Timeline for A walk on a $2D$ Poisson process in which every step goes to the nearest unvisited point: expected distance from origin after $365$ steps?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| S Feb 14 at 13:19 | history | bounty ended | Dan | ||
| S Feb 14 at 13:19 | history | notice removed | Dan | ||
| Feb 14 at 13:19 | vote | accept | Dan | ||
| Feb 9 at 21:07 | comment | added | Dan | @EricTowers Nearest to my uncle. | |
| Feb 9 at 19:29 | comment | added | Eric Towers | Bob Dobbs is commenting on the ambiguity in "nearest unvisited village". Nearest to your uncle or nearest to you (presumably, the starting village)? | |
| Feb 9 at 5:51 | answer | added | mjqxxxx | timeline score: 5 | |
| Feb 9 at 0:16 | comment | added | Dan | @mjqxxxx Thanks, I have edited. | |
| Feb 9 at 0:16 | history | edited | Dan | CC BY-SA 4.0 |
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| Feb 8 at 19:08 | comment | added | mjqxxxx | And as @achillehui says, if you slightly modify the process so that each step must be within an angle $2\pi / 3$ of the previous one, then you find $E'(N) \approx 0.69 \sqrt{N}$ instead... an increase of around $60\%$. | |
| Feb 8 at 16:39 | comment | added | mjqxxxx | Your derivation of the expected distance for fixed-size steps isn't quite correct. You should be finding $E(N) \approx \sqrt{\pi N} /4 \approx 0.44311 \sqrt{N}$ when the step size is $1/2$. | |
| S Feb 7 at 7:20 | history | bounty started | Dan | ||
| S Feb 7 at 7:20 | history | notice added | Dan | Draw attention | |
| Feb 6 at 8:33 | comment | added | achille hui | IMHO, someone should do a simulation and verify as number of day $n$ changes, the average distance of uncle from home do scale like $\sqrt{\frac{n}{\pi\alpha}}$ for some constant $\alpha \in (0,1)$. Personally, I bet $\alpha$ should be somewhat between $0.4 - 0.6$. The first estimate corresponds to a restrict random walk where at each step, the direction of movement is at an angle at most $120^\circ$ from previous movement. The second estimate corresponds to the case in each movement of a typical distance $r$, your uncle cannot revisit a circle of same radius at the current position. | |
| Feb 6 at 1:45 | history | edited | Dan | CC BY-SA 4.0 |
edited title
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| Feb 5 at 9:30 | comment | added | Dan | @quasi Good point. I have editted. | |
| Feb 5 at 9:30 | history | edited | Dan | CC BY-SA 4.0 |
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| Feb 5 at 9:22 | comment | added | Bob Dobbs | He will make a spiral with radius R. | |
| Feb 5 at 9:05 | comment | added | quasi | If the country is unbounded, what does it mean for the villages to be uniformly distributed? | |
| Feb 5 at 7:59 | history | edited | Dan | CC BY-SA 4.0 |
edited title
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| Feb 5 at 7:19 | history | asked | Dan | CC BY-SA 4.0 |