Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Characteristic polynomial of generalized Ewens random permutations

Quentin François

TL;DR

This work proves that the characteristic polynomial of permutation matrices sampled from a generalized Ewens distribution converges inside the unit disk to a random holomorphic function F, given by the exponential of a Poisson series. The limit is F(z)=exp(- sum_{k>=1} (z^k/k) X_k) with X_k = sum_{l|k} l Y_l and Y_l Poisson, capturing the cycle-structure-driven randomness of the model. The proof combines tightness via second-moment estimates grounded in singularity analysis with convergence of traces to Poisson-based limits, and yields a Poisson-series and infinite-product representation that parallels Gaussian holomorphic chaos in structure and universality. These results extend the holomorphic-chaos framework to generalized Ewens measures and connect permutation-based models to log-correlated random analytic fields.

Abstract

We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure, according to certain weights assigned to each cycle length. The proof is based on uniform control of the characteristic polynomial using results from the singularity analysis of generating functions, together with the convergence of traces to explicit random variables expressed via a Poisson family. The limit function is the exponential of a Poisson series which has already appeared in the case of uniform permutation matrices. It is the Poisson analog of the Gaussian Holomorphic Chaos, related to the limit of characteristic polynomials for other matrix models such as Circular Ensembles, i.i.d. matrices, and Gaussian elliptic matrices.

Characteristic polynomial of generalized Ewens random permutations

TL;DR

This work proves that the characteristic polynomial of permutation matrices sampled from a generalized Ewens distribution converges inside the unit disk to a random holomorphic function F, given by the exponential of a Poisson series. The limit is F(z)=exp(- sum_{k>=1} (z^k/k) X_k) with X_k = sum_{l|k} l Y_l and Y_l Poisson, capturing the cycle-structure-driven randomness of the model. The proof combines tightness via second-moment estimates grounded in singularity analysis with convergence of traces to Poisson-based limits, and yields a Poisson-series and infinite-product representation that parallels Gaussian holomorphic chaos in structure and universality. These results extend the holomorphic-chaos framework to generalized Ewens measures and connect permutation-based models to log-correlated random analytic fields.

Abstract

We show the convergence of the characteristic polynomial for random permutation matrices sampled from the generalized Ewens distribution. Under this distribution, the measure of a given permutation depends only on its cycle structure, according to certain weights assigned to each cycle length. The proof is based on uniform control of the characteristic polynomial using results from the singularity analysis of generating functions, together with the convergence of traces to explicit random variables expressed via a Poisson family. The limit function is the exponential of a Poisson series which has already appeared in the case of uniform permutation matrices. It is the Poisson analog of the Gaussian Holomorphic Chaos, related to the limit of characteristic polynomials for other matrix models such as Circular Ensembles, i.i.d. matrices, and Gaussian elliptic matrices.

Paper Structure

This paper contains 9 sections, 10 theorems, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Convergence of the characteristic polynomial
  4. Method of proof
  5. Proof of Theorem \ref{['th:conv']}
  6. Tightness: proof of Theorem \ref{['th:tightness']}
  7. Convergence of traces: proof of Theorem \ref{['th:conv_of_traces']}
  8. Poisson Expression: proof of Theorem \ref{['th:poisson_expression']}
  9. Poisson Multiplicative Function

Key Result

Theorem 2.2

Let $\Theta = (\theta_k)_{k \geqslant 1}$ be a sequence of positive real numbers such that $g_\Theta \in F(r, \gamma, K)$ for some $r > 0$, $\gamma > 0$ and $K \in \mathbb C$. We have the convergence in law, for the topology of local uniform convergence in $\mathbb D$: where with $(Y_\ell)_{\ell \geqslant 1}$ independent Poisson random variables with parameter $\frac{\theta_\ell}{\ell}r^\ell$.

Figures (2)

  • Figure 1: Phase portrait of $p_n$ for an Ewens matrix of size $n=10000$ with parameter $\theta=100$ (left) and phase portrait of the limit function with same parameter (right). The unit circle is represented in red.
  • Figure 2: Integration contours $\Gamma$ and $\Gamma'$.

Theorems & Definitions (19)

  • Definition 1.1: Generalized Ewens measure, Nikeghbali_Zeindler
  • Definition 2.1: Logarithmic class function
  • Theorem 2.2: Convergence of the characteristic polynomial
  • Remark 2.3: Outside region
  • Remark 2.4: Extension to sums of permutations
  • Lemma 3.1: Tightness and convergence of coefficients imply convergence of functions
  • Theorem 3.2: Tightness
  • Theorem 3.3: Convergence of coefficients
  • Theorem 3.4: Poisson expression for $F$
  • Lemma 4.1: Uniform control
  • ...and 9 more