Second channel video: https://youtu.be/KnWK7xYuy00
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"A drunk man will find his way home, but a drunk bird may get lost forever." What is this sentence about?
In 2D, the random walk is "recurrent", i.e. you are guaranteed to go back to where you started; but in 3D, the random walk is "transient", the opposite of "recurrent". In fact, for the 2D case, that also means that you are guaranteed to go to ALL places in the world (the only constraint is, of course, time). [Think about why.]
Markov chains are also an important tool in modelling the real world, and so I feel like this is a good excuse for bringing it up.
At the end, I also compare this phenomenon to Stein's paradox - in both cases, there is a cutoff between 2 and 3 dimensions, and they have similar intuitive explanation - is that a coincidence?
Video chapters:
00:00 Introduction
00:59 Chapter 1: Markov chains
03:20 Chapter 2: Recurrence and transience
10:08 Chapter 3: Back to random walks
Further reading:
Larry Brown’s paper: http://stat.wharton.upenn.edu/~lbrown/Papers/1971b%20Admissible%20estimators,%20recurrent%20diffusions,%20and%20insoluble%20boundary%20value%20problems.pdf
Using electric circuits to prove recurrence / trasience: https://math.dartmouth.edu/~pw/math100w13/mare.pdf
More complicated, but more general proof: https://sites.math.washington.edu/~morrow/336_19/papers19/Legrand.pdf
Actual probability for 3D random walk to come back: https://mathworld.wolfram.com/PolyasRandomWalkConstants.html
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If you want to know more interesting Mathematics, stay tuned for the next video!
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