Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the sampling entropy of permutons

Balázs Maga

TL;DR

The paper develops a comprehensive theory of sampling entropy for permutons, revealing distinct asymptotic regimes: linear, linear with a logarithmic factor, and log-periodic growth in random constructions. It connects sampling entropy to absolute continuity, measure-preserving dynamics, and random automorphisms of $d$-ary trees, establishing that $H_n(\mu)$ can scale as $\Theta(n\log n)$ when $\mu$ has an ac part, while in many singular settings the limit of $H_n(\mu)/n$ can equal a dynamical entropy (e.g., $h_{KS}(f)$) or exhibit asymptotically log-periodic behavior. The work also demonstrates almost-sure convergence under a perturbed random model and shows strong concentration phenomena, illustrating rich interactions between combinatorial permutation structure, ergodic theory, and probabilistic self-averaging. These results illuminate when sampling entropy aligns with classical entropy notions and when it exhibits more intricate oscillatory or self-similar behavior, with implications for understanding limit objects in permutation combinatorics and related stochastic processes.

Abstract

For a permuton $μ$ let $H_n(μ)$ denote the Shannon entropy of the sampling distribution of $μ$ on $n$ points. We investigate the asymptotic growth of $H_n(μ)$ for a wide class of permutons. We prove that if $μ$ has a non-vanishing absolutely continuous part, then $H_n(μ)$ has a growth rate $Θ(n \log n)$. We show that if $μ$ is the graph of a piecewise continuously differentiable, measure-preserving function $f$, then $H_n(μ)/n$ tends to the Kolmogorov--Sinai entropy of $f$. Using genericity arguments, we also prove the existence of function permutons for which $H_n(μ)$ does not converge either after normalizing by $n$ or by $n\log n$. We study the sampling entropy of a natural family of random fractal-like permutons determined by a sequence of i.i.d. choices. It turns out that for every $n$, $H_n(μ)/n$ is heavily concentrated. We prove that the sequence $H_n(μ)/n$ either converges or has deterministic log-periodic oscillations almost surely, and argue towards the conjecture that in nondegenerate case, oscillation holds. On the other hand, for a straightforward random perturbation of the model $\tildeμ$ of $μ$, we prove the almost sure convergence of $H_n(\tildeμ)/n$.

On the sampling entropy of permutons

TL;DR

The paper develops a comprehensive theory of sampling entropy for permutons, revealing distinct asymptotic regimes: linear, linear with a logarithmic factor, and log-periodic growth in random constructions. It connects sampling entropy to absolute continuity, measure-preserving dynamics, and random automorphisms of -ary trees, establishing that can scale as when has an ac part, while in many singular settings the limit of can equal a dynamical entropy (e.g., ) or exhibit asymptotically log-periodic behavior. The work also demonstrates almost-sure convergence under a perturbed random model and shows strong concentration phenomena, illustrating rich interactions between combinatorial permutation structure, ergodic theory, and probabilistic self-averaging. These results illuminate when sampling entropy aligns with classical entropy notions and when it exhibits more intricate oscillatory or self-similar behavior, with implications for understanding limit objects in permutation combinatorics and related stochastic processes.

Abstract

For a permuton let denote the Shannon entropy of the sampling distribution of on points. We investigate the asymptotic growth of for a wide class of permutons. We prove that if has a non-vanishing absolutely continuous part, then has a growth rate . We show that if is the graph of a piecewise continuously differentiable, measure-preserving function , then tends to the Kolmogorov--Sinai entropy of . Using genericity arguments, we also prove the existence of function permutons for which does not converge either after normalizing by or by . We study the sampling entropy of a natural family of random fractal-like permutons determined by a sequence of i.i.d. choices. It turns out that for every , is heavily concentrated. We prove that the sequence either converges or has deterministic log-periodic oscillations almost surely, and argue towards the conjecture that in nondegenerate case, oscillation holds. On the other hand, for a straightforward random perturbation of the model of , we prove the almost sure convergence of .

Paper Structure

This paper contains 19 sections, 23 theorems, 179 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main results
  3. Insights and further speculation
  4. The organization of the paper
  5. Preliminaries
  6. Permutons
  7. Basic properties of entropy
  8. Permutons from automorphisms of the $d$-ary tree
  9. Entropy of convex combinations of pre-permutons
  10. Absolutely continuous permutons
  11. Measure-preserving functions with regularity properties
  12. Entropy of generic functions
  13. Permutons from random automorphisms of the $d$-ary tree
  14. Patterns of the random automorphism
  15. Self-averaging sequences and log-periodicity
  16. ...and 4 more sections

Key Result

Theorem 1.1

If $\mu$ is absolutely continuous with respect to the Lebesgue measure, then More specifically, if the absolutely continuous part of $\mu$ is denoted by $\mu_{\mathrm{ac}}$, then

Figures (2)

  • Figure 1: (Left) The third approximation $\mu_3$ of an outcome of $\mu_F$ for $d=2$. The filled squares form the support, restricted to any of these $\mu_3$ is uniform. Shaded squares visualize $\mu_1, \mu_2$. (Right) The third approximation of $\mu_{F, \mathrm{Unif}}$, where the random permutations are the same as on the left, but the sidelengths vary.
  • Figure 2: Visualizing $g$ (red) and $h$ (green) for $f$ (blue) being the symmetric tent map. The interval $J_i$ ($i=1, \dots, 5)$ is fixed to be $J_i = [0.2\cdot(i-1), 0.2\cdot i]$, while $\alpha=0.9$.

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.9
  • Proposition 2.1: HOPPEN201393
  • Lemma 2.2
  • Lemma 2.3
  • ...and 54 more