[Paper Review] Cutoff for the Transposition Walk on Permutations with One-Sided Restrictions

Olena Blumberg|arXiv (Cornell University)|Feb 22, 2012

semigroups and automata theory6 references3 citations

TL;DR

This paper establishes chi-squared cutoff for the random transposition walk on permutations with one-sided interval restrictions using spectral analysis via Hanlon's diagonalization. It proves that under certain conditions on the restriction matrix, the walk exhibits sharp mixing in a window much smaller than the mixing time, resolving a conjecture by Diaconis and Hanlon on the discrepancy between chi-squared and total variation mixing behavior.

ABSTRACT

This paper explores the mixing time of the random transposition walk on permutations with one-sided interval restrictions. In particular, we're interested in the notion of cutoff, a phenomenon which occurs when mixing occurs in a window of order smaller than the mixing time. One of the main tools of the paper is the diagonalization obtained by Hanlon; the use of the spectral information is inspired by the famous paper of Diaconis and Shahshahani on the mixing time of the random transposition walk on the entire symmetric group. The diagonalization allows us to prove chi-squared cutoff for a broad class of one-sided restriction matrices. Furthermore, under an extra condition, the walk also undergoes total variation cutoff. Finally, a large collection of examples which undergo chi-squared cutoff but in which total variation mixing occurs substantially earlier and without cutoff is produced. These results resolve a conjecture of Diaconis and Hanlon from Section 5 of Hanlon's paper.

Motivation & Objective

• To analyze the mixing time of the random transposition walk on permutations with one-sided interval restrictions.
• To determine under what conditions the walk exhibits cutoff, particularly distinguishing between chi-squared and total variation cutoff.
• To resolve a conjecture by Diaconis and Hanlon regarding the discrepancy between chi-squared and total variation mixing times in restricted permutation walks.
• To extend spectral techniques from the symmetric group to restricted permutation structures using Hanlon’s diagonalization.
• To characterize the conditions under which total variation mixing occurs earlier and without cutoff, despite chi-squared cutoff.

Proposed method

• Utilizes Hanlon’s diagonalization of the transition matrix to obtain exact eigenvalues and eigenvectors for one-sided restriction matrices.
• Applies spectral techniques inspired by Diaconis and Shahshahani’s work on the symmetric group to bound the chi-squared distance.
• Employs the chi-squared distance as a proxy for total variation, leveraging its amenability to spectral analysis.
• Imposes a two-step restriction structure (each S(i) is either [1,n] or [a,n]) to ensure vertex transitivity, enabling analysis using only eigenvalues.
• Derives precise asymptotic bounds on the mixing time by analyzing the behavior of the largest eigenvalues as n → ∞.
• Uses combinatorial arguments to compare chi-squared and total variation cutoff, particularly under varying f(n) and g(n) growth conditions.

Experimental results

Research questions

• RQ1 Under what conditions on the restriction matrix does the random transposition walk on one-sided interval-restricted permutations exhibit chi-squared cutoff?
• RQ2 When does total variation mixing occur substantially earlier than chi-squared mixing, and under what conditions does it fail to exhibit cutoff?
• RQ3 What is the precise relationship between the growth rates of f(n) and g(n) in two-step restriction matrices and the occurrence of cutoff in total variation?
• RQ4 Can the spectral framework used for two-step matrices be extended to a broader class of one-sided restriction matrices?
• RQ5 What structural properties of the restriction matrix ensure that chi-squared and total variation mixing times coincide?

Key findings

• For a broad class of two-step one-sided restriction matrices, the random transposition walk exhibits chi-squared cutoff under the conditions that f(n) → ∞ and f(n)/n → 0 as n → ∞.
• Total variation cutoff occurs if and only if f(n) and g(n) are commeasurable in the limit, i.e., lim f(n)/g(n) exists and is finite.
• When g(n) = 1, total variation mixing occurs significantly earlier than chi-squared mixing and does not exhibit cutoff, even though chi-squared cutoff holds.
• The condition f(n)/n → 0 is primarily a technical assumption to simplify calculations, but is necessary for the dominant eigenvalue to be driven by the first f(n) rows.
• The eigenvalue analysis confirms that f(n) governs the mixing time, and if f(n) + g(n) > n, the mixing is instead driven by the last g(n) columns, invalidating the argument.
• The paper resolves a conjecture by Diaconis and Hanlon by constructing explicit examples where chi-squared cutoff occurs but total variation mixing lacks cutoff and happens much earlier.