长程扩张图：构造与截断
Long-range expanders: construction and cutoff
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中文总结 AI 辅助
本文证明每个Ramanujan图具有长程扩张，构造对数围长的显式长程扩张图，并确立固定度长程扩张图序列上随机游走的截断现象及其高斯极限轮廓，为传递谱扩张图的截断猜想提供中间进展。
中文摘要 AI 辅助
长程扩张是有限度正则图上的树状体积增长条件，由Dodos、Tikhomirov、Tyros及作者引入，用于证明定量非线性Poincaré不等式和非嵌入定理。受几何群论和非线性泛函分析中关于对数围长超扩张图的基本问题启发，我们的第一个主要结果是：每个Ramanujan图都具有长程扩张。这给出了具有对数围长的显式长程扩张图（如Lubotzky--Phillips--Sarnak图），并建立了严格的层次结构：Ramanujan扩张蕴含长程扩张，长程扩张蕴含谱扩张，且两个逆命题均不成立。随后，我们从动力学角度比较了不同的扩张概念。我们的第二个主要结果确立了在任意固定度的长程扩张图序列上随机游走的截断现象，并给出显式的高斯极限轮廓。特别地，截断位置和轮廓与Ramanujan图的情形一致。这为Ramanujan图上的截断（由Lubetzky和Peres证明）与传递谱扩张图上截断的长期猜想之间提供了中间进展。
英文摘要
Long-range expansion is a combinatorial graph property introduced to construct metric spaces with strong quantitative obstructions to low-distortion embeddings. Such extremal constructions are motivated by fundamental questions in metric geometry and nonlinear functional analysis. However, the only known constructions of long-range expander sequences are random regular graphs; the problem of constructing explicit sequences, especially with logarithmic girth, remains open. We resolve this question in strong form by proving that the Ramanujan condition implies long-range expansion. In particular, classical Ramanujan constructions (such as LPS graphs) give explicit long-range expander sequences with logarithmic girth. We also clarify that long-range expansion implies spectral expansion, and both implications are strict. While long-range expansion, the Ramanujan property, and spectral expansion are strictly separated, it is \textit{a priori} not clear how robust these gaps are. We next compare them from a dynamics perspective: how do the different notions of expansion affect the mixing rate of random walks? Our second main result is that, from a dynamics perspective, LRE and the Ramanujan property are equivalent. We show that the random walk on any LRE sequence has the same cutoff location and Gaussian limit profile as on a Ramanujan graph. This incidentally offers intermediate progress between cutoff for Ramanujan graphs---proven by Lubetzky and Peres---and the long-standing conjecture of cutoff for transitive spectral expanders.
发表机构
- University of Texas at Austin（德克萨斯大学奥斯汀分校）
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