Read-back: random_transpositions_mixing

For every real number $\delta$ with $\delta > 0$, there exists a natural number $N$ such that for every natural number $n \ge N$ the following inequality holds:

where the left-hand side is a natural number cast to a real, $\log$ is the natural logarithm, and the objects involved are defined as follows. $P_n$ is the "random transpositions" transition matrix on the group of permutations of the $n$-element set $\{0,\dots,n-1\}$: its entry in row $a$, column $b$ is $\mu(b\,a^{-1})$, where $\mu$ assigns weight $1/n$ to the identity permutation, weight $2/n^2$ to every transposition of two distinct indices, and $0$ to every other permutation (so the entry is $1/n$ if $b = a$, $2/n^2$ if $b = \tau a$ for some transposition $\tau$, and $0$ otherwise). The mixing time $t_{\mathrm{mix}}(P_n)$ is defined as the least natural number $t$ (an infimum over naturals) such that

where the supremum ranges over all permutations $x$ (as starting states), $u$ is the uniform distribution on the permutation group, giving each of the $n!$ permutations mass $1/n!$, and $d_{TV}(\mu,\nu)$ is defined as the supremum, over all finite subsets $A$ of the state space, of $\left|\sum_{x\in A}\mu(x) - \sum_{x\in A}\nu(x)\right|$ (the sup-over-events form of total-variation distance, with no factor $\tfrac12$; the suprema here are real-valued suprema, which default to $0$ for sets of values that are empty or unbounded above). By the natural-number infimum convention, if no $t$ satisfies the displayed threshold condition then $t_{\mathrm{mix}}(P_n) = 0$, and the claimed inequality would then hold trivially for $n \ge 1$. The theorem is purely asymptotic: for each $\delta > 0$ it only asserts the bound for all sufficiently large $n$ (some threshold $N$, about which nothing quantitative is said), and it gives an upper bound only — no matching lower bound, and no claim about small $n$.