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An Erd\H os Rényi Law for the Longest Consecutive Monotone Block in a Random Permutation

Anant Godbole

TL;DR

The paper establishes an Erdős-Rényi law for the longest consecutive monotone block in a random permutation by modeling the permutation with i.i.d. Uniform$[0,1]$ variables. It develops a Poisson-approximation framework for counts of monotone blocks, introduces strict monotone blocks to obtain tractable bounds, and leverages auxiliary variables together with the Borel-Cantelli lemmas to prove that the longest monotone block scales as $\frac{\ln n}{\ln\ln n}$ almost surely. The approach combines Stein-Chen Poisson approximation, auxiliary block-counts, and careful asymptotics to translate local occurrence probabilities into a global almost-sure growth law. Potential impact includes a deeper understanding of monotone structures in random permutations and avenues for refining probabilistic limit laws for permutation patterns.

Abstract

The Erd\H os-Rényi law states that given a sequence $\{X_j\}_{j=1}^\infty$ of i.i.d.~($p$) coin-tosses, the longest run $L_n$ of heads in the first $n$ coin tosses approaches $\log_{1/p}n$ almost surely. In this paper we explore a formulation of this result in the case of random permutations and prove an Erd\H os-Rényi law for the longest consecutive monotone block in a random permutation.

An Erd\H os Rényi Law for the Longest Consecutive Monotone Block in a Random Permutation

TL;DR

The paper establishes an Erdős-Rényi law for the longest consecutive monotone block in a random permutation by modeling the permutation with i.i.d. Uniform variables. It develops a Poisson-approximation framework for counts of monotone blocks, introduces strict monotone blocks to obtain tractable bounds, and leverages auxiliary variables together with the Borel-Cantelli lemmas to prove that the longest monotone block scales as almost surely. The approach combines Stein-Chen Poisson approximation, auxiliary block-counts, and careful asymptotics to translate local occurrence probabilities into a global almost-sure growth law. Potential impact includes a deeper understanding of monotone structures in random permutations and avenues for refining probabilistic limit laws for permutation patterns.

Abstract

The Erd\H os-Rényi law states that given a sequence of i.i.d.~() coin-tosses, the longest run of heads in the first coin tosses approaches almost surely. In this paper we explore a formulation of this result in the case of random permutations and prove an Erd\H os-Rényi law for the longest consecutive monotone block in a random permutation.
Paper Structure (6 sections, 10 theorems, 69 equations)

This paper contains 6 sections, 10 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Poisson Approximation
  3. Auxiliary Variables
  4. Main Result
  5. Future Work
  6. Funding

Key Result

Proposition 1.1

Given a sequence $\{X_j\}_{j=1}^\infty$ of i.i.d. ($p$) coin-tosses, we have (with $L_n$ denoting the longest pure run of heads in the first $n$ tosses) the hierarchy of results etc. holds, as opposed to the much sharper minimal growth rate criterion for two constants $\alpha$ and $\beta$.

Theorems & Definitions (15)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • ...and 5 more