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Elegant six-page proof reveals the emergence of random structure

quantamagazine.org

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

#111

Post by oh_my_goodness »

_8j50 wrote: post #101

Ok, please ELI5 this for me, The very statement that anything at all is random is completley absurd to me, especially from an academic context. Are they using a definition of random that is equivalent to "nearly impossible to predict"? , I mean, yes, from the perspective of a limited observer random things can exist, but in an absolute sense, for something to be random then even with the knowledge of all things past…

Lemme go through details, not because anyone doesn't know the details, but to get them written down and discussable.

"Random" means I draw up a list of all possible outcomes. The probability of an event is defined as the fraction of outcomes in which that event is true.

Example. Choose two numbers "randomly" between 1 and 3. What's the probability that the two numbers are equal? The list of outcomes is: (1,1),(1,2),(1,3),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3). There are 9 possible outcomes. In 3 outcomes the first and second numbers are equal. So the probability is 3/9 = 1/3.

The math part is just bookkeeping.* The math isn't random, it just uses the rule above. And there is no claim (within the math) that things are or aren't "really random" in the actual world.

*Can be very slippery bookkeeping. People who know what they're doing get wrong answers all the time. But bookkeeping.

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

#112

Post by scoot »

feral wrote: post #86
fjfaase wrote: post #74

A nice 'visual' illustration of the threshold is found in Percolation theory https://en.wikipedia.org/wiki/Percolation_theory where if you have an infinite grid of square rooms with either a wall or a door between each pair of rooms, that there is shift about what is connected when the percentage of door gets just over 50% as is visualized in this animation: https://en.wikipedia.org/wiki/Percolation_theory#/media/Fil…

And, to give an example we've probably all become familiar with, this this is equivalent to how a disease becomes an epidemic as the R value crosses 1.

That seems more like a truism: If on average one case leads to more than one other (R > 1), the number of cases will grow. The R number therefore appears to be nothing more than a measure of growth or decline. What am I missing?

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

#113

Post by siver_john »

_8j50 wrote: post #104
d--b wrote: post #103

Earlier quoted context omitted.

This is mathematics, not physics. You can define stuff that don't "really" exist in the world, but that work for the purpose of calculating stuff. "randomness" has as much right to exist in a mathematical sense as a "point" or a "plane".

How can you prove something that isn't real? How is randomness defined as a concept in mathematics?

As long as you can clearly define your starting axioms you can mathematically prove or disprove anything. The physical example would be to just say, suppose a universe with the same properties of ours exist but where the gravitational constant is twice as large prove whether X can occur in that universe.

As far as how randomness is defined I believe that might be field dependent but this is me getting out of my depth (I've only taken a few courses on combinatorics).

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

#114

Post by pietroppeter »

ccbccccbbcccbb wrote: post #84
fjfaase wrote: post #74

A nice 'visual' illustration of the threshold is found in Percolation theory https://en.wikipedia.org/wiki/Percolation_theory where if you have an infinite grid of square rooms with either a wall or a door between each pair of rooms, that there is shift about what is connected when the percentage of door gets just over 50% as is visualized in this animation: https://en.wikipedia.org/wiki/Percolation_theory#/media/Fil…

I recommend everyone interested in percolation theory to also check https://meltingasphalt.com/interactive/going-critical/ and play with critical thresholds interactively!

Great link, thanks for sharing!

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

#115

Post by ArtWomb »

jchallis wrote: post #35

Jinyoung Park's Path to Math video on IAS is a really wonderful 3 minute story of how she came to study threshold behavior in random discrete structures, just like this work. Highly recommended. https://www.ias.edu/ideas/paths-math-jinyoung-park

Math is one of those fields where there seems to be a clear correlation betwixt youth and creativity. It's just the consequence of having fresh eyes looking at an ancient problem ;)

But experience is clearly necessary due to the shear vastness of the topic. I find history and philosophy of math to be just as engaging and often crucial. Understanding the origins of probability in 17th century gambling houses, for example, sheds enormous light on the subject.

To that end, I think something like "Mathematical Evolutions" might be the best place to start:

"To tell the tale of The Isometric Problem, one must begin by quoting Virgil..."

https://www.maa.org/sites/default/files/pdf/upload_library/2...

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

#116

Post by Ar-Curunir »

urthor wrote: post #71
marai2 wrote: post #69

Earlier quoted context omitted.

Since we regularly get comments on HN about people wanting to study math in some capacity and about the challenges in doing so - I thought this was a really quote from this video: Jinyoung (after having been a secondary school Math teacher for 7 years): "... there was a big obstacle in studying mathematics or pursuing my career in mathematics, which was me, myself. Because I just couldn't stop thinking that oh I'm to…

From the link: "Jinyoung is a first-generation college graduate, and after seven years as a secondary school teacher in South Korea, she went on to earn a mathematics Ph.D. from Rutgers University. Jinyoung’s story demonstrates the importance of role models at all levels of one’s education and the fact that it is never too late to begin anew."

And yet people will shit on DEI efforts. Incorporating more folks into the scientific endeavor helps combat things like stagnation in science precisely because it reduces academic incest.

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

#117

Post by Ar-Curunir »

ThinkBeat wrote: post #105

I notice that most links on physics, astronomy and math cover pre-prints. Would it not be prudent to wait for the final version. Is all the peer reviews and "fact checking" done prior to the preprint?

For a big result like this, enough experts read the paper to catch glaring mistakes. Sort of a de facto peer review

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

#119

Post by raxxorraxor »

fjfaase wrote: post #74

A nice 'visual' illustration of the threshold is found in Percolation theory https://en.wikipedia.org/wiki/Percolation_theory where if you have an infinite grid of square rooms with either a wall or a door between each pair of rooms, that there is shift about what is connected when the percentage of door gets just over 50% as is visualized in this animation: https://en.wikipedia.org/wiki/Percolation_theory#/media/Fil…

Sudden thresholds were just recently relevant with disease transmission and recoverability too.

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

#120

Post by omnicognate »

leephillips wrote: post #118

How can a Hamiltonian cycle be increasing? If you have one and you add an edge, then you no longer have a chain of edges that passes through every vertex exactly once, because two vertices will have two edges.

The chain of edges forming the cycle doesn't have to include all edges in the graph. There just has to exist a set of edges that form a Hamiltonian cycle. Adding further edges doesn't change that (but adding further vertices would).

Edit: Put another way, it's not the Hamiltonian cycle itself that's increasing, it's the property of there existing a Hamiltonian cycle in the graph.

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