You need to know: Eucledean space , norm of vector , matrix, multiplication of matrix by a vector, norm of matrix A, basic probability theory, independence, convergence in distribution.
Background: Fix a sequence of positive numbers. For every positive integer n, let be a random matrix whose entries are such that (i) for all i and j; (ii) if either or ; and (iii) all such that are selected independently at random, each equal to or with equal probabilities. Random matrices selected in accordance to these rules are called random band matrices.
The Theorem: On 22nd June 2009, Sasha Sodin submitted to arxiv a paper in which he proved that if then converges in distribution to .
Short context: The study of random matrices is one of the central topics in probability theory, and random band matrices is an important model having physical applications. In 2008, Khorunzhiy proved the statement of the Theorem under stronger assumption that , and conjectured that weaker assumption suffices. The Theorem proves this conjecture. This result is sharp, because Bogachev, Molchanov, and Pastur proved in 1991 that the conclusion of the Theorem fails if .
Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.13 of this book for an accessible description of the Theorem.
Theorems of the 21st century 