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Uniqueness of the second eigenspace of the interchange process

Dennis Belotserkovskiy, Joe P. Chen

TL;DR

This work provides a complete characterization of the 2nd eigenspace of the interchange process on finite connected weighted graphs, proving that it is uniquely determined by the 2nd eigenvectors of the random-walk Laplacian on all graphs except the 4-cycle with uniform weights. The authors develop and deploy an octopus induction scheme within the representation theory of the symmetric group, using Specht modules and Schur reductions to compare eigenvalues across vertex removals via the Schur complement. They establish precise equalities and inequalities that govern when nonstandard irreps can contribute to the 2nd eigenvalue, and they prove a detailed kernel analysis of the octopus operator, including general and special-case descriptions of ker Δ and ker C'. A byproduct is a clear accounting of how often the 2nd eigenvalue of the exclusion process matches the random-walk 2nd eigenvalue, tying multiplicities to Kostka numbers and Specht-module decompositions. The results illuminate the algebraic structure underlying the interchange and exclusion processes and solidify the exceptional status of the 4-cycle in this spectral-geometry context.

Abstract

The spectral gap theorem of Caputo, Liggett, and Richthammer states that on any connected graph equipped with edge weights, the 2nd eigenvalue of the interchange process equals the 2nd eigenvalue of the random walk process. In this work we characterize the 2nd eigenspace of the interchange process. We prove that this eigenspace is uniquely determined by the 2nd eigenvectors of the random walk process on every connected weighted graph except the $4$-cycle with uniform edge weights. The key to our proof is an induction scheme on the number of vertices, and involves the octopus (in)equality, representation theoretic computations, and graph Laplacian computations.

Uniqueness of the second eigenspace of the interchange process

TL;DR

This work provides a complete characterization of the 2nd eigenspace of the interchange process on finite connected weighted graphs, proving that it is uniquely determined by the 2nd eigenvectors of the random-walk Laplacian on all graphs except the 4-cycle with uniform weights. The authors develop and deploy an octopus induction scheme within the representation theory of the symmetric group, using Specht modules and Schur reductions to compare eigenvalues across vertex removals via the Schur complement. They establish precise equalities and inequalities that govern when nonstandard irreps can contribute to the 2nd eigenvalue, and they prove a detailed kernel analysis of the octopus operator, including general and special-case descriptions of ker Δ and ker C'. A byproduct is a clear accounting of how often the 2nd eigenvalue of the exclusion process matches the random-walk 2nd eigenvalue, tying multiplicities to Kostka numbers and Specht-module decompositions. The results illuminate the algebraic structure underlying the interchange and exclusion processes and solidify the exceptional status of the 4-cycle in this spectral-geometry context.

Abstract

The spectral gap theorem of Caputo, Liggett, and Richthammer states that on any connected graph equipped with edge weights, the 2nd eigenvalue of the interchange process equals the 2nd eigenvalue of the random walk process. In this work we characterize the 2nd eigenspace of the interchange process. We prove that this eigenspace is uniquely determined by the 2nd eigenvectors of the random walk process on every connected weighted graph except the -cycle with uniform edge weights. The key to our proof is an induction scheme on the number of vertices, and involves the octopus (in)equality, representation theoretic computations, and graph Laplacian computations.

Paper Structure

This paper contains 31 sections, 42 theorems, 154 equations, 2 tables.

Table of Contents

  1. Introduction
  2. A routine exercise and a surprise finding
  3. Some previously known eigenspace results
  4. The spectral gap theorem revisited
  5. Setup and main results
  6. Operators on the symmetric group algebra
  7. Symmetric group and Young tableaux
  8. Permutation modules and Specht modules
  9. The branching rule for restricted and induced representations
  10. Main results
  11. The octopus induction scheme
  12. Equality in \ref{['ineq:octo1']}
  13. Equality in \ref{['ineq:octo2']}
  14. Equality in \ref{['ineq:Schur']}
  15. Proof of the main thm:main
  16. ...and 16 more sections

Key Result

Proposition 1.1

For $1\leq k \leq \lfloor \frac{n}{2}\rfloor-1$ and $u\in \mathbb{R}^{G\choose k}$, define $u^\uparrow \in \mathbb{R}^{G\choose {k+1}}$ by If $A^{(k)} u = \lambda u$ for some eigenvalue $\lambda \in \mathbb{R}$, then $A^{(k+1)} u^\uparrow = \lambda u^\uparrow$.

Theorems & Definitions (87)

  • Proposition 1.1
  • proof
  • Example 1.2
  • Proposition 2.1: Sagan*Theorem 2.4.6 and Corollary 2.4.7
  • Proposition 2.2: Sagan*Theorem 2.5.2
  • Proposition 2.3: Branching rule Sagan*Theorem 2.8.3
  • Theorem 2.4: Uniqueness of the 2nd eigenspace of $L_G$
  • Corollary 2.5
  • proof
  • Example 2.6: Multiplicity of the 2nd eigenvalue in the exclusion process
  • ...and 77 more