Percolation: a Mathematical Phase Transition
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- čas přidán 2. 05. 2024
- -----SOURCES------------------------
Percolation - Béla Bollobás and Oliver Riordan
Cambridge University Press, New York, 2006.
Sixty Years of Percolation - Hugo Duminil-Copin
www.ihes.fr/~duminil/publi/20...
Percolation - Geoffrey Grimmett
volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 1999.
-----NOTES-------------------------
Note at 10:42 - The uniqueness of the infinite cluster is known for the d-dimenional lattice since the works of Aizenman, Kesten and Newman - [Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation (1987)] and Burton and Keane - [Density and uniqueness in percolation (1989)]. It does not hold in general: when the graph in question is a regular tree for example, there are always infinitely many clusters during the supercritical phase.
The two last results shown here are only known for site percolation (in which vertices are open or closed instead of edges) in the triangular lattice, where a scaling limit for the boundaries of critical clusters was proved to exist (more on that in the third note). It is believed that these results are universal, that is, valid in great generality for planar percolation processes near criticality.
The third result is from an appendix by Gábor Pete in the paper [Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome? (2017)] by Ahlberg and Steif. Consider an n by n box, and the event where there exists a left-right crossing of said box. Recall the uniform coupling from the video: intuitively, the result is saying that the point at which this crossing emerges in the uniform coupling is with high probability inside an interval of size n^{-3/4} around 1/2.
The fourth result is saying that the average size of the cluster of the origin (or any other given point) goes to infinity as we let p approach the critical parameter like a specific power of the distance between p and p_c. This power is called a critical exponent. The existence of these exponents was proved by Smirnov and Werner in the paper [Critical exponents for two-dimensional percolation (2001)].
Note at 10:52 - Hugo Duminil-Copin has several major contributions to the study of processes arising in statistical physics, including Bernoulli percolation. Among his works on Ising and Ising-like processes we can cite [Random Currents and Continuity of Ising Model’s Spontaneous Magnetization (2015)] with Aizenman and Sidoravicius and [Sharp phase transition for the random-cluster and Potts models via decision trees (2019)] with Raoufi and Tassion.
Note at 12:38 - In the triangular lattice site percolation, Stanislav Smirnov proved the conformal invariance of crossing probabilities at criticality (see www.unige.ch/~smirnov/papers/... for an overview), which led to the proof of the existence of scaling limits of exploration curves as Schramm-Loewner evolution processes. See [Critical percolation in the plane (2009)] by Smirnov. This provided a deep understanding of the critical phase in the triangular lattice site percolation, which to this day is not extended to the square lattice.
Note at 17:52 - It is not at all obvious that the probability of being connected to infinity is continuous above criticality. This result can be proved in the d-dimenional hypercubic lattices using the uniqueness of the infinite cluster, and more generally it was proved for transitive graphs (intuitively, graphs in which all vertices look the same) by Häggström, Peres and Schonmann in [Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness (1999)].
-----SECTIONS-----------------------
0:00 Introduction
1:37 Definition - Bernoulli Percolation
5:23 Definition - Uniform Coupling
7:56 Exploration - High-Resolution Square Grid
9:40 Exploration - Questions and Kesten's Theorem
10:58 Exploration - Ising Model
11:54 Exploration - Critical Percolation
12:50 Exploration - Three-Dimensional Cubic Lattice and Beyond
14:13 Proof - Theorem Statement
15:14 Proof - Simplifications
16:29 Proof - Definition of Critical Parameter
18:41 Proof - Critical Parameter is Greater Than Zero
20:44 Proof - Duality Definition
21:56 Proof - Critical Parameter is Less Than One
25:16 Proof - Summary and Idea for Kesten's Theorem
26:11 Conclusion
-----CREDITS------------------------
Caio Alves - writing, 3D animation
Aranka Hrušková - writing, clarinet
Vilas Winstein - writing, 2D animation, editing, voice-over
Special thanks to Anisah Awad, Gábor Pete, Jyotsna Sreenivasan, Angie Zavala
This video is an entry in the second Summer of Mathematics Exposition (#SoME2)
The photographs used in this video are licensed under the Creative Commons Attribution-ShareAlike license:
creativecommons.org/licenses/...
