[Paper Review] Representing Random Permutations as the Product of Two Involutions

Charles Burnette, Eric Schmutz|arXiv (Cornell University)|Jul 21, 2015

Stochastic processes and statistical mechanics20 references3 citations

TL;DR

This paper establishes that the number of ways to express a random permutation of $n$ elements as a product of two involutions, denoted $\mathbf{N}_n(\sigma)$, is asymptotically lognormal as $n \to \infty$. The proof hinges on showing that $\mathbf{N}_n(\sigma)$ is well-approximated by the product of cycle lengths $\mathbf{B}_n(\sigma)$, whose asymptotic lognormality was established by Erdřs and Turán, thereby transferring the lognormal limit law to $\mathbf{N}_n$.

ABSTRACT

An involution is a permutation that is its own inverse. Given a permutation $σ$ of $[n],$ let $\mathbf{N}_{n}(σ)$ denote the number of ways to write $σ$ as a product of two involutions of $[n].$ If we endow the symmetric groups $S_{n}$ with uniform probability measures, then the random variables ${\mathbf N}_{n}$ are asymptotically lognormal. The proof is based upon the observation that, for most permutations $σ$, $\mathbf{N}_{n}(σ)$ can be well approximated by $\mathbf{B}_{n}(σ),$ the product of the cycle lengths of $σ$. Asymptotic lognormality of $\mathbf{N}_{n}$ can therefore be deduced from Erdős and Turán's theorem that $\mathbf{B}_{n}$ is itself asymptotically lognormal.

Motivation & Objective

• To determine the asymptotic distribution of $\mathbf{N}_n(\sigma)$, the number of ways to write a random permutation $\sigma \in S_n$ as a product of two involutions.
• To resolve Lugo's conjecture that $\mathbf{N}_n$ is asymptotically lognormal under uniform random permutation measure.
• To establish a connection between $\mathbf{N}_n(\sigma)$ and the product of cycle lengths $\mathbf{B}_n(\sigma)$, showing that $\mathbf{N}_n(\sigma)$ is well-approximated by $\mathbf{B}_n(\sigma)$ for most permutations.
• To leverage known asymptotic lognormality of $\mathbf{B}_n(\sigma)$ to deduce the same limit law for $\mathbf{N}_n(\sigma)$.

Proposed method

• Uses the explicit formula $\mathbf{N}_n(\sigma) = \prod_{k=1}^n \sum_{j=0}^{\lfloor c_k/2 \rfloor} \frac{k^{c_k - j} c_k!}{2^j j! (c_k - 2j)!}$, where $c_k$ is the number of $k$-cycles in $\sigma$, to analyze the structure of factorizations.
• Establishes that $\mathbf{N}_n(\sigma) \geq \mathbf{B}_n(\sigma) = \prod_k k^{c_k}$, using a constructive factorization argument based on cycle decomposition.
• Applies probabilistic bounds on cycle counts: $\mathbb{P}_n(c_k \geq 2 \text{ for some } k \geq \xi) = O(1/\xi)$ and $\mathbb{P}_n(c_k \geq \xi \text{ for some } k \leq \xi) = O(\xi/e^\xi + \xi/n)$, showing that most permutations have few long cycles.
• Uses the bound $\mathbf{N}_n(\sigma) \leq (c \xi^\xi)^\xi \mathbf{B}_n(\sigma)$ with high probability for $\xi = \sqrt{\log n}$, which grows slower than $\sigma_n = \sqrt{\frac{1}{3}\log^3 n}$.
• Applies the continuity of the standard normal CDF $\Phi(x)$ and the known asymptotic lognormality of $\mathbf{B}_n(\sigma)$, with $\mu_n \sim \frac{1}{2}\log^2 n$ and $\sigma_n^2 \sim \frac{1}{3}\log^3 n$, to transfer the limit law to $\mathbf{N}_n(\sigma)$.
• Combines upper and lower tail bounds to show $\mathbb{P}_n(\log \mathbf{N}_n \leq \mu_n + x\sigma_n) \to \Phi(x)$, proving asymptotic lognormality.

Experimental results

Research questions

• RQ1 Is the number of ways to write a random permutation as a product of two involutions, $\mathbf{N}_n(\sigma)$, asymptotically lognormal?
• RQ2 How does $\mathbf{N}_n(\sigma)$ relate to the product of cycle lengths $\mathbf{B}_n(\sigma)$, and can this approximation be quantified in probability?
• RQ3 What is the typical order of magnitude of $\mathbf{N}_n(\sigma)$ for most permutations $\sigma$?
• RQ4 Why is the average $\mathbb{E}_n[\mathbf{N}_n]$ misleadingly large, and how does the distribution concentrate around $\exp(\frac{1}{2}\log^2 n)$?
• RQ5 Can the asymptotic lognormality of $\mathbf{B}_n(\sigma)$ be extended to $\mathbf{N}_n(\sigma)$ despite the more complex combinatorics of factorizations?

Key findings

• The random variable $\mathbf{N}_n$, counting the number of factorizations of a uniformly random permutation $\sigma \in S_n$ into two involutions, is asymptotically lognormal: $\mathbb{P}_n(\log \mathbf{N}_n \leq \mu_n + x\sigma_n) \to \Phi(x)$ as $n \to \infty$.
• For most permutations $\sigma$, $\mathbf{N}_n(\sigma)$ satisfies $e^{(\frac{1}{2}-\epsilon)\log^2 n} < \mathbf{N}_n(\sigma) < e^{(\frac{1}{2}+\epsilon)\log^2 n}$, indicating concentration on the log scale around $\exp(\frac{1}{2}\log^2 n)$.
• $\mathbf{N}_n(\sigma)$ is well-approximated by $\mathbf{B}_n(\sigma) = \prod_k k^{c_k}$, the product of cycle lengths of $\sigma$, with high probability when $n$ is large.
• The approximation error is controlled via bounds on cycle counts: $\mathbf{N}_n(\sigma) \leq (c \xi^\xi)^\xi \mathbf{B}_n(\sigma)$ with probability $1 - O(1/\xi + \xi/n + \xi/e^\xi)$ for $\xi = \sqrt{\log n}$.
• The asymptotic lognormality of $\mathbf{B}_n(\sigma)$, established by Erdřs and Turán with $\mu_n \sim \frac{1}{2}\log^2 n$ and $\sigma_n^2 \sim \frac{1}{3}\log^3 n$, implies the same limit law for $\mathbf{N}_n(\sigma)$.
• The maximum value of $\mathbf{N}_n(\sigma)$ is $|\mathcal{T}_n|$, attained only by the identity permutation, while the minimum is $n-1$, attained by permutations with a single $(n-1)$-cycle.