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Your question is essentially covered by Cor 9.5 in [1] which implies that as long as the ratio of self-loops to the original degree is bounded above and below, the mixing time of this modified walk is equivalent (up to constants) to the mixing time of the lazy walk on $G_1$. [1] Peres, Yuval, and Perla Sousi. "Mixing times are hitting times of large ...

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