What is...the drunken bird constant?

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.
This time.
What is...the drunken bird constant? Or: Coming home, or not…?
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Slides.
www.dtubbenhaue...
TeX files for the presentation.
github.com/dtu...
Thumbnail.
Main discussion.
Section 5.9 from vignette.wikia... (this is Mathematical Constants - S.R. Finch)
mathworld.wolf...
Background material.
www.ime.unicam...
en.wikipedia.o...
math.stackexch...
math.stackexch...
en.wikipedia.o...
people.duke.ed...
mathworld.wolf...
/ random-walk-a-comprehe...
blogs.sas.com/...
mathstrek.blog...
/ the-drunkards-walk-exp...
mathworld.wolf...
www.bragitoff....
math.stackexch...
mathoverflow.n...
• Visualizing Random Wal...
Computer talk.
demonstrations...
demonstrations...
demonstrations...
demonstrations...
demonstrations...
Pictures used.
Picture created using reference.wolf...
A variation of mathematica.st...
Another variation of the same wonderful post
i.pinimg.com/o...
Picture created using reference.wolf...
RUclips and co.
• What is a Random Walk?...
• Why Do Random Walks Ge...
#combinatorics
#dynamics
#mathematics

[@peabrainiac6370]

When looking at the graph near the end of the video, it would've also been interesting to see how close this probability gets to the trivial lower bound of 1/(2d) for larger d - intuitively I'd expect it to get rather close, but with how complicated the integral formula is it's quite cumbersome to try plotting it myself. Nonetheless a neat video though of course ^^'

[@VisualMath]

Interesting question, I don’t know but I would bet that this is known. In any case, I plotted the first 50 integrals versus 1/(2d) and they seem to get arbitrary close. I would guess that they are asymptotically the same for dimension → infinity 🤔