Great video! Congratulations on your outstanding work. If I weren't already in love with percolation, your presentation would surely win me over 😉. Hugo Duminil-Copin
I am a simple man, I see percolation, I want coffee, I click like
Absolutely amazing video. I studied chemical engineering in college and I've always found the idea of phase transitions a bit mystifying. How can individual atoms and molecules coordinate to create such different structures on a macroscopic scale with just local interactions? And why does that transition happen so dramatically and suddenly? This is such a great demonstration of how a phase transition can happen even with just a couple relatively simple rules.
This could be adjusted to our voting system! Distracted individuals vs. unified people!
@@Snowflake_tv If applying these sorts of ideas to social graphs, say a communications network, it becomes clear that it would be trivially easy to prevent widespread acceptance of any 'new idea'. It wouldn't be picky, you wouldn't get to choose which new idea would be made to remain isolated, but it would cement the 'status quo' in place so long as the communication network you're modifying remains the primary network through which ideas circulate in the society. It would be a very 'quiet' tyranny and potentially impossible to detect as different from normalcy. By their very nature, the weakest links between people are between those connected to different clusters (highly interconnected groups). In order for an idea to "spread widely" enough through the society, making the bridge between mostly-disconnected groups would be necessary... but one of the easiest things to prevent. Familiar with the 'Kevin Bacon' phenomenon where you can connect anyone in entertainment to Kevin Bacon with very few 'hops' through the graph of shared appearances? To change that from "very few hops" to "extremely many hops" requires only removing a dozen to 20 possible hops.
@@DustinRodriguez1_0 Right! Imagine if a central body came to decide on some 'p' value in times of insubordination in social media, and somehow developed legal mechanisms to force social networks' models to operate below that 'p' value threshold (video recommendations, suggested posts, automatic ads, etc.). It would be kind of like 'social containment'. Having said that, I am sure there are also some less macabre applications in the field of disease control.
@@DustinRodriguez1_0 That's a very interesting point.
@Daniel Fernandez This video reminds me of hydrogen bridge linkage between water (H2O) molecules to resemble the edges in the graph. They form and they break with temperature. Unfortunately however, no snow flakes or typical crystals show up in this rectangular grid. Maybe if the grid was different and used a more hexagonal structure, this would become visible. It would be interesting to repeat this whole animation with other tilings like triangle or hexagons to cover the plane. This means, each node having a degree other than 4.
Congrats on winning one of the #SOME2 prizes!
In the 80's, I made a visualization where the screen is covered in square tiles, where each tile has two quarter circles drawn on it, centered at opposite corners. A random tiling of the two possible tile orientations gives a percolation of 50% which exhibits fractal tendencies.
I shaded the different connected regions in different colors.
Then I added the ability to rotate a tile by clicking on it, causing regions to split and merge.
I tinkered around with making a game out of it, but never completed that. It remained an interesting "toy" in all the experiments though.
This is indeed fascinating both mathematically and aesthetically. Simple rules, complex results.
Congrats to your team to becoming one of the winners of the SoME2!
Congratulations!! This was one of the first videos I saw a few weeks ago and it's fantastic!
you did a great job hooking interest from the initial question. often the first 30 seconds of a video or essay or whathaveyou are what matter most and you knocked it out of the park.
Dude, this is the best explanation for critical phase transition that I have ever seen. You are amazing!
I absolutely love it. Thank you for such a great video!
straight up one of the most interesting math videos around; keep it up!!
Hey, I'm at the university of Geneva and one of the teachers is Hugo Dominil-Copin, it's a shame that I do not have him as a teacher yet but hopefully it will come ! thanks to your video I now have a basic notion of percolations, great job !
