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computational_physics
lectures
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2470b43b
Commit
2470b43b
authored
Mar 22, 2021
by
Michael Wimmer
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fix figure link
parent
6972f322
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#58151
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src/proj2-monte-carlo.md
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@@ -100,7 +100,9 @@ is trained on dimensions $\leq 3$.
Let us look at a simple example. Consider two hybercubes in $d$ dimensions. The two hybercubes have side lengths $L$, so their
volume is $L^d$. We now consider that the two hypercubes are shifted slightly with respect to each other, such that their overlap
in every Cartesian coordinate direction is $
\v
arepsilon L$ with $
\v
arepsilon < 1$, as schematically sketched in 2D below:
![
Vanishing overlap in high dimensions
](
highd-overlap.svg
)
![
Vanishing overlap in high dimensions
](
figures/highd-overlap.svg
)
We can now compute the ratio of the overlap volume to the volume of the hybercubes and find
$$
\f
rac{(
\v
arepsilon L)^d}{L^d} =
\v
arepsilon^d
\x
rightarrow{d
\r
ightarrow
\i
nfty} 0!$$
So even for a slight shift, the overlap ratio will decrease exponentially with dimension.
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