Iterated Decomposition of Biased Permutations Via New Bounds on the Spectral Gap of Markov Chains
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The spectral gap of a Markov chain can be bounded by the spectral gaps of constituent "restriction" chains and a "projection" chain, and the strength of such a bound is the content of various decomposition theorems. In this paper, we introduce a new parameter that allows us to improve upon these bounds. We further define a notion of orthogonality between the restriction chains and "complementary" restriction chains. This leads to a new Complementary Decomposition theorem, which does not require analyzing the projection chain. For ϵ-orthogonal chains, this theorem may be iterated O(1/ϵ) times while only giving away a constant multiplicative factor on the overall spectral gap. As an application, we provide a 1/n-orthogonal decomposition of the nearest neighbor Markov chain over k-class biased monotone permutations on [n], as long as the number of particles in each class is at least Clog n. This allows us to apply the Complementary Decomposition theorem iteratively n times to prove the first polynomial bound on the spectral gap when k is as large as Θ(n/log n). The previous best known bound assumed k was at most a constant.
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