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The spectral gap is an important value to evaluate the mixing time of thecontinuous time Markov processes. Caputo, Ligget and Richthammer, in theirseminal paper \arXiv:0906.1238, proved Aldous' spectral gap conjecture whichasserts that spectral gaps of the random walk and the interchange process onthe common weighted graph are equal. In this paper, we introduce a generalframework to compare the spectral gaps of two processes, and to prove ananalogue of Aldous' spectral gap conjecture. In our framework, to sort out theproblem, we will define the functor from the category of the weighted group tocategory of the weighted complete directed graph, and construct the weightedcomplete directed graph associated with the generalized exclusion process bythe quotient of the weighted complete directed graph associated with theinterchange process on extended graph by the weighted subgroup. Moreover, as anapplication of our framework, we prove the analogue of Aldous' spectral gapconjecture for the generalized exclusion process under appropriate assumptionswhich show that the spectral gap of the generalized exclusion process is equalto that of the random walk.

The Analogue of Aldous' spectral gap conjecture for the generalized exclusion process