This is amazing !! :) Love you all ! You are doing so well
Wow. I just stumbled upon this video. I know you couch a lot of what you said in "this is not necessary for the math that we're going to do" but -- very seriously -- it really helped in understanding what you were showing. This is fantastic. I've actually recommended this to friends that don't give 2 whatevers about it because it just looks so good! This is going to bleed out there even if you don't get credit. Seriously great presentation here, kudos
It never even occurred to me that a video on percolation theory might be posted on CZcams. This video just popped up among my recommendations. Lucky me! Thank you for taking the time and trouble to produce and post this video.
great work! , the visualizations are outstanding, and your explanation was nicely ordered.
I had a percolation assignment in my computer science class. Found the idea cool, and glad I found this video.
What was your assignment ? Curious, have you tried to have computer do an assignment ? One day-- To figure out why primal and dual permutations are impossible ? 25:45 onward.
Congrats on winning the contest!
Super cool video, amazing how you can bring the main Ideas of such a complicated subject across
This is an amazing, topical video. thank you!
This is a really beautiful subject that I hadn't known about before! And very nice proof at the end!
Glad you enjoyed!
@@SpectralCollective non ci credo poco assemblato con una ex prendemmo di tutto ti mando un po' che volevo chiederti un consiglio da darci il tuo indirizzo di spedizione sono un paio che si poteva chiama la canzone non mi piace scrocchia di il tuo luogo e orario per me dicevo ieri al primo anno della facoltà e la mia C'è un sacco e si dovrebbe essere battutacce di tutto ma se tu ti svegli che ci sto a fare una prova con la passione di un benzinaio di che ring e di solito provano che ci siamo detti per le monodose non ci sarò per il capannone diviso fra uffici della pallavolista 🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣
What a fantastically produced video! As a student of probability theory this semester, I adore the topic and cannot wait to learn more. Thank you!
Very succinct and pretty modelling. Thanks for the upload!
Thanks for this insight. It makes a lot clear and most interestingly why Bernoulli is a fascinating mathematician.
Beautiful subject! I admire the way you presented it.
Congrats on the win
Outstanding work! Thank you very much for your videos.
Thank you for watching!
excellent video!!! congratulations for the work!!
Great presentation! I love the visual intuition.
Thank you for the high-quality video!
That was awesome. I've fooled with graph theory and so run across a few simple questions on random graphs, but I'd never even heard of this topic to my memory. In under half an hour you defined it clearly and even gave a taste of the flavor in the topic's proofs.
Thank you!
Wow! Great video on a great topic. Thank you!
Incredible video !
I loved the video. Also this channel seems to be a gem! I'm so glad I found it.
Thanks for this, I’ve had thoughts on phase transitions since heat transfer in college, namely the growth of clusters and how they “compete”
It's time for The Percolator™
Edit: I am reminded of the Physics study of the transition gradient between Laminar and Turbulent flow
Congrats on being a contest winner! It was well-earned. This video was fascinating. It might be a good example to include in a video about emergence, which is one of my favorite subjects.
This channel is wonderful!
Thanks for sharing such an insightful video!
Fascinating subject and so lucidly explained. Just subscribed! Hope you keep making videos!
just beautiful.
Excellent video. Very interesting, informative and worthwhile video.
Really excellent video. Please keep it up!
Really an excellent video, great job!
Truly magnificent stuff here
Work of art!
Fascinating insight on the probabilistic approach of this microscopic phenomenon! Plus a beautiful and crystal clear proof of a math theorem in a CZcams video, which sounds pretty much like a "Truth AND Dare" challenge! Thank you.
What a beautiful proof due to Peierls, and a terrific explanation of it by yourself. Thank you for the last 27 minutes.
Such a brilliant explanation thank you so much !
Thank you for watching!
incredible video!!
Very interesting introduction to this topic. Well made. Thank you.
What a great video, thank you!
Great video! Your explanation is very clear. Nice proof at the end.
Beautiful video!
Amazing video! Animation and voice are great! Do more!
Great video! I have heard about percolation in relationship to this-year Fields Medal but I haven't dive into it. Even though I am not a huge fan of probability theory by the way you presented the topic it seems to be a really fascinating subject. I was quite upset when the video ended because I was so impressed by it. I would love to see more!
Thank you!
