Open Research Index

Microscopic boundary, prefixes and longest runs

Brice Pouly

Palomar Registrypalomar/PALOMAR-2026-09-20-000011Sep 20, 2026math.NTscore 0

The exact left-border probability is 2^(-pi(L)), and the microscopic non-vacancy probability satisfies 0 <= q_L/2^(-pi(L)) - 1 <= 2^(-pi(L)/256) for sufficiently large L. The results give conditional localization and uniqueness, stability under changes of mesoscopic cutoff, and a quantitative Poisson law for runs contained in a finite prefix. They yield the critical longest-run distribution and asymmetric almost-sure envelopes for all integer prefix lengths. The exact border identity needs no literature hypothesis. The asymptotic results use, as applicable, scalar Stein bounds, the prime…

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