Mixing time of the Card-Cyclic-to-Random shuffle

Abstract

The Card-Cyclic-to-Random shuffle on $n$ cards is defined as follows: at time $t$ remove the card with label $t$ mod $n$ and randomly reinsert it back into the deck. Pinsky introduced this shuffle and asked how many steps are needed to mix the deck. He showed $n$ steps do not suffice. Here we show that the mixing time is on the order of $\Theta(n \log n)$.