Asymptotic Face Distributions in Random Reduced $\mathfrak s\mathfrak l_3$ Webs

Abstract

We study the distribution of interior faces in uniformly random reduced $\mathfrak s \mathfrak l_3$ webs. Using Tymoczko's bijection between $3\times n$ standard Young tableaux and reduced webs, this problem can be reformulated in terms of constrained lattice paths and associated $m$-diagrams. We develop a framework that expresses crossing probabilities in the $m$-diagram as solutions to discrete Dirichlet problems on the triangular lattice, which are evaluated through solutions to lattice Green's functions. From this we obtain explicit limiting formulas for the frequencies of interior faces of each type. As an application, we analyze faces at a distance at least $d$ from the boundary. We prove that almost all interior faces far from the boundary are hexagons, while faces of size $6+2k$ occur with probability $O(d^{-2k})$.