Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

David Kogan

TL;DR

The paper develops a discrete-harmonic framework to analyze interior-face statistics of uniform random reduced $\mathfrak{sl}_3$ webs via Tymoczko’s $3\times n$ tableau bijection and constrained lattice paths. By translating crossing configurations to axis-hitting problems and solving Dirichlet problems with lattice Green's functions on triangular lattices and wedges, it derives explicit limiting face-type frequencies and shows bulk webs are predominantly hexagonal, with larger faces decaying as powers of depth $d$. The approach yields exact constants for key crossing events (e.g., $\frac{243\sqrt{3}}{40\pi}-3$ for a first arc closing without further crossings) and provides a general method to compute face distributions from local crossing data. These results illuminate the typical geometry of large random reduced $\mathfrak{sl}_3$ webs and establish precise asymptotics for interior-face sizes in the bulk. The techniques open avenues for applying discrete potential theory to combinatorial models arising from representation theory and planar graphs.

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 $O(d^{-2k})$.

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

TL;DR

The paper develops a discrete-harmonic framework to analyze interior-face statistics of uniform random reduced webs via Tymoczko’s tableau bijection and constrained lattice paths. By translating crossing configurations to axis-hitting problems and solving Dirichlet problems with lattice Green's functions on triangular lattices and wedges, it derives explicit limiting face-type frequencies and shows bulk webs are predominantly hexagonal, with larger faces decaying as powers of depth . The approach yields exact constants for key crossing events (e.g., for a first arc closing without further crossings) and provides a general method to compute face distributions from local crossing data. These results illuminate the typical geometry of large random reduced webs and establish precise asymptotics for interior-face sizes in the bulk. The techniques open avenues for applying discrete potential theory to combinatorial models arising from representation theory and planar graphs.

Abstract

We study the distribution of interior faces in uniformly random reduced webs. Using Tymoczko's bijection between standard Young tableaux and reduced webs, this problem can be reformulated in terms of constrained lattice paths and associated -diagrams. We develop a framework that expresses crossing probabilities in the -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 from the boundary. We prove that almost all interior faces far from the boundary are hexagons, while faces of size occur with probability .

Paper Structure

This paper contains 14 sections, 20 theorems, 81 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Interior Faces of sl3 webs and m-diagrams
  3. Tymoczko's Bijection
  4. Interior Faces in an m-diagram
  5. Crossing and Face Probabilities
  6. Bijection from subpaths and partial SYT to partial m-diagram
  7. Calculating crossing Probabilities
  8. Calculating face Probabilities
  9. Green Function
  10. Green's Function on the Full Lattice
  11. Green's Function in Wedge
  12. Point Mass boundary condition
  13. Full boundary condition
  14. Faces at Distance d from the Boundary

Key Result

Theorem 1.1

Let $d \to \infty$ with $n \to \infty$ and $d = o(n)$. For faces at distance at least $d$ from the boundary in a uniformly random reduced $\mathfrak s\mathfrak l_3$ web, we have:

Figures (16)

  • Figure 1: Zoomed $O(1)$ window of a reduced $\mathfrak{sl}_3$ web in the upper half–plane. The boundary lies on the real line. Moving rightward, i.i.d. boundary moves open and close strands, producing the local web configuration.
  • Figure 2: Uniform random reduced web with $n=800$ ($2400$ boundary vertices)
  • Figure 3: A $3\times3$ standard Young tableaux (a) and its $m$-diagram (b). Red arcs join second-row entries to first-row entries; blue arcs join third-row entries to second-row entries.
  • Figure 4: Example of the map from an $m$-diagram (a) to its associated reduced $\mathfrak{sl}_3$ web (b) for a $3\times 3$ tableau. Blue arcs join second-row entries to first-row entries; red arcs join third-row entries to second-row entries. Replacing middle vertices with trivalent Y’s, orienting edges, and resolving crossings yields the reduced web. Arrows denote orientations.
  • Figure 5: Interior faces of size $4$ in a $m$-diagram (corresponding to $6$ in the reduced web). Black vertices $p, q$ denote the leftmost and rightmost crossings, which expand to green edges in the web. There are either one or two arc segments above the face.
  • ...and 11 more figures

Theorems & Definitions (52)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: SYT to $m$-diagrams
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Theorem 3.1
  • Remark 1
  • ...and 42 more