A 30-year mystery is solved: if you want to mix things up as fast as possible, nothing beats pure, total randomness.
March 30, 2026
Original Paper
A proof of Fill's spectral gap conjecture
arXiv · 2603.26303
The Takeaway
It sounds intuitive, but mathematicians struggled for decades to prove that any bias or pattern in how you swap objects actually slows down the process of reaching equilibrium. This proof confirms that 'perfect' chaos is the most efficient path to a well-shuffled state.
From the abstract
We prove a quantitative lower bound on the spectral gap of the adjacent-transposition chain on the symmetric group with a general probability vector. As a consequence, among all regular probability vectors, the spectral gap of the transition matrix is minimised by the uniform probability vector, i.e., $p_{i,j}\equiv {\frac 1 2}$ for all $i \ne j$. A second consequence is a uniform polynomial bound on the inverse spectral gap in the regular case. This resolves a longstanding conjecture known as F