Thank you for your work! Statistical Physics is sometimes hard, counter-intuitive and a mess, but it is even harder not to find it beautiful especially when explained so clearly. Kudos on the animation and the overall style of the video
I've thought of this concept before, but I never had a name for it! Cool!
This is brilliant... thank you
Wow! Amazing video! I feel like I learned so much mathematics knowledge from you guys! Keep it up!
Thank you for these kind words!
this is a very elaborative exposition.
11:21: a cool procedural way to produce the infamous DOOM FIREBLU texture!
Wow thank you, great explanation!
Amazing video!! Really accesible explanation, even for people like me who never really liked statistics much
Excellent video!
It's an excellent explanation. And congratulations! for winning the SoME2.
Fascinating video. One of the few in this challenge to have a white background!
Great video, thank you so much!
Awesome. For me, a section that links percolation to some actual scientific phenomenons will add a lot to this video.
Thank you so so so much!
@10:57 when you pronounced Ising I was very confused, I even kind of remember being told that Ising was British.
Nope, the English speaking world has lied to me. He's German, and the i in his name is the proper i sound and not the English ai.
Good on ya for saying it right and correcting me
This has kind of set my mind on fire this evening. I'm thinking of it from a soil permiability in civil/geo tech engingeering point of view.
Permiability (k) for a given material is in [m/s], so wrapped up in that is the P of the material and the length of the path approximating a straight line velocity for say a pond on the surface.
Free draining gravel versus super fine particle clays, high to low P
Congratulations 👏👏👏 🎉🎉🎉
I freaking love all the great math channels i'm discovering through SoME2! Excited to be one of your first 4 thousand subscribers, I'm sure there will be many, many more to come!
Thank you very much! Some time ago I've written a code for generating random graphs(Erdos-Renyi model) and noticed that when I set number of edges big enough, graph always became as one big connected component plus several lone nodes. I don't have enough math background to explain this and even considered my code works wrong, but now I see the same principle of percolation, just on different topology(not a grid).
This is amazing! This is my second year in a bioinformatics undergrad, so Bernoulli appearing was quite a welcomed surprise. During the video a recurring phrase appeared in my mind: above a probability of 1/2, there is a “more likely than not” probability of a given path of length L to be of length L + 1 when the percolating graph is being constructed. I think that idea offers a good intuition as to why there is a non-0 likelihood of an infinite subgraph above a Pc > 1/2. Let me know if that makes sense!
love the clarinet
Brilliant video. I love the background music.
Really nice video. Quite a good estimate of p_c for a 25 minutes video that takes us from 0 to the finish line :)
Gracias por tu video.
Fascinating and very well-made video.
When I look at the animation, it actually looks like there are TWO phase transitions and THREE phases. The 1st phase is just static (as in the random patterns you seen on an analog television that is not tuned to a broadcast station), that is, random at the smallest scale and no structure beyond that smallest scale. The 3rd phase is the one you identified, where the is just one solid block with small, scattered flaws. Between these two, there is a 2nd phase, there are numerous medium-scale blobs, the size of which increase with p. But maybe there is no mathematical distinction between my perceived 1st and 2nd phases. It did look like static over a range of p-values though, not just at p=0.
You’re right that near-critical behavior is qualitatively different than very subcritical or very supercritical behavior. But it’s a bit tougher to pin down exactly what’s going on, and it’s more of a continuous change than the phase transition at p=1/2, where the probability of an infinite cluster jumps from 0 to 1.
Something you might find interesting is that there are other networks where there are three different phases with sharp transitions between them. For instance, in some rapidly growing (technical term: nonamenable) networks (ie not like any finite-dimensional grid), there is a phase with no infinite cluster, then a phase with infinitely many different infinite clusters, and finally a phase with a unique infinite cluster. You can see a hint at this behavior in the network at around 13:30 in the video (although in this case, like all branching trees the final phase turns out to be trivial, there will be infinitely many infinite clusters all the way up until p=1).
Water can exist in solid and liquid form next to each other. The middle of a phase transition is just the middle of a phase transition, not a new phase.
