Percolation: a Mathematical Phase Transition

[@HUGODUMINIL]

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

[@jonathan3372]

Is this the real Hugo Duminil-Copin?! This is amazing.

[@shashanks.k855]

wooo..w! Great work @SpectralCollective indeed deserving of the celebrity praise, very greatful to the percolation communities communication efforts. Thank you thank you.

[@haabyalexis]

​@@jonathan3372 it seems like he is real, his channel is ten years old, and I doubt anyone would have used his name ten years ago . I do find it amazing , as someone who loves maths (but isn't the best at it ), to stumble upon one of the greatest mathematicians of our time , especially one I've seen in real life before.

[@detritic]

I am a simple man, I see percolation, I want coffee, I click like

[@danielfernandez9929]

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.

[@눈꽃Snowflake]

This could be adjusted to our voting system! Distracted individuals vs. unified people!

[@DustinRodriguez1_0]

@@눈꽃Snowflake 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.

[@maxfierro551]

@@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.

[@HenriFaust]

@@DustinRodriguez1_0 That's a very interesting point.

[@koeniglicher]

@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.

[@RaphaelAndrieux]

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 !

[@gira5759]

Congrats to your team to becoming one of the winners of the SoME2!

[@kevinbyrne4538]

It never even occurred to me that a video on percolation theory might be posted on KZbin. This video just popped up among my recommendations. Lucky me! Thank you for taking the time and trouble to produce and post this video.

[@ourstori1873]

This is indeed fascinating both mathematically and aesthetically. Simple rules, complex results.

[@gaggablagblag9997]

Dude, this is the best explanation for critical phase transition that I have ever seen. You are amazing!

[@amguardia]

KZbin's compression algorithm hates this video.

[@A.R.T.C.R.E.W]

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.

[@QuantumHistorian]

Congrats on winning one of the #SOME2 prizes!

[@bruno_523]

Congratulations!! This was one of the first videos I saw a few weeks ago and it's fantastic!

[@nathanderhake839]

I had a percolation assignment in my computer science class. Found the idea cool, and glad I found this video.

[@joestitz539]

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.

[@nathanderhake839]

​@@joestitz539 My assignment was a bit different: There was an n by n grid with each square randomly colored black or white (with probability p that a square is colored white). A grid is said to have white squares percolate if there is an adjacent path with white squares from the top to the bottom of the grid. As n goes to infinity, the threshold was about p=0.59, which was interesting, and unlike in this video, for some p values (between 0.59 and 1-0.59) it was likely for neither the black squares or the white squares to percolate. Haven't really looked into why primal and dual percolations are 0 probability to have together in this instance.

[@ophiuchus203]

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

[@SpectralCollective]

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

[@눈꽃Snowflake]

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...

[@tbraghavendran]

Are you guys from Stanford 🤔

[@RadicalCaveman]

@@눈꽃Snowflake Think how irritated your tenant is by you! Why would anyone sympathize with a landlord?

[@눈꽃Snowflake]

@@RadicalCaveman 🥲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.

[@JohnDlugosz]

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.

[@williamwazere]

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

[@BartvandenDonk]

Thanks for this insight. It makes a lot clear and most interestingly why Bernoulli is a fascinating mathematician.

[@webx135]

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.

[@j.s.42822]

What a beautiful proof due to Peierls, and a terrific explanation of it by yourself. Thank you for the last 27 minutes.

[@eli0damon]

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.

[@SpectralCollective]

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).

[@thewhitefalcon8539]

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.

[@nuggetx4086]

Congrats on the win

[@johnchessant3012]

This is a really beautiful subject that I hadn't known about before! And very nice proof at the end!

[@SpectralCollective]

Glad you enjoyed!

[@lucacomoda7704]

@@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 🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣🤣

[@simons408]

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.

[@눈꽃Snowflake]

Do you know how to download the program to visualize? I'd like to modify it into a network of nodes on a sphere.

[@brendawilliams8062]

It surprises me that this is used for a computer program. It looks more like the circuit

[@redcoat4348]

Congrats on winning the contest!

[@sirknightartorias68]

Wow!! A whole semester worth of class in a beautiful way put together.

[@SpectralCollective]

Glad you enjoyed it!

[@airazure2050]

Awesome. For me, a section that links percolation to some actual scientific phenomenons will add a lot to this video.

[@zuzaaa1998]

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!

[@SpectralCollective]

Thank you!

[@riccardoformenti4332]

straight up one of the most interesting math videos around; keep it up!!

[@arongil]

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!

[@mathboy8188]

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!

[@kangkanbhakta123]

It's an excellent explanation. And congratulations! for winning the SoME2.

