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

TL;DR

This paper determines that the mixing time of the Card-Cyclic-to-Random shuffle is on the order of n log n steps, providing a precise asymptotic estimate for how quickly the deck becomes well-shuffled.

Contribution

The paper establishes the exact order of the mixing time for the Card-Cyclic-to-Random shuffle, confirming it is Θ(n log n), which was previously unknown.

Findings

Mixing time is Θ(n log n) steps.

The shuffle does not mix in fewer than linear steps.

The result confirms the conjectured order of mixing time.

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 $\Theta(n \log n)$.