Randomly rotating an object in a high dimensional space takes exactly n squared log n steps to reach total chaos, solving a forty year old mystery.
April 29, 2026
Original Paper
Kac's walk on rotation matrices mixes in $n^2 \log n$ steps
arXiv · 2604.23828
The Takeaway
Kac’s walk is a famous mathematical model used to track how random movements lead to a state of total disorder. Forty years of work failed to prove exactly how many steps are required for a high dimensional system to reach equilibrium. The proof finally confirms that the process takes exactly n squared log n steps to finish. Solving the problem required a new framework that treats random walks like a form of fluid calculus. This provides a definitive speed limit for how fast information spreads through a rotating system. It has direct implications for how we simulate everything from gasses to high speed computer networks.
From the abstract
Kac's walk on the rotation group, introduced by Hastings in 1970, is an important high-dimensional Markov chain with applications in statistical physics, statistics, cryptography, and computational science. Despite its simple transition rules, determining its total-variation mixing time has remained a challenging problem for decades. A key obstacle is that the walk is not conjugation-invariant, placing it beyond the reach of classical Fourier-analytic techniques that apply to many related random