Partial shuffles by lazy swaps

TL;DR

This paper investigates the minimum number of random transpositions needed to achieve uniform distribution of elements or pairs in a set, providing exact and asymptotic bounds and confirming a conjecture.

Contribution

It establishes new bounds for the number of transpositions required for uniformity, including exact results for specific cases and a surprising non-quadratic bound for pair distribution.

Findings

Minimum for full element uniformity is about (1/2) n log_2 n for powers of 2

O(n log^2 n) transpositions suffice for pairwise uniformity

Only 2n-3 transpositions are needed to make a specific pair uniformly distributed

Abstract

What is the smallest number of random transpositions (meaning that we swap given pairs of elements with given probabilities) that we can make on an $n$-point set to ensure that each element is uniformly distributed -- in the sense that the probability that $i$ is mapped to $j$ is $1/n$ for all $i$ and $j$? And what if we insist that each pair is uniformly distributed? In this paper we show that the minimum for the first problem is about $\frac{1}{2} n \log_2 n$, with this being exact when $n$ is a power of $2$. For the second problem, we show that, rather surprisingly, the answer is not quadratic: $O(n \log^2 n)$ random transpositions suffice. We also show that if we ask only that the pair $1,2$ is uniformly distributed then the answer is $2n-3$. This proves a conjecture of Groenland, Johnston, Radcliffe and Scott.