[@shokan7178]

This is amazing !! :) Love you all ! You are doing so well

[@Dark_Souls_3]

Thanks for this, I’ve had thoughts on phase transitions since heat transfer in college, namely the growth of clusters and how they “compete”

[@DoctorSoctopus]

it’s time for the percolator

[@redpepper74]

This was really neat and also very soothing? Love the oboe in the background :)

[@arankahruskova4433]

It's a clarinet 😉

[@redpepper74]

@@arankahruskova4433 dang and I was so sure lol

[@xenonchikmaxxx]

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).

[@Ivan_1791]

I loved the video. Also this channel seems to be a gem! I'm so glad I found it.

[@kushnpush6480]

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.

[@rosshoyt2030]

👍

[@nickgreefpool]

great work! , the visualizations are outstanding, and your explanation was nicely ordered.

[@boblenaif]

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_am_who_I_am_who_I_am]

Absolutely amazing presentation!

[@sword199723]

This channel is wonderful!

[@EPMTUNES]

Fascinating video. One of the few in this challenge to have a white background!

[@forthrightgambitia1032]

Came here for perculation, got a map of the Holy Roman Empire at 12:28.

[@miloszforman6270]

Indeed there was a kind of percolation going on within this empire due to marriages between all these princes (mergers) and divisions due to succession, governed by statistical processes. Nice observation.

[@KiyakChannel]

Beautiful subject! I admire the way you presented it.

[@appidydafoo]

It's time for The Percolator™ Edit: I am reminded of the Physics study of the transition gradient between Laminar and Turbulent flow

[@Saxysellig]

Fascinating insight on the probabilistic approach of this microscopic phenomenon! Plus a beautiful and crystal clear proof of a math theorem in a KZbin video, which sounds pretty much like a "Truth AND Dare" challenge! Thank you.

[@konee0]

Wow! Amazing video! I feel like I learned so much mathematics knowledge from you guys! Keep it up!

[@SpectralCollective]

Thank you for these kind words!

[@mongostylelord]

Outstanding work! Thank you very much for your videos.

[@SpectralCollective]

Thank you for watching!

[@official-obama]

8:45 in most situations, not including varying pressure and supercooling

[@MatildaHinanawi]

I've thought of this concept before, but I never had a name for it! Cool!

[@stevefrayne]

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!

[@A.R.T.C.R.E.W]

Excellent presentation, very easy to follow the rigor in large part due to the elegant visualization techniques used. I wonder if this percolation framework can be used to characterize how the mind characterizes, organizes, and seeks out concepts. For example, one well-recognized benefit of maintaining a daily gratitude practice is that by intentionally seeking out things to be grateful for, your brain is more likely to do so spontaneously during day to day living. There are obviously a lot of very simplifying assumptions here, but suppose it looks like this: say that an infinite cluster past the critical probability corresponds to the spontaneous, subconscious emergence of a concept in the mind (gratitude in this case). Conscious instances of reinforcing the concept could be thought of as incrementing the probability, up to the critical value where it will more often manifest on its own. Great video.

[@actuallyandroid5700]

Very succinct and pretty modelling. Thanks for the upload!

[@idolgin776]

Great presentation! I love the visual intuition.

[@matthewrberning]

absolutely fantastic video, thank you for sharing this

[@elgajd]

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.

[@cineblazer]

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!

[@abdefsdf]

This was AMAZING. Thank you for all the hard work. What a beautiful problem. If you have time it would be amazing to learn about the ising model as well. Thank you again.

[@SpectralCollective]

I do plan to cover the ising model in the (possibly distant) future…

[@juancristi376]

Beautiful video!

[@Krunschy]

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!

[@SpectralCollective]

Glad you enjoyed it!

[@grezamisoit]

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!

[@kech-agmaio8620]

I absolutely love it. Thank you for such a great video!

[@EnergiaRocket]

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.

[@klemensfloge4946]

Super cool video, amazing how you can bring the main Ideas of such a complicated subject across

[@uku5840]

Thanks! Now I finally understand what people mean when they say someone is "off the perc"

[@mattb5816]

Wow, not many mathematics videos get my brain moving like this. When you mentioned it was known that there will never be more than one infinite cluster, that blew my mind. Intuitively (given the square grid), it seems like you could have two or four infinite clusters. I still haven't wrapped my head around why this is true. I've narrowed my intuition down to: what if all edges have a weight of 1? As you increase p, you will always have an infinite grid of unconnected nodes, but the moment you hit 1, all nodes become part of the same graph. Literally as I wrote those last two sentences, I realized where my intuition went wrong. However, when p reaches 1 and all nodes suddenly have to connect (this situation is what made me realize it totally could be just one colour), how do we guarantee they must all be the same colour, and how do we determine which colour dominates?

