Through Thick and Thin<\/strong><\/p>\n Anderson thought of a material as a grid of points that an electron can randomly hop around on. If an electron hops around a lot, the material conducts. If it can\u2019t hop around, it insulates.<\/p>\n To understand the electron\u2019s overall behavior, you can use an array of numbers called a matrix to compute lists of values. These lists of values are called the eigenfunctions.<\/p>\n In a relatively pure material, nearly all of the eigenfunctions have values that are, on average, very small. This tells you that the electron has a relatively equal chance of hopping to a variety of different spots on the grid. It\u2019s delocalized.<\/p>\n Horng-Tzer Yau has spent decades studying the interplay of randomness and order in matrices.<\/p>\n Courtesy of Horng-Tzer Yau<\/p>\n Anderson said that for matrices that describe materials with enough randomness, each eigenfunction should see some of its values suddenly get very large, while others drop to zero. This means that the electron is now trapped in a particular region of the grid. It\u2019s localized.<\/p>\n The problem is that it\u2019s very difficult to calculate the eigenfunctions for the type of matrix that Anderson used. They sit awkwardly out of reach of the standard methods.<\/p>\n That\u2019s where band matrices come in.<\/p>\n \u201cBand\u201d refers to a sprinkling of numbers along the diagonal of the matrix \u2014 a hallmark of Anderson\u2019s matrices. If the only nonzero numbers in a matrix are on the line that runs from its upper lefthand corner to its lower righthand corner, that matrix has a band of width 1. Additional nonzero numbers around that diagonal make the band wider. The matrices in Anderson\u2019s model always have a very thin band. The eigenfunctions of such thin matrices are hard to calculate.<\/p>\n The band width acts like a reflection of how far the electron can move: If the band is wider, the electron can teleport to more distant points on the grid. (This is not a very realistic model, but it\u2019s still a useful one.)<\/p>\n In Anderson\u2019s matrices, some entries are random, and others aren\u2019t. But in 1990, physicists noticed that band matrices where all the entries were random also displayed a localization transition: Wide bands meant delocalized electrons, thinner bands localized ones. Unlike in Anderson\u2019s model, this transition was slow, rather than sudden. But researchers could still detect a threshold \u2014 a band width that separated delocalized states from localized ones. And so, as with Anderson\u2019s model, mathematicians wanted to pinpoint that threshold. That is, they wanted to find the slimmest possible band matrices in which the eigenfunctions\u2019 values remained small.<\/p>\n That was still difficult to do, because the thinner the band, the harder it is to analyze the matrix\u2019s eigenfunctions. But it was potentially easier than computing the eigenfunctions for Anderson\u2019s thin-banded matrices. And if mathematicians could prove this new threshold, maybe it would help them make sense of those more difficult matrices.<\/p>\n Out of Control<\/strong><\/p>\n The physicists who uncovered the transition in band matrices started out with an even simpler model. They imagined a material like an infinitely thin wire \u2014 a one-dimensional version of the problem. They then used numerical experiments to pinpoint the precise threshold between localization and delocalization.<\/p>\n Jun Yin, a former physicist, initially hoped to focus his research on the quantum behavior of gases. But he was soon drawn to a new problem, which involved random matrices and a model of semiconductors.<\/p>\n But these experiments were not the same as mathematical proof. \u201cThey are based on totally uncontrolled approximations which, while plausible, are often very difficult to make rigorous,\u201d said Antti Knowles<\/a> of the University of Geneva. So mathematicians kept band matrices on their to-do list, hoping to turn the theories into theorems. Among them were Yau and his then-postdoc, Jun Yin<\/a>.<\/p>\n Yin joined Yau\u2019s group in 2008, after finishing a Ph.D. in Princeton University\u2019s physics department. The pair started with the one-dimensional case. By 2013, working with Knowles and L\u00e1szl\u00f3 Erd\u0151s<\/a>, they were able to prove that most eigenfunctions are delocalized<\/a> once the band is very wide. But this width was still much greater than the threshold that physicists had predicted.<\/p>\n![\"A](\"https:\/\/www.europesays.com\/us\/wp-content\/uploads\/2025\/08\/Horng-TzerYau_coHorng-TzerYau.webp.webp\")
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