Why Do Random Walks Get Lost in 3D?

In this video, we try to gain some intuition for why symmetric random walks are recurrent in 1 and 2D, but transient in 3D. This was proved by mathematician George Pólya in 1921.

Links for further reading:
Awesome demos of random walks by Russell Lyons (https://rdlyons.pages.iu.edu/rw/rw.html)
Analysis of high-dimensional random walks by Gregory Lawler, including expected escape time (http://www.math.uchicago.edu/~lawler/reu1)
Lecture on random walks by Andrej Košmrlj, including some physics and biology connections (https://www.princeton.edu/~akosmrlj/MAE545_S2018/lecture17_slides.pdf)

Special thanks to Ryan Adams, Elmira Amirloo, Alex Beatson, and Yair Shenfeld for feedback on this video.

Some of the animations in this video were created with 3Blue1Brown's manim library (https://github.com/3b1b/manim​).

Music: Trinkets by Vincent Rubinetti

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Видео Why Do Random Walks Get Lost in 3D? канала Ari Seff