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Today I came across an interesting question on the mathoverflow: what are the biggest problems in probability theory? In the answers, there is one about self-avoiding walks. And the most famous scientist in this field, as far as I know, is Gordon Slade. And several days ago, I also saw a post of this subject. At that time, I have no idea about this area, so I did not have anything feeling about this and then I just skipped it. Now I think I have realized the importance of this field in the whole probability theory. Thus I have to know something about this, at least getting to know what it is. Here I want to share you with the materials I have collected. (a recent post about the talk given by Professor Slade) (a post about this area) (a post about the winners in icm2010, including this area)

I have noticed this concept before. Since I am just new in Probability field, so you should forgive me that I just noticed this academic area several months ago and did not realize the importance of it. Today I attended the regular colloquium of my department and the speaker, Zbigniew J. Jurek, gave a lecture about The Random Integral Representation Conjecture. In this talk, he mentioned free probability. Moreover, he also joked that free statistics will come into being.

I also fount a useful link about the survey of free probability. I hope it will be useful for you. Terry Tao also have a post about this.

General philosophy of probability theory
Probability is central to science, more than any other part of math. It enters statistics, physics, biology, and even medicine as we will see when and if we discuss tomography. This is the broad view.
There is also a narrow view – one needs to understand it before one can effectively apply it and it has many subtleties. Possibly this is due to the fact that probability, stochasticity, or randomness, may not actually exist! I think it mostly exists in our uncertainty about the world. The real world seems to be deterministic (of course one can never test this hypothesis). It is chaotic and one uses probabilistic models to study it mainly because we don’t know the initial conditions. Einstein said that ”god does not play dice”. My own view is that the world may be deterministic, but I like to think I have free will. I believe that probability should be regarded only as a model of reality.

From the notes of Lawrence A. Shepp

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