I took a course on advanced statistical physics with a big focus on the Ising model and man, did this bring back some memoties
absolutely fantastic .
Congrats!
Very nice video. These clusters can also be used to significantly speed up statistical simulations of the Ising model near the critical point; this is the Wolff algorithm.
One person in our team, Caio Alves, has given an online course in percolation theory in the past, and the (hopefully) self-contained slides can be found here: sites.google.com/view/caioalves/percolation-spring-2021
Thanks. I'll learn it when time comes. I'm irritated by a dependent tenant, right now, My energy is being distracted from learn something...
Are you guys from Stanford 🤔
@@Snowflake_tv Think how irritated your tenant is by you! Why would anyone sympathize with a landlord?
@@NuisanceMan 🥲I'm still suffering from the same problem. Trust me, I did my best to offer what she has wanted. She just has wanted far more than the price.
Just came back to this video when I remembered how impressive this video was to me. Just Bernoulli percolation per se is already interesting in and of itself, but once he mentioned how this relates to states of matter, I truly had my mind blown!
Beautiful clarinet music.
Wow!. That seemes like something I want to know more about in the future.
Outstanding video. Thanks so much for sharing. I seriously hope you will make more of these if you enjoyed the fun of making and sharing it. I really like three elements of the video most of all.
1. that you spent a good long time cycling through the p value “art”
2. that you did a nice slow long zoom in the fractal demo
3. Wonderful passive classical music behind it.
These are all independent of the actual analysis but they are what make the video pleasant on top of fun and intellectually rewarding. Too many creators discount opportunities to make videos pleasant since that’s not essential to the lecture content. I’m glad you didn’t chose to go the extra mile. My only suggestion of areas to improve is you could have narrowed your p range over time in #1 (above) to from 0.5 +/- X where X shifts from 0.5 a few times (you did that) then drops to 0.4 then 0.3 then 0.25 (you did that) then continues to drop more slowly as it narrows to X approaches 0.01 or so. That way we can get a “zoomed in in time” look at behavior closer and closer to the boundary. This is intended as constructive criticism but please know I love what you did and I’m very thankful you gave us the treasure.
Overall your visualizations were excellent!
I just wrote my bachelors thesis about percolation transition. The visualisation you did to explain the model is very nice. I would not say that the proof for p_c = 1/2 is too overly complicated using the dual lattice.
Do you know how to download the program to visualize? I'd like to modify it into a network of nodes on a sphere.
It surprises me that this is used for a computer program. It looks more like the circuit
Mesmerizing....
Excellent video thanks
Amazing
fascinating!!!
This is a nice video. Thankyou
thank you vilas for bringing your video to my attention. i enjoyed it very much. and it brought to mind a question i had, even from back in my days of physics and math - although it has become more sharp or clear in my mind in recent years: the assumption of independence. within mathematical modelling that is used to simplify the math. and yet, with the physical reality of what guatama called 'dependence co-arising' or what heisenberg called 'the uncertainty principle', that is an assumption that will forever keep the model outside the bounds of the experience of the real material world. have mathematical modelling been done to assume that the 'decison-action' of gate affects that of neighbouring gates? guy from oaxaca.
I'm discovering your channel. I watch this video. I'm subscirbing. No proof is necessary, its just logical. Thank you for this great work!
Whew! I'll be honest, when I clicked I had nooo idea the cluster rabbit fuck hole I had stepped into, which Lord knows I'll never escape from. But I can honestly say that the colored visual representation of transition shown around the 8-10 minute mark of the video were absolutely breathtaking and almost hypnotic. I replayed that visual masterpiece at least 20 times and it was just as satisfying every single time. I slowed the video down and it was even more satisfying. Eventhough 99% of the information in this video flew miles above my head, it was certainly worth the gut wrenching anxiety I felt desperately attempting to grasp the concepts you were reeling out like it was nothing. All things considered, very satisfied with my decision to click. Now whenever I use my bong with a percolator I will think of this video. Thank you for sharing this with everyone as I'm sure it was no easy feat to produce this video. Thumbs all the way up.
👍