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Stream: deprecated: physics

Topic: Weird checkerboard result

Eric Forgy (Dec 01 2020 at 21:51):

I was doing some tinkering and found a weird result.

Think of a Feynman checkerboard:

image.png

I wanted to create a series that counted the number of paths connecting $(0,0)$ to $(0,2n)$ by the number of turns, i.e.

s_1 = # of paths with 1 turn
s_2 = # of paths with 2 turns
s_3 = # of paths with 3 turns
...
s_2n = # of paths with 2n turns

After fiddling around with some Julians, we found this is a known series.

n=1: [2]
n=2: [2,2,2]
n=3: [2,4,8,4,2]
n=4: [2,6,18,18,18,6,2]
n=5: [2,8,32,48,72,48,32,8,2]
...

I was curious about finding the "average number of turns" so I converted this series to probabilities by dividing the series by the sum of the series.

The result is pretty weird.

The average number of turns

$\langle Turns(n)\rangle = \sum_{i}^{2n} i*p_i = n$

where

$p_i = \frac{s_i}{\sum s_i}$

is the corresponding normalized term in the series.

It almost seems like this series could possibly alternatively be defined as the symmetric series whose corresponding probabilities give

$\text{"Expectation of 1:2n"} = n.$

I share this just to see if anyone might have some intution to share. It kind of boggles my mind