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Full Resolution to Papikian's Conjecture

Paul Vollrath

TL;DR

The paper resolves Papikian's conjecture on the spectrum of the signed up-down walk on the spherical building by modeling the walk as a weighted graph on $i$-simplices and passing to a quotient $X^i/H$ under the automorphism group stabilizer. A height-profile quotient identifies the vertex set with $i$-flags of subsets of $[n]$, proving the number of distinct eigenvalues is independent of the thickness $q+1$. In the $q\to\infty$ limit, the adjacency structure becomes a sparse matrix $D_i$ that decomposes into blocks corresponding to characteristic flags, each block matching a signed up-down walk on a complete complex of reduced dimension. Consequently, the asymptotic spectrum of the walk is the union of block spectra, yielding the limiting positive eigenvalues $n-1, n-2, \dots, n-i$ (together with $0$). This approach provides a general, automorphism-driven framework for spectral analysis of spherical buildings and their high-dimensional expander properties.

Abstract

We prove Papikian's conjecture on the spectrum of the signed up-down walk on the spherical building. Namely, we show that in the spherical building of dimension n-2 and thickness q + 1, the number of distinct eigenvalues is independent of q and for q going to infinity the positive eigenvalues converge to n-1, ... , n-i.

Full Resolution to Papikian's Conjecture

TL;DR

The paper resolves Papikian's conjecture on the spectrum of the signed up-down walk on the spherical building by modeling the walk as a weighted graph on -simplices and passing to a quotient under the automorphism group stabilizer. A height-profile quotient identifies the vertex set with -flags of subsets of , proving the number of distinct eigenvalues is independent of the thickness . In the limit, the adjacency structure becomes a sparse matrix that decomposes into blocks corresponding to characteristic flags, each block matching a signed up-down walk on a complete complex of reduced dimension. Consequently, the asymptotic spectrum of the walk is the union of block spectra, yielding the limiting positive eigenvalues (together with ). This approach provides a general, automorphism-driven framework for spectral analysis of spherical buildings and their high-dimensional expander properties.

Abstract

We prove Papikian's conjecture on the spectrum of the signed up-down walk on the spherical building. Namely, we show that in the spherical building of dimension n-2 and thickness q + 1, the number of distinct eigenvalues is independent of q and for q going to infinity the positive eigenvalues converge to n-1, ... , n-i.
Paper Structure (15 sections, 35 theorems, 53 equations)

This paper contains 15 sections, 35 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Basics
  3. Simplicial Complexes & the Signed Up-Down Walk
  4. The Spherical Building
  5. Invariance of $\Delta_i^+$ under Automorphisms
  6. Weighted Graphs and Their Quotients
  7. The Edge Weights of $X^i$
  8. The Proof
  9. The Quotient
  10. The Limit
  11. The Block Decomposition
  12. The Spectrum of a Simplex
  13. The Asymptotic Eigenvalues
  14. Appendix
  15. Counting Subspaces

Key Result

Theorem 1.1

For the signed up down walk on an $n$-dimensional building of type $A$ at level $i$ the number of distinct eigenvalues is bounded by the expression which is independent of the thickness $q+1$. Further, when fixing the dimension $n$ and for the thickness going to infinity, the non-zero eigenvalues of the signed up-down walk approach $n+1,\dots, n+1-i$ at level $i$.

Theorems & Definitions (77)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 67 more