A central limit theorem for a generalization of the Ewens measure to random tuples of commuting permutations

1. Problem Statement and Context

The Core Problem:
The paper investigates the statistical behavior of the number of joint orbits, denoted as $\kappa_{\ell,n}(\sigma_1, \dots, \sigma_\ell)$, for a random ordered $\ell$-tuple of pairwise commuting permutations $(\sigma_1, \dots, \sigma_\ell)$ in the symmetric group $S_n$.

• Commuting Tuples: The set of interest is $C_{\ell,n} = \{(\sigma_1, \dots, \sigma_\ell) \in S_n^\ell \mid \sigma_i \sigma_j = \sigma_j \sigma_i\}$.
• Joint Orbits: The random variable $K_{\ell,n}$ counts the number of orbits of the subgroup generated by these permutations acting on the set $[n] = \{1, \dots, n\}$.
• Generalization of Ewens Measure: The authors introduce a probability measure $P_{\ell,n,x}$ on $C_{\ell,n}$ where a tuple is weighted by $x^{\kappa_{\ell,n}}$.
  • When $\ell=1$, this recovers the classic Ewens measure on $S_n$ (where the weight is $x^{\text{number of cycles}}$).
  • When $x=1$, this corresponds to the uniform distribution over commuting tuples.
  • For $\ell \ge 2$, the structure of joint orbits corresponds to "twisted discrete $\ell$-dimensional tori."

The Goal:
To prove a Central Limit Theorem (CLT) for $K_{\ell,n}$ as $n \to \infty$ for any fixed $\ell \ge 1$ and $x > 0$, determining the asymptotic mean, variance, and the Gaussian limiting distribution.

2. Methodology

The proof relies on analytic combinatorics and saddle point analysis, avoiding direct probabilistic coupling arguments (though the authors discuss the potential for such proofs).

A. Generating Functions

The authors utilize the ordinary generating function for the polynomials $H_{\ell,n}(x)$, which encode the distribution of $K_{\ell,n}$. Based on a formula by Bryan and Fulman, the generating function is an infinite product:
$$G_\ell(x, z) = \sum_{n=0}^\infty H_{\ell,n}(x) z^n = \prod_{\delta_1, \dots, \delta_{\ell-1}=1}^\infty \left(1 - z^{\delta_1 \cdots \delta_{\ell-1}}\right)^{-x \delta_1^{\delta_2-1} \cdots \delta_{\ell-2}^{\delta_{\ell-1}-1}}$$
This is a holomorphic function for $|z| < 1$.

B. Saddle Point Analysis

To extract the coefficients $H_{\ell,n}(x)$ (and subsequently the moments of $K_{\ell,n}$), the authors use Cauchy's integral formula:
$$H_{\ell,n}(x) = \frac{1}{2\pi i} \oint_{C(r)} z^{-n} G_\ell(x, z) \frac{dz}{z}$$
They optimize the radius of integration $r = e^{-t}$ (where $t > 0$) to minimize the modulus of the integrand. This leads to a "saddle point" equation determining $t_n$ such that $n = x Z^{[\ell]}_\ell(t_n)$, where $Z^{[\ell]}_\ell(t)$ is a specific multiple series related to the Riemann zeta function.

C. Asymptotic Analysis of $Z$-functions

A significant portion of the technical work involves analyzing the asymptotic behavior of the functions:
$$Z^{[\ell]}_{\alpha_1, \dots, \alpha_\ell}(t) = \sum_{\delta_1, \dots, \delta_\ell=1}^\infty \delta_1^{\alpha_1-1} \cdots \delta_\ell^{\alpha_\ell-1} e^{-\delta_1 \cdots \delta_\ell t}$$
as $t \to 0^+$. The authors establish that for specific parameters, these behave like $t^{-\alpha}$, involving products of Riemann zeta functions $\zeta(k)$. Specifically, they define a constant:
$$K_\ell = (\ell-1)! \zeta(2)\zeta(3)\cdots\zeta(\ell)$$

D. Decomposition of the Integral

The integral for the coefficient is split into two parts:

1. Major Arc ($|\theta| \le t_n$): Where the integrand is approximated by a Gaussian. The authors perform a Taylor expansion of the logarithm of the generating function around the saddle point.
2. Minor Arc ($|\theta| > t_n$): They prove that the contribution from this region decays exponentially fast and is negligible compared to the major arc.

