Not sure why you have to read 3/4 of the article to get to a _link_ to a pdf which _only_ has the _abstract_ of the actual paper:
N. Benjamin Murphy and Kenneth M. Golden* (golden@math.utah.edu), University of
Utah, Department of Mathematics, 155 S 1400 E, Rm. 233, Salt Lake City, UT 84112-0090.
Random Matrices, Spectral Measures, and Composite Media.
"We consider composite media with a broad range of scales, whose
effective properties are important in materials science, biophysics, and
climate modeling. Examples include random resistor networks, polycrystalline media, porous bone, the brine microstructure of sea ice, ocean eddies, melt ponds on the surface of Arctic sea ice, and the polar ice packs themselves. The analytic continuation method provides Stieltjes integral representations for the bulk transport coefficients of such systems, involving spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices or discretizations of continuum systems, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. In this lecture we will discuss various implications and applications of these integral representations. We will also discuss computations of the spectral measures of the operators, as well as statistical measures of their eigenvalues. For example, the effective behavior of composite materials often exhibits large changes associated with transitions in the connectedness or percolation properties of a particular phase. We demonstrate that an onset of connectedness gives rise to striking transitional behavior in the short and long range correlations in the eigenvalues of the associated random matrix. This, in turn, gives rise to transitional behavior in the spectral measures, leading to observed critical behavior in the effective transport properties of the media."
In this lecture we will discuss computations of the spectral measures of this operator which yield effective transport properties, as well as statistical measures of its eigenvalues.
>The data seem haphazardly distributed, and yet neighboring lines repel one another, lending a degree of regularity to their spacing
Wow, that kind of reminds me of the process of evolution in that it seems so random and chaotic at the most microscopic scales but at the macroscopic, you have what seems some semblance of order. The related graph also sprung to mind just how very like organisms repel (less tolerance to inbreeding) but at the same time species breed with like species and only sometimes stray from that directive. What is the pattern that underlies how organisms determine production or conflict with other organisms and can we find universality in it?
I guess it's called "universality" for a reason. I suppose if we look hard enough, we'll see it in more things. I read the article and I'm hoping some brilliant minds out there can dissect musical tastes in the same way. I'd love to see if it could relate to what we find harmonious in music and what we find desynchronous via different phase, frequency and amplitude properties.
> I guess it's called "universality" for a reason.
> I'm hoping some brilliant minds out there can dissect musical tastes
There has to be some reason there are "Top 10" listings for video games, music, art, tv, movies, anime, vacation destinations, toys, interior designs, historical buildings in NYC, et. al.
Certainly there is a great deal of variance in the order and membership of these lists, but you do find a lot in common. Without some underlying pattern or bias, I don't think we'd see this in so many places so consistently.
I am fairly convinced there is something to do with biological efficiency around information theory that drives our aesthetic preferences.
Today I was thinking about how observing the macroscopic is not a neutral process, it involves processing more and more information the further you zoom out. Perhaps there's something about these "zooming out" kinds of processes that resembles the law of large numbers but more broadly?
No, it is a hypothesis I formulated here after reading the article. I did a quick check on google scholar but I didn't hit any result. The more interesting question is, if true, what can you do with this information. Maybe it can be a way to evaluate a complete program or specific heap allocator, as in "how fast does this program reach universality". Maybe this is something very obvious and has been done before, dunno, heap algos are not my area of expertise.
didn't realize this post got traction, it seems like it was HN pooled, I came across this article and related topics after trying to search what would be rigorous and closest to the phenomenon of the unreasonable effectiveness of mathematics by wigner, renormalization groups were the closest that I came across, the reason why the post title doesn't match the story title is likely due to the story being switched to a more detailed article I considered posting, the title is from a quanta video covering universality, linked below
The article has a graphic contrasting a "Random" distribution vs. a "Universal" distribution vs. a "Periodic" distribution. I'm guessing the "Random" distribution is actually a Poisson distribution, as that arises naturally in several cases.
But the big question is, does this "Universal" distribution match up to any well known probability distribution? Or could it be described by a relatively simple probability distribution function?
I think you mean a Poisson process rather than a Poisson distribution. The Poisson distribution is a discrete distribution on the non-negative integers. The Poisson process’s defining characteristic is that the number of points in any interval follows the Poisson distribution.
There have been a large variety of point processes explored in the literature, including some with repulsion properties that give this type of “universality” property. Perhaps unsurprisingly one way to do this is create your point process by taking the eigenvalues of a random matrix, which falls within the class of determinantal point processes [1]. Gibbs point processes are another important class.
Just a layman: the graphic suggested to me that you might take the lines and their deviation from a periodic distribution. The random distribution is clearly further from periodic, the universal one closer. I wondered if there was some threshold that determined random vs. universal.
It's not that a random shuffling of songs doesn't sound random enough, it's that certain reasonable requirements besides randomness don't hold. For example, you'd not want hear the same track twice in a row, even though this is bound to happen in a strictly random shuffling.
