Pictured: Mathematicians Horng-Tzer Yau (right) and Jun Yin (left).

Harvard Math Professor Horng-Tzer Yau has spent decades studying the interplay of randomness and order in matrices. UCLA Math Professor Jun Yin joined Yau’s group after completing his Ph.D in Physics at Princeton University in 2008. They first tackled the one-dimensional case together, proving that most eigenfunctions are delocalized once the band is very wide. It was the biggest proof of a delocalization phenomenon since Anderson introduced his model.

“For years, they explored all sorts of ways to show that the eigenfunctions remain small for smaller band widths. They even took a detour into a seven-dimensional version of the problem, a setting which has little bearing on physics but which they hoped would yield mathematical insight.

But after a decade of work, they had only gotten a smidge closer to their goal.

It seemed they’d tried everything. Then, in the spring of 2024, they realized that a method they’d previously dismissed might be useful after all.”

Read the full Quanta Magazine here: https://www.quantamagazine.org/new-physics-inspired-proof-probes-the-borders-of-disorder-20250815/

Read the full paper, “Delocalization of One-Dimensional Random Band Matrices”, here: https://arxiv.org/abs/2501.01718

Read the full paper, “Delocalization of Two-Dimensional Random Band Matrices”, here: https://arxiv.org/abs/2503.07606