Anti-concentration with respect to random permutations

TL;DR

This paper extends anti-concentration from sums with independent signs to sums formed by random permutations, studying $S = \sum_{i=1}^n w_{\pi_i} v_i$ with a random permutation $\pi$. The authors develop a general bound for $\mathbb{P}( w_\pi \cdot v \in I )$ when $w$ and $v$ are increasing with $\Delta(v)>0$, tying the bound to the gaps in $v$ and the spread of $w$ via parameters like $\Delta(v)$ and $w_{n-i_2}-w_{i_1}$. The core methodology combines Rogozin’s concentration bound, probabilistic lemmas on random subset sums (with and without replacement), and a combinatorial Sperner/Dilworth framework to control the number of component configurations that yield a given sum, yielding precise decay rates and useful corollaries. These results yield new, sharp anti-concentration statements in the permutation setting with potential implications for random polynomials, random matrix theory, and related combinatorial probability problems.

Abstract

Classical anti-concentration results focus on the random sum $S := \sum _{i=1}^n ξ_i v_i$, where $ξ_i$ are independent random variables and $v_i$ are real numbers. In this paper, we prove new concentration results concerning the random sum $S := \sum_{i=1}^n w_{π_i } v_i $, where $w_i , v_i$ are real numbers and $π$ is a random permutation.

Theorems & Definitions (14)

• Theorem 1.1: Erdős-Littlewood-Offord
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4: Anti-concentration for random permutations
• Corollary 1.5
• Corollary 1.6
• Lemma 1.7: Random subset sum with replacement
• Lemma 1.8: Random subset sum without replacement
• Theorem 2.1: Rogozin
• proof : Proof of Lemma \ref{['randomset1']}