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The Universality Phenomenon

Tháng Hai 6, 2013

In the last 8 years or so,  I spent lots of time looking  at something called “The Universality Phenomenon”  from probability and mathematical physics. Typically, one looks at a lage system consisting of many independent particles (or variables). It has been observed that the outcome of the system, in most cases, do not depend too much on the fine details of the behavior of individual particles.

The most famous “universality” theorem  is the Central Limit Theorem. In its normalized form, it says that if \xi_1, \dots, \xi_n are iid random variables with mean 0 and variance 1, then

S_n:= \frac{\xi_1 +\dots +\xi_n}{\sqrt n} \rightarrow N(0,1), the normal distribution. Notice that this statement does not take into account any other distributional information of the \xi_i (such as third moment, the median, tail decay, etc); thus the  theorem holds for the big class all random variables with mean 0 and variance 1, raging from a very continuous one such as gaussian, to a very discrete one such as Bernoulli (\pm 1 variable).

Universality has been proved, or conjectured, for a  wide class of functions. Instead of taking the sum, one can consider any other functions of the \xi_i (such as higher degree polynomials, for instance). The hard cases are when the functions are defined only implicitly, such as in the case of random matrix theory. Here the \xi_i (or more conveniently \xi_{ij})  are the entries of the matrix, and one may care about the, say, kth largest eigenvalue, or the number of eigenvalues in a  given small disk or interval. True,  these are still well-defined functions of the entries, but there are no way to write them down in a nice, explicit,  form as in the central limit theorem. Terence Tao blog has many detailed discussions on  problems of this type, so I do not labor further here.

Recent years witness  a tremendous progress in this area, and there is an explosion in term of  connections to other areas to mathematics as well.   This recent article of Natalie Wolchover does a very good job of  giving an comprehensive overview. A very recent, and quite fascinating connection has emerged in the work of Keneth Golden and his student Ben Murphy concerning in the melting of polar ice-caps. (Ken has made about 15 trips to both poles for his studies and the penguins on his website look especially cute :=)).

Still, there are so much to be done and so many mysteries remain. For instance, we still very far from understand why roots of the zeta function follow the law of  eigenvalues of a  random matrix (going back to the famous Montgomery-Dyson tea conversation at the IAS). Of course, to study such a thing, one first needs to assume the Riemann hypothesis (that all roots are on the same line). We are nowhere near it, either. But, well, better be prepared.

A small puzzle: Olydzko famously checked the Montgomery-Dyson conjeture by numerically computed the first 10^{20} roots of the zero function (well, not so sure about the exponent 10, but something of this magnitude). But given that computers only have finite precision, how can he be sure that they are indeed on the line ? (In other words: how can we be sure that it is indeed on the line, and not of distance, say, 10^{-100} from it.)

Btw, an even larger prime has been found today…



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7 phản hồi
  1. Minh permalink

    Could you please name me a few big guys who are in pursuit of the Riemann hypothesis?

  2. Manjunath Krishnapur permalink

    Hi Van, nice to see your blog!

    Re your puzzle: If you take a small circle centered on the critical line, if there is a zero off the line inside this circle, then there will be two (by the symmetry about the critical line). Hence, if using the argument principle you find the integral of f’/f around the circle to be 1.003, then you can be pretty sure that the number of zeros inside is 1 and not 2 and hence it must be exactly on the line?

  3. Hùng permalink

    theo em hiểu thì Odlyzko không tính 10^20 không điểm đầu tiên. việc tính N không điểm đầu tiên chỉ làm được cho N khoảng 10^13. cái Odlyzko làm là tính các không điểm trong 1 khoảng nhỏ [T-H, T+H], với H nhỏ hơn nhiều so với T (H = O(\sqrt{T})), kiểu 1 tỷ không điểm quanh độ cao 10^20.

    thứ nữa sai số trong tính toán của Odlyzko là sai số về độ cao, chứ các điểm tính đều trên đường \sigma = 1/2. cụ thể thì các tính toán kiểu này đều trên hàm Hardy Z(t) thực, trong đó Z(t) = |\zeta(1/2 + it)|, và xem xem khi nào thì hàm Z(t) đổi dấu.

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