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We analyze the absolute spectral gap of Markov chains on graphs obtained froma cycle of $n$ vertices and perturbed only at approximately $n^{1/\rho}$ randomlocations with an appropriate, possibly sparse, interconnection structure.Together with a strong asymmetry along the cycle, the gap of the resultingchain can be bounded inversely proportionally by the longest arc length (up tologarithmic factors) with high probability, providing a significant mixingspeedup compared to the reversible version.

Improved Mixing Rates of Directed Cycles with Additional Sparse Interconnections