering. Direct partition and quadratic-reconstruction proofs provide alternative routes. For each specified observation, the target calculations identify its limit and the dependencies within the joint constructions.