The variance of the current across the characteristic in ASEP is of order t^(2/3)

You need to know: Set of integers, probability, independent random variables, exponential distribution, expectation, variance. For , denote the largest integer in if , and the smallest integer in if .

Background: At time , let us put, for each , a particle in site x, independently and with the same probability . Then, for each particle z, independently generate a random variable from exponential distribution with parameter 1, wait for time , and then jump to an adjacent site right or left, with probabilities and , 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 , let , where is the number of particles that began in at time but lie in at time t, and is the number of particles that began in at time but lie in at time t.

The Theorem: On 15st August 2006, Márton Balázs and Timo Seppäläinen submitted to arxiv a paper in which they proved that, for , there exists positive constants and , such that , for all .

Short context: ASEP is one of the simplest models for diffusion processes – see here for its the 2-dimensional version. Random variable has the meaning of the total net particle current seen by an observer moving at speed v during time interval . In 1994, Ferrari and Fontes proved that , which implies that the variance grows linearly in time, except when , which is called the characteristic speed. for this v is called the current across the characteristic, and The Theorem proves that its variance grows as .

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 10.5 of this book for an accessible description of the Theorem.

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