Proposition 8.11 -- random transpositions lower bound

The random transpositions shuffle of a deck of $n$ cards picks two cards independently and uniformly at random and swaps them: the identity is applied with probability $1/n$ and each transposition with probability $2/n^2$. Its stationary distribution is uniform. For a tolerance $\varepsilon$, the mixing time $t_{\mathrm{mix}}(\varepsilon)$ is the first $t$ at which $\max_x\|P^t(x,\cdot)-\mathrm{unif}\|_{TV}\le\varepsilon$, where $\|\mu-\nu\|_{TV}=\max_A|\mu(A)-\nu(A)|$ is the total variation distance.

The theorem (Proposition 8.11 of Levin–Peres–Wilmer) asserts: for every $n\ge2$ and every $0<\varepsilon<1$,

So order $\tfrac12\,n\log n$ shuffles are necessary. The obstruction is the number of fixed points: until almost every card has been touched at least once — a coupon-collector event taking $\tfrac12 n\log n$ pair draws — the deck has many more cards in their original position than a uniform ordering would.