Limit profile for the transpose top-2 with random shuffle

TL;DR

This paper determines the limit profile for the transpose top-2 with random shuffle on the alternating group $A_n$ by comparing it to the random walk on $A_n$ generated by all $3$-cycles, using a non-commutative Fourier-analytic analogue of the comparison method. The main result shows the total-variation distance at time $(n-3/2)\,\log n + c n$ converges to the Poisson-driven profile $d_{TV}(\mathrm{Poi}(1+e^{-c}),\mathrm{Poi}(1))$ for all real $c$, and the authors also derive the complete spectrum of the alternating group graph. The work provides a detailed spectral decomposition via partitions of $n$, distinguishing self-conjugate and non-self-conjugate cases, and demonstrates the method in a non-commuting setting where standard comparison fails. These results give precise mixing-time behavior (cutoff and limit profile) for TT2R on $A_n$ and yield explicit spectral information for the associated interconnection networks.

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).

Figures (3)

• Figure 1: Young diagrams with $4$ boxes.
• Figure 2: Standard Young tableaux of shape $(3,1)$.
• Figure 3: Example of $\text{CPar}(4)$ and $\text{NCPar}(4)$.

Theorems & Definitions (28)

• Theorem 1.1
• Definition 2.1
• Remark 2.1
• Definition 2.2
• Theorem 2.1
• proof
• Remark 2.2
• Lemma 3.1
• Lemma 3.2
• proof