顺从Cayley图的Lifshitz指数与van Hove指数的相等性
Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
摘要：
本文研究了周期Laplace算子和随机Laplace算子在顺从有限生成群的Cayley图上的低能渐近性。对于周期算子的渐近性的特点是范霍夫指数或零诺维科夫-舒宾不变量。我们考虑的随机模型是根据Cayley图的一个邻接Laplacian在站点上或边渗流子图给出的。谱分布的渐近行为是指数的，其特征在于Lifshitz指数。我们表明，对于相邻拉普拉斯算子的两个不变量/指数重合。结果也适用于更一般的对称转移算子。对于组合拉普拉斯算子，谱分布函数的低能渐近性具有不同的普适性，这实际上可以建立在没有顺从性假设的准传递图上。后者的结果也适用于长程键渗流模型。
英文摘要：
We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov-Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.
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