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Consider the unit disk $C \subseteq \mathbb R^2$, and imagine the following process:

Repeat $N$ times:

  • Take a strand of spaghetti (the length is fixed!) and randomly place it in the disk so that it forms a smooth curve with the 2 endpoints pined on at points on the circumference $\partial C$.

It's required that as $N \rightarrow \infty$, the distribution of spaghetti material in $C$ becomes uniform in one of the following senses:

  • A) Length: For every subregion $S$ of $C$, the ratio total length within $S$ to the total length within $C$ uniformly approaches the ratio of area of $S$ to area of $C$.

  • B) Particle count: For every subregion $S$ of $C$, the ratio of the total amount of spaghetti material within $S$ to the total amount of spaghetti material within $C$ uniformly approaches the ratio of area of $S$ to area of $C$, where we define the density of spaghetti material at a point to be the number of spaghetti strands passing through that point.

Question

  • How would the sampling be done so as to achieve this?
  • Can one succinctly characterize all the possible sampling procedures that can achieve this ?
  • How about making the convergence to uniformity as rapid as possible.
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    $\begingroup$ I don't understand whether I can control the shape of each spaghetto inside $C$, or whether it sort of hangs independent of my will, only depending on the location of the endpoints. $\endgroup$
    – Ben McKay
    Mar 2, 2017 at 19:00
  • $\begingroup$ @BenMcKay thanks for the quick feedback. For the shape of the spagetti inside $C$, you can do as you want, as long as it remains "smooth" (no kinks which would ruin the spagetti, etc. :) ). $\endgroup$
    – dohmatob
    Mar 2, 2017 at 19:06
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    $\begingroup$ If by "the distribution in $C$ becomes uniform" you require only that for every subregion $S$ of $C$ the ratio (total length within $S$ / total length within $C$) uniformly approaches (area of $S$ / area of $C$), then there is a huge amount of freedom in how you distribute the spaghetti. For example, you could make all spaghetti pieces vertical, and distribute them evenly across the width of the circle. Did you mean to require something more specific? For example, are both ends of each spaghetti piece to be chosen independently, say, from the same distribution? Are the lengths of pieces fixed? $\endgroup$
    – Robin Saunders
    Mar 2, 2017 at 19:32
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    $\begingroup$ I'm also not clear what you mean by "diffusion" in the title. $\endgroup$
    – Robin Saunders
    Mar 2, 2017 at 19:33
  • $\begingroup$ @RobinSaunders I forgot to mention that the spaghettis all have the same fixed length (so you can't just deploy them vertically across the disk, as you obviously won't have uniformity). Yes unformity is to be understood in the sense you stipulate. I'm editing the question to make it clearer $\endgroup$
    – dohmatob
    Mar 3, 2017 at 6:44

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