  A lazy transposition $(a,b,p)$ is the random permutation that equals the
identity with probability $1-p$ and the transposition $(a,b)\in S_n$ with
probability $p$. How long must a sequence of independent lazy transpositions be
if their composition is uniformly distributed? It is known that there are
sequences of length $\binom{n}2$, but are there shorter sequences? This was
raised by Fitzsimons in 2011, and independently by Angel and Holroyd in 2018.
We answer this question negatively by giving a construction of length $\frac23
\binom{n}2+O(n\log n)$, and consider some related questions.