E. Moment Generating Functions

Using the Curtiss continuity theorem, the authors analyze the moment generating function of the normalized variable. By showing that the log-moment generating function converges to $s^2/2$, they establish convergence in distribution to the standard normal $N(0,1)$ and convergence of moments.

3. Key Contributions and Results

A. Main Theorem (Theorem 1.1)

For $\ell \ge 2$ and $x > 0$, as $n \to \infty$:

1. Asymptotic Mean:
   $$\mathbb{E}[K_{\ell,n}] \sim (x K_\ell)^{1/\ell} \frac{1}{\ell-1} n^{\frac{\ell-1}{\ell}}$$
2. Asymptotic Variance:
   $$\text{Var}(K_{\ell,n}) \sim (x K_\ell)^{1/\ell} \frac{1}{\ell(\ell-1)} n^{\frac{\ell-1}{\ell}}$$
3. Central Limit Theorem:
   The normalized variable converges in distribution and moments to a standard Gaussian:
   $$\frac{K_{\ell,n} - \mathbb{E}[K_{\ell,n}]}{\sqrt{\text{Var}(K_{\ell,n})}} \xrightarrow{d} N(0, 1)$$

Note: For $\ell=1$, the power law $n^{(\ell-1)/\ell}$ is interpreted as $\ln n$, recovering the classic result by Goncharov (uniform case) and Hansen (Ewens measure).

B. Technical Constants

The paper introduces the constant $K_\ell$ involving products of zeta values. For example:

• $\ell=2$: $K_2 = \zeta(2) = \pi^2/6$.
• $\ell=3$: $K_3 = \zeta(2)\zeta(3)$.

C. Connection to Log-Concavity

The CLT provides weak confirmation for a conjecture by the first author regarding the log-concavity of the coefficients $A(\ell, n, k)$ (the number of tuples with $k$ orbits). Since the Gaussian density is log-concave, the limiting distribution suggests the underlying discrete distribution shares this property.

D. Relation to Other Fields

The authors situate their work within several broader contexts:

• Random Matrices: Commuting permutations are discrete analogues of random commuting unitary matrices.
• Statistical Mechanics: The weights relate to "polymer gas" models.
• Number Theory: The weights for $\ell=2$ involve the sum of divisors function $\sigma(j)$, linking the problem to the Riemann Hypothesis.
• Group Theory: The problem is equivalent to studying random homomorphisms $\phi: \mathbb{Z}^\ell \to S_n$.

4. Significance

1. Generalization of Classical Results: It extends the famous CLT for the number of cycles in a single random permutation (Goncharov) and the Ewens measure (Hansen) to the multi-dimensional setting of commuting tuples.
2. New Asymptotic Regime: Unlike the $\ell=1$ case where the growth is logarithmic ($\ln n$), for $\ell \ge 2$, the number of orbits grows as a power law $n^{1 - 1/\ell}$. This represents a distinct phase in the statistical behavior of these combinatorial structures.
3. Self-Contained Proof: The authors provide a rigorous, self-contained proof using saddle point analysis, making the complex asymptotics accessible without requiring deep prior knowledge of specific advanced techniques like Meinardus' theorem (though they relate to it).
4. Bridge Between Fields: The work connects combinatorics (permutations), probability (limit theorems), number theory (zeta functions), and mathematical physics (random matrix models and polymer gases).

5. Future Directions

The paper outlines several open problems:

• Large Deviations: Studying the probability of $K_{\ell,n}$ deviating significantly from the mean.
• Local CLT: Refining the result to approximate the probability mass function pointwise, not just the distribution.
• Probabilistic Proofs: Finding a coupling argument (generalizing the Feller coupling) to prove the CLT without heavy analytic machinery.
• Generalized Weights: Extending the measure to allow weights depending on the size or shape of the orbits, not just the count.
• Random Commuting Matrices: Using these discrete results to inform the study of continuous random commuting matrices.