Random shuffling of songs usually refers to a randomized ordering of a given set of songs, so the same song can’t occur twice in a row if the set only contains unique items. People don’t usually mean an independent random selection from the set each time.
If the list of songs is random shuffled, you can only hear the same song twice if there is a duplicate or if you've cycled through the whole list. That's why you shuffle lists instead of randomly selecting list elements.
>For example, you'd not want hear the same track twice in a row, even though this is bound to happen in a strictly random shuffling.
Why would it be? A random shuffling of a unique set remains a unique set.
It's only when "next song is picked at random each time from set" which you're bound to hear the same song twice, but that's not a random playlist shuffling (shuffling implies the new set is created at once).
You could think of it as wanting your desire to hear the song again build up to a sufficient level to make it worth a relisten, sort of how a bus driver might want potential passengers to accumulate at a bus stop before picking them up, and therefore delay arrival. Very plausible to me that a good music randomization would have similar statistics if you phrase it right.
Song shuffling has been broken for ages now. It used to work correctly, like shuffling and dealing a deck of cards, only reshuffling and redealing when the entire deck has been dealt (or the user initiates a reshuffle).. Now it's just randomly jumping around a playlist, sometimes playing the same song more than once before all the songs are played once. I have a feeling that money is involved somehow, as with everything else that's been enshittified.
Readit News
N. Benjamin Murphy and Kenneth M. Golden* (golden@math.utah.edu), University of Utah, Department of Mathematics, 155 S 1400 E, Rm. 233, Salt Lake City, UT 84112-0090. Random Matrices, Spectral Measures, and Composite Media.
Quanta is not doing hypey PR research press releases, these are substantive articles about the ongoing work of researchers.
"We consider composite media with a broad range of scales, whose effective properties are important in materials science, biophysics, and climate modeling. Examples include random resistor networks, polycrystalline media, porous bone, the brine microstructure of sea ice, ocean eddies, melt ponds on the surface of Arctic sea ice, and the polar ice packs themselves. The analytic continuation method provides Stieltjes integral representations for the bulk transport coefficients of such systems, involving spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices or discretizations of continuum systems, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. In this lecture we will discuss various implications and applications of these integral representations. We will also discuss computations of the spectral measures of the operators, as well as statistical measures of their eigenvalues. For example, the effective behavior of composite materials often exhibits large changes associated with transitions in the connectedness or percolation properties of a particular phase. We demonstrate that an onset of connectedness gives rise to striking transitional behavior in the short and long range correlations in the eigenvalues of the associated random matrix. This, in turn, gives rise to transitional behavior in the spectral measures, leading to observed critical behavior in the effective transport properties of the media."
In this lecture we will discuss computations of the spectral measures of this operator which yield effective transport properties, as well as statistical measures of its eigenvalues.
So a lecture and not a paper, sadly.
Deleted Comment
Wow, that kind of reminds me of the process of evolution in that it seems so random and chaotic at the most microscopic scales but at the macroscopic, you have what seems some semblance of order. The related graph also sprung to mind just how very like organisms repel (less tolerance to inbreeding) but at the same time species breed with like species and only sometimes stray from that directive. What is the pattern that underlies how organisms determine production or conflict with other organisms and can we find universality in it?
I guess it's called "universality" for a reason. I suppose if we look hard enough, we'll see it in more things. I read the article and I'm hoping some brilliant minds out there can dissect musical tastes in the same way. I'd love to see if it could relate to what we find harmonious in music and what we find desynchronous via different phase, frequency and amplitude properties.
> I'm hoping some brilliant minds out there can dissect musical tastes
There has to be some reason there are "Top 10" listings for video games, music, art, tv, movies, anime, vacation destinations, toys, interior designs, historical buildings in NYC, et. al.
Certainly there is a great deal of variance in the order and membership of these lists, but you do find a lot in common. Without some underlying pattern or bias, I don't think we'd see this in so many places so consistently.
I am fairly convinced there is something to do with biological efficiency around information theory that drives our aesthetic preferences.
- https://www.quantamagazine.org/the-universal-pattern-popping...
- https://www.quantamagazine.org/tag/universality/
- https://en.wikipedia.org/wiki/Universality_class
- https://en.wikipedia.org/wiki/Renormalization_group
-https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...
But the big question is, does this "Universal" distribution match up to any well known probability distribution? Or could it be described by a relatively simple probability distribution function?
There have been a large variety of point processes explored in the literature, including some with repulsion properties that give this type of “universality” property. Perhaps unsurprisingly one way to do this is create your point process by taking the eigenvalues of a random matrix, which falls within the class of determinantal point processes [1]. Gibbs point processes are another important class.
[1] https://en.wikipedia.org/wiki/Determinantal_point_process
I wonder if the semi-random "universality" pattern they talk about in this article aligns more closely with what people want from song shuffling.
Why would it be? A random shuffling of a unique set remains a unique set.
It's only when "next song is picked at random each time from set" which you're bound to hear the same song twice, but that's not a random playlist shuffling (shuffling implies the new set is created at once).