The anti-concentration phenomenon with respect to random permutations

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.