Timeline for How quickly can irreducible aperiodic convex combinations of permutation matrices converge to the stationary distribution?

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Aug 6, 2021 at 13:49	comment	added	Joseph Van Name		Unless I am missing something, the final paragraph in your answer where you use the fact that $\lambda_{\star,A}\geq\frac{1}{r}$ or $\lambda_{\star,A}\geq\frac{1}{r}$ only applies when $A=\frac{1}{r}(A_{1}+\dots+A_{r})$ for permutation matrices $A_{1},\dots,A_{r}$. What about the more general case of convex combinations when $A=t_{1}A_{1}+\dots+t_{r}A_{r}$ where $A_{1},\dots,A_{r}$ are permutation matrices and $t_{1}+\dots+t_{r}=1$ and $t_{i}\geq 0$ for each $i$?
Aug 6, 2021 at 13:42	comment	added	Joseph Van Name		+1. I would personally show that the matrix corresponding to the Markov chain where you multiply by r and add a random number from 0 to r-1 is the average of r many permutation matrices by the following fact: If $A$ is a doubly stochastic matrix where each entry of $A$ is either $0$ or $1/r$, then $A$ can be written as the average of $r$ permutation matrices.
Aug 6, 2021 at 1:08	history	answered	Will Sawin	CC BY-SA 4.0