[@sadkritx6200]

The colours are there for visual understanding and like he said, it's not important for the mathematical proof. The thing that matters is whether when p=1 do all the dots connect or not. I feel like the water analogy works better in this case . As when p=1 water can flow through all the points like a "pipe system". And even if you want to choose a colour, you can always choose the one with the infinite grid because there always exists a single infinite grid after the critical parameter.

[@Anonymous-s7j1y]

Thank you for the high-quality video!

[@prdoyle]

I'd be interested to know where this falls down in the 3D case. Is it that there's no useful definition of a dual grid in 3D?

[@SpectralCollective]

Yes. There is a dual structure but it consists of two-dimensional faces instead of edges. The special thing about the square grid is that the dual structure is the same as the primal one

[@KekusMagnus]

I took a course on advanced statistical physics with a big focus on the Ising model and man, did this bring back some memoties

[@MooImABunny]

@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

[@dbum896]

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!

[@tremkl]

When you first started using a lot of math to prove p =/= 0, I was like “Isn’t it obvious if p = 0, l = 0?” It was really interesting, though, that through the more arbitrary proof you used, you both absolutely proved 1/3 < p < 2/3, and also layer the foundation for the intuition of why p=.5, even if it would take a lot more math to absolutely prove it. Fun aside, I recently watched a Numberphile video where someone talked about a program that they used to generate random mazes with particular properties, and it was clearly built on these same principles. It created the primal graph, and then made the maze out of the duality.

[@alejrandom6592]

13:08 it's obviously just 1/(2^(n-1)) for n dimensions

[@alejrandom6592]

I'm just kidding

[@metrictensor9730]

I am physics grad student and I have joined a group which works on quantum critical phenomenons.. renormalization Group etc which can also be applied to percolation

[@prikarsartam]

this is a very elaborative exposition.

[@adefossez]

Really nice video. Quite a good estimate of p_c for a 25 minutes video that takes us from 0 to the finish line :)

[@ylanhernandezmotte]

Such a brilliant explanation thank you so much !

[@SpectralCollective]

Thank you for watching!

[@ipudisciple]

15:15 I think this argument is easier: The existence of an infinite cluster is unaffected by finite changes, so (Kolmagorov) its probability P_p[∃∞] is 0 or 1. By your independent edge value argument, P_p[∃∞] is weakly monotone. Obviously P_0[∃∞]=0 and P_1[∃∞]=1. Therefore there is a critical probability p_c.

[@atakansaracyakupoglu9930]

Brilliant video. I love the background music.

[@ndmath]

266K+ views, you basically have won math KZbin.

[@pacificll8762]

Incredible video !

[@paulofreitasgomes6389]

excellent video!!! congratulations for the work!!

[@ethos8863]

At about 10:02 and my current thought is that this "phase transition" is just the observation of exponential growth, as the speed at which they merge depends on the amount of them.

[@evanbasnaw]

Grant Sanderson sent me here. Great video. Love the style. :)

[@kameramann7824]

Amazing video!! Really accesible explanation, even for people like me who never really liked statistics much

[@f1f1s]

11:21: a cool procedural way to produce the infamous DOOM FIREBLU texture!

[@taliastocks]

As an electrical engineer, I found your open/closed convention confusing! Loved the video. -Talia

[@crabcrab2024]

Wow! Great video on a great topic. Thank you!

[@SatisfyingWhirlpools]

14:58 in the theorem, there is a probability for p< and p>, but what about p=??? Is there a probability somewhere in the middle that an infinite graph exists?

[@shortnotes-bds2621]

I really wanted to know what Hugo Dominil Copin was working on. This was a great introduction.

[@SpectralCollective]

I’m glad you enjoyed it!

[@GodelsLaw]

Excellent video!

[@riverground]

Wow!. That seemes like something I want to know more about in the future.

[@robertschlesinger1342]

Excellent video. Very interesting, informative and worthwhile video.

[@iwasjason]

Fascinating subject and so lucidly explained. Just subscribed! Hope you keep making videos!

[@cowboycurtis2099]

imagine how understanding this concept and social networking/information could augment one's ability. Lucy!!! You have some percolating to do!! Oh Ricky!

[@_iyalei]

Fascinating subject, i was surprised that the mathematical formalisms needed to prove that are less advanced than i thought. Great presentation, especially the style with the soft music in the background and the visualisations tightly following the argument!

[@1PercentPure]

This is an amazing, topical video. thank you!

[@SpectralCollective]

Thanks for watching!

[@1PercentPure]

@@SpectralCollective here i am a year later rewatching it and realizing that you replied to me : )

[@SirGarthur]

I thought i was going to learn about bongs, but this is even better