存在谱隙时的调和测度
Harmonic measure in the presence of a spectral gap
摘要：
研究了有限图的调和测度，重点研究了扩张子，即正谱隙。它表明，如果谱间隙是积极的，然后对所有的集合，不是太大的调和测度从一个统一的开始点是不超过一个常数因子的统一措施。对于大的集合，有一个严格的对数校正因子。我们还表明，积极的频谱间隙不允许一个固定的比例的调和测度的集合上支持的小子集，相反，在欧几里得空间的情况。结果是定量的光谱间隙的函数，也适用于当光谱间隙衰减到0的图形的大小增长到无穷大。作为一个应用，我们考虑有限图上的扩散限制聚集（DLA）模型，得到聚集体增长率的上界。
英文摘要：
We study harmonic measure in finite graphs with an emphasis on expanders, that is, positive spectral gap. It is shown that if the spectral gap is positive then for all sets that are not too large the harmonic measure from a uniform starting point is not more than a constant factor of the uniform measure on the set. For large sets there is a tight logarithmic correction factor. We also show that positive spectral gap does not allow for a fixed proportion of the harmonic measure of sets to be supported on small subsets, in contrast to the situation in Euclidean space. The results are quantitative as a function of the spectral gap, and apply also when the spectral gap decays to 0 as the size of the graph grows to infinity. As an application we consider a model of diffusion limited aggregation, or DLA, on finite graphs, obtaining upper bounds on the growth rate of the aggregate.
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