The anti-concentration phenomenon with respect to random permutations

TL;DR

This paper investigates anti-concentration phenomena in the symmetric group, providing structural characterizations, bounds, and applications to random polynomials and matrices, extending prior work from product spaces to permutation-based models.

Contribution

It introduces a systematic study of anti-concentration in permutation spaces, including inverse theorems, bounds, and applications to polynomials and matrices, which were not previously explored.

Findings

Established near-optimal structural characterization of vectors with high concentration.

Proved polynomial decay bounds for the probability of specific sums in permutation models.

Showed that certain random permutation matrices are nonsingular with high probability.

Abstract

The anti-concentration phenomenon in probability theory has been intensively studied in recent years, with applications across many areas of mathematics. In most existing works, the ambient probability space is a product space generated by independent random variables. In this paper, we initiate a systematic study of anti-concentration when the ambient space is the symmetric group, equipped with the uniform measure. Concretely, we focus on the random sum $S_{\pi} = \sum_{i=1}^{n} w_i\, v_{\pi(i)}$, where $w=(w_1,\dots,w_n)$ and $v=(v_1,\dots,v_n)$ are fixed vectors and $\pi$ is a uniformly random permutation. The paper contains several new results, addressing both discrete and continuous anti-concentration phenomena. On the discrete side, we establish a near-optimal structural characterization of the vectors $w$ and $v$ under the assumption that the concentration probability $\sup_x P(S_{\pi}=x)$ is polynomially large. On the continuous side, we study the small-ball event $|S_{\pi}-L|\le \delta$. Our results exhibit sub-gaussian decay in $L$. Our results have applications in various areas. First, we use our inverse theorems to derive and strengthen a number of previous anti-concentration bounds. In particular, we show that if both $w$ and $v$ have distinct entries, then $\sup_x P(S_{\pi}=x) \le n^{-5/2+o(1)}$. Next, we apply our new results to study random polynomials, and prove that the number of extremal points of random permutation polynomials is bounded by $O(\log n)$, extending results of S{\"o}ze~\cite{Soze1, Soze2}. In the final application, we prove that random matrices whose rows are independent random permutations of a fixed non-degenerate vector are nonsingular with high probability.