路径最大化图索引随机游走的期望值域
Paths maximize the expected range of graph-indexed random walks
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中文总结 AI 辅助
本文证明路径最大化图索引随机游走的期望值域，确立BHM猜想期望形式，并由此推导出LNR猜想，证明经Lean 4验证。
中文摘要 AI 辅助
我们证明，在所有同阶连通二分图中，一条路径最大化了均匀选择的图同态映射到整数（其中一个顶点固定在零）的期望值域。这确立了Benjamini--Häggström--Mossel猜想的期望形式。证明过程在每个二分划分类上对同态进行限制和重新缩放，然后收缩所得高度函数为常数的边。对这些零边秩的定量估计补偿了简单随机游走期望值域中的奇偶项，从而允许对顶点数进行归纳。随后我们证明BHM不等式蕴含了Loebl--Nešetřil--Reed不等式（针对任意连通图上均匀选择的整数1-Lipschitz函数），并因此作为BHM的推论得到LNR猜想。该证明通过与OpenAI GPT-6 Astra交互获得，并由作者验证。主要结果也已在Lean 4中形式化并检查通过。
英文摘要
We prove that a path maximizes the expected range of a uniformly chosen graph homomorphism into the integers, with one vertex pinned at zero, among all connected bipartite graphs of the same order. This establishes the expectation form of the Benjamini--Häggström--Mossel conjecture. The proof restricts and rescales a homomorphism on each bipartition class, then contracts the edges on which the resulting height function is constant. A quantitative estimate for the rank of these zero edges compensates for a parity term in the expected range of a simple random walk, allowing an induction on the number of vertices. We then prove that the BHM inequality implies the Loebl--Ne\v set\v ril--Reed inequality for uniformly chosen integer 1-Lipschitz functions on arbitrary connected graphs, and hence obtain the LNR conjecture as a corollary of BHM. The proof was obtained through interaction with OpenAI GPT-6 Astra and verified by the author. The main results have also been formalized and checked in Lean~4.
