稀疏随机着色中彩色等差数列的阈值与波动

中文摘要 AI 辅助

本文研究了$[n]:=\{1, 2, \ldots, n\}$的稀疏随机着色中具有指定颜色模式的等差数列的阈值与波动，其中$[n]$的每个元素根据给定的概率向量独立着色。对于任何可容许的有序颜色调色板，我们确定了彩色等差数列出现的完整多参数阈值区域。该阈值由两种竞争机制控制：全局一阶矩条件和局部颜色可用性条件，从而产生一个多面体可满足性区域，并具有分段多面体阈值曲面。在可满足性区域内，我们建立了给定长度的彩色等差数列数量的渐近正态性，并给出了Wasserstein距离下的显式收敛速率。在阈值曲面上，我们识别出三种不同的渐近状态：泊松、具有混合泊松跳跃的复合泊松以及具有均匀跳跃的复合泊松，在适当归一化后成立。这些结果在统一框架下完整描述了稀疏随机着色下一般彩色等差数列的阈值和波动行为，该框架插值了二项随机子集中的经典无色/单色等差数列与多色（包括彩虹）等差数列。

英文摘要

In this paper, we derive thresholds and fluctuations for arithmetic progressions with prescribed color patterns in sparse random colorings of $[n]:=\{1, 2, \ldots, n\}$, where each element of $[n]$ is colored independently according to a given probability vector. For any admissible ordered palette of colors, we determine the full multi-parameter threshold region for the appearance of a colorful arithmetic progression. The threshold is governed by two competing mechanisms: a global first-moment condition and a local color-availability condition, resulting in a polyhedral satisfiability region, with a piecewise-polyhedral threshold surface. In the satisfiability region we establish asymptotic normality for the number of colorful arithmetic progressions of a given length, with an explicit rate of convergence in Wasserstein distance. On the threshold surface, we identify three distinct asymptotic regimes: Poisson, compound Poisson with mixed Poisson jumps, and compound Poisson with uniform jumps, after an appropriate normalization. These results provide a complete description of the threshold and fluctuation behavior of general colored arithmetic progressions under sparse random colorings, in a unified framework that interpolates between classical uncolored/monochromatic progressions in binomial random subsets and multicolored, including rainbow, arithmetic progressions.

发表机构

• University of Pennsylvania（宾夕法尼亚大学）
• National University of Singapore（新加坡国立大学）
• Stanford University（斯坦福大学）
• Indian Statistical Institute, Kolkata（印度统计学院，加尔各答）

机构由 AI 辅助整理，请以论文原文为准。

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29 pages, 1 figure