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首页> 外文期刊>SIAM Journal on Discrete Mathematics >INTERLACINGS FOR RANDOM WALKS ON WEIGHTED GRAPHS AND THE INTERCHANGE PROCESS
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INTERLACINGS FOR RANDOM WALKS ON WEIGHTED GRAPHS AND THE INTERCHANGE PROCESS

机译：加权图与互换过程的交互

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摘要

We study Aldous's conjecture that the spectral gap of the interchange process on a weighted undirected graph equals the spectral gap of the random walk on this graph. We present a conjecture in the form of an inequality and prove that this inequality implies Aldous's conjecture by combining an interlacing result for Laplacians of random walks on weighted graphs with representation theory. We prove the conjectured inequality for several important instances. As an application of the developed theory, we prove Aldous's conjecture for a large class of weighted graphs, which includes all wheel graphs, all graphs with four vertices, certain nonplanar graphs, certain graphs with several weighted cycles of arbitrary length, and all trees. Caputo, Liggett, and Richthammer have recently resolved Aldous's conjecture, after independently and simultaneously discovering the key ideas developed in the present paper.
机译：我们研究了Aldous的猜想，即加权无向图上交换过程的光谱间隙等于该图上随机游走的光谱间隙。我们以不等式的形式给出一个猜想，并通过将加权图上随机游走的拉普拉斯算子的交织结果与表示理论相结合，证明该不等式暗示了Aldous的猜想。我们证明了几个重要实例的猜想不等式。作为发达理论的一种应用，我们证明了Aldous对于大量加权图的猜想，其中包括所有轮图，所有具有四个顶点的图，某些非平面图，某些具有任意长度的加权周期的图以及所有树。卡普托，利吉特和里希哈默尔在独立并同时发现本文提出的关键思想之后，最近解决了阿尔多斯的猜想。

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