Abstract:We study endpoint distributions of balanced affine weighted Motzkin paths. In the balanced case, the generating-function equation has Pearson-type characteristic geometry. We show that this geometry controls the terminal-height law globally: the characteristic escape time determines the limiting cumulant generating function, the large-deviation rate function, and the ray-scale asymptotics. Thus the usual Gaussian window is only the local quadratic approximation to a global Pearson-driven profile. For finite sizes, we prove a uniform Daniels saddlepoint approximation in the one-dominant-singularity regimes and identify the exceptional antipodal case requiring a lattice/interference correction.
From: Alexander Omelchenko [view email]
[v1]
Sat, 24 Jan 2026 23:55:18 UTC (157 KB)
[v2]
Sat, 13 Jun 2026 17:26:51 UTC (113 KB)
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