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Elegant six-page proof reveals the emergence of random structure

quantamagazine.org

41–50 of 178 posts

Re: Elegant six-page proof reveals the emergence of random structure

#41

Here's a direct link to the preprint if you want to skip the fluff: https://arxiv.org/abs/2203.17207

Actually, the "fluff" in this case is outstanding. The paper itself doesn't provide much context, but the Quanta piece does a great job at explaining why the result is important.

Re: Elegant six-page proof reveals the emergence of random structure

#43
What level of expertise is this proof accessible to? Could a final year undergraduate understand this perhaps with a bit of help or background reading?

I am talking about reading and understanding it, not necessary validating the correctness.

Re: Elegant six-page proof reveals the emergence of random structure

#44

Here's a direct link to the preprint if you want to skip the fluff: https://arxiv.org/abs/2203.17207

Actually, the "fluff" in this case is outstanding. The paper itself doesn't provide much context, but the Quanta piece does a great job at explaining why the result is important.

I agree its a pleasant read, but like other Quanta articles, it spends too long on the journey and not enough time on the precise details for my liking. There are few symbolic expressions to be found. Others agree, which is why I shared the link.

Plus, calling it 'fluff' is surely the mildest of sassy descriptions, no?

Re: Elegant six-page proof reveals the emergence of random structure

#45

I wish Quanta was a print publication I could subscribe to. Definitely the types of articles I'd like to sit down and read not on a computer.

Here Quanta's editor-in-chief explains his thoughts on a print version [2018]: https://www.publishersweekly.com/pw/by-topic/columns-and-blo...

Re: Elegant six-page proof reveals the emergence of random structure

#46
post #42

> 2 (of a scientific theory or solution to a problem) pleasingly ingenious and simple: the grand unified theory is compact and elegant in mathematical terms. A 6 page proof described as "elegant" must be incredibly dense.

The proof section is 3.5 pages. That's really pretty short, as far as modern math results go. I get your point, though.

Re: Elegant six-page proof reveals the emergence of random structure

#47
I wonder if this has implications for evolution of life. Life is about DNA which is about a graph of nucleoids connecting to each other in specific structures. But DNA must have evolved out of random structures and random mutations. So if there's a threshold at which structures become inevitable, then DNA has a chance to be born?

Re: Elegant six-page proof reveals the emergence of random structure

#48

What level of expertise is this proof accessible to? Could a final year undergraduate understand this perhaps with a bit of help or background reading? I am talking about reading and understanding it, not necessary validating the correctness.

I think so. To be honest not 100% sure but it seems you would need a bit probability theory + understanding what graphs/hypergraphs are. So my first impression is, it is very accessible, which I would also expect from an elegant proof...

Re: Elegant six-page proof reveals the emergence of random structure

#50

What level of expertise is this proof accessible to? Could a final year undergraduate understand this perhaps with a bit of help or background reading? I am talking about reading and understanding it, not necessary validating the correctness.

I would say yes because I am very close to understanding it :) Although I have a math teacher degree, we didn't really learn that much higher level stuff and also it's been twenty years since I touched anything like this. Notation wise I am missing A△B and not much else. For the theorems mentioned, I am reasonably sure if I worked through https://www.ihes.fr/~duminil/publi/2017percolation.pdf I would understand this paper in question.

Edit: ah! I paged through the percolation introduction and it mentions A∆B ∶= (B ∖ A) ∪ (A ∖ B) so that's the symmetric difference between A and B. And it seems the first ten pages is enough, we only need to get to the Russo theorem.

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