Total variation distance d(t) drops sharply at mixing time t* ~ n log n
Random walk on hypercube {0,1}ⁿ: at each step, flip one coordinate uniformly at random.
Total variation distance d(t) = ½ Σ|P^t(x,·) − π(·)| from uniform distribution π = 1/2ⁿ.
Cutoff phenomenon: d(t) ≈ 1 for t < t*, drops sharply to 0 near t* = ¼n ln n.
Exact formula: d(t) = (1/2)‖P^t(e₀,·) − Unif‖_TV — computed via eigenvalues (1 − 2/n)^k.