Critical Poisson fields for long constant stretches of a random completely multiplicative function

Source abstract

Let f be a Rademacher random completely multiplicative function. We study the left endpoints of long constant stretches. A refined weighted two-window relation estimate, combined with conditioning on the small primes, yields total-variation Poisson approximation in a dyadic block at the critical scale. The same lattice comparison controls positions, exact excesses, the threshold staircase, and a discrete Poisson–Gaussian bridge. On a finite prefix, the exact left-border event has probability q_L = 2^(-π(L))(1 + O(2^(-π(L)/256))). Combining this estimate with mesoscopic negligibility and a relative bulk lattice-process approximation yields an unconditional microscopic–bulk crossover for the first contained exceedance. An accompanying technical companion supplies the detailed arithmetic, rank, threshold, and relative-error verifications.