An interesting spectral gap problem, from Jim Fill

TL;DR

This paper discusses a spectral gap problem related to matrix properties, presenting unpublished notes from 2003, with recent progress on a related conjecture and influence on subsequent research.

Contribution

It compiles and analyzes early unpublished notes on a spectral gap problem, providing insights and progress on a conjecture, and highlighting its influence on later studies.

Findings

Progress on a conjecture related to the spectral gap problem

Analysis of the spectral structure of matrix K

Influence on subsequent research in the field

Abstract

At the request of Laszlo Babai, founder and an editor of the free online journal Theory of Computing (ToC), theoryofcomputing.org, in August, 2025, I am posting on the arXiv, essentially unedited and not updated, a combination of two closely related sets of unpublished notes from 2003. ToC is keen on publishing links to all bibliography items, and a paper soon to be published there makes progress on a conjecture in my 2003 notes. The sections "The problem", "Evidence in favor of the conjecture", "Facts about the spectral structure of the matrix $K$", and "Stronger conjectures" previously formed a document entitled "An interesting spectral gap problem, from Jim Fill". The sections "Introduction: Self-organizing lists" and "The move-ahead-$1$ (MA1) rule" formed a document entitled "Background on the gap problem". The two sets of notes have inspired some research, including (as one example, with no attempt here at a literature survey) a 2022 paper by Bhakta, Miracle, Randall, and Streib cited in this arXiv document.