You need to know: Set of points with integer coefficients, probability, independent random variables, exponential distribution, covariance of random variables X and Y.
Background: At time , let us put, for each , a particle in site x, independently and with probability . Then, for each particle p, independently generate a random variable from exponential distribution, wait for time , and then jump to an adjacent site up, down, or right, with probabilities , respectively, provided that the target site is unoccupied (and otherwise stay). This action is then repeated and performed in parallel for all particles. We call this asymmetric simple exclusion process (ASEP). For any , let be the number of particles (which can be or ) in position x at time t. Next define , where is the first coordinate of x.
The Theorem: On 31st January 2002, Horng-Tzer Yau submitted to the Annals of Mathematics a paper in which he proved the existence of a constant such that, for sufficiently small ,
Short context: ASEP is one of the simplest models for modelling diffusion process, function defined above is known as diffusion coefficient, and understanding how fast grows with t is important for estimating the speed of diffusion process. The Theorem implies that grows approximately as for large t, and can therefore be called “ law for ASEP”.
Links: The original paper is available here. See also Section 4.3 of this book for an accessible description of the Theorem.
Theorems of the 21st century 
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