The lack of a so-called big problem in probability theory seems to point out suggest the richness of the subject itself. One of the most fascinating subfields is the determination of convergence rate of finite state space Markov chains. Many convergence problem even on finite groups have exhausted current analytic techniques. For instance, intuitions from coupon collector's problem suggests that the random adjacent transposition walk exhibits cutoff in total variation convergence to the uniform measure on the symmetric group, and the upper and lower bounds gap is only a factor of 2. There are many tools one could employ to study such problems, such as representation theory and discretized version of inequalities from PDE theory, which makes the problems solutions very attractive in my opinioncreative.
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Post Made Community Wiki by François G. Dorais♦
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The lack of a so-called big problem in probability theory seems to point out the richness of the subject itself. One of the most fascinating subfields is the determination of convergence rate of finite state space Markov chains. Many convergence problem even on finite groups have exhausted current analytic techniques. For instance, intuitions from coupon collector's problem suggests that the random adjacent transposition walk exhibits cutoff in total variation convergence to the uniform measure on the symmetric group, and the upper and lower bounds gap is only a factor of 2. There are many tools one could employ to study such problems, such as representation theory and discretized version of inequalities from PDE theory, which makes the problems very attractive in my opinion. |
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