Limit profile for the transpose top-2 with random shuffle

TL;DR

This paper analyzes the limit profile of a lazy random walk on the alternating group generated by specific 3-cycles, using Fourier analysis and spectral methods, and compares it to a broader class of random walks.

Contribution

It determines the limit profile for the transpose top-2 with random shuffle and provides the complete spectrum of the alternating group graph, answering open questions.

Findings

Derived the limit profile via Fourier analysis comparison.

Obtained the complete spectrum of the alternating group graph.

Answered a previously open question about the spectrum.

Abstract

The transpose top-$2$ with random shuffle (J. Theoret. Probab., 2020) is a lazy random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(\star,n-1,n)$ and $(\star,n,n-1)$. We obtain the limit profile of this random walk by comparing it with the random walk on $A_n$ generated by all $3$-cycles. Our method employs a non-commutative Fourier analysis analogue of the comparison method introduced by Nestoridi (Electron. J. Probab., 2024). We also give the complete spectrum of the alternating group graph, thus answering a question of Huang and Huang (J. Algebraic Combin., 2019).