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

Thanks for watching! These videos do take a lot of work, but I plan to continue participating in the summer of math exposition so probably will keep making about one video per year

Thank you!

Yes, good point! Actually, for a 52 card deck it actually does shift the amount of shuffles by about 52. But that’s a little trickier to prove

@@SpectralCollective Going from 242 shuffles to ~190 is perhaps not that much of an improvement, subjectively speaking ;)

Yes, great point! I wasn’t able to go into detail on that, but I believe that in most situations, the “pathological game” event is *not* the event which causes the distance after 3 or 4 shuffles to be high. What I mean is that the event which gives a witness to the distance being high is one which is much more complicated than anything which is relevant to these games. I don’t have a good source for this, plus I didn’t know exactly how to say it without getting too long-winded, which is why it isn’t in the video, but I think I will add a clarification in the description now. Thanks!

@@SpectralCollective Thank you for your reply. I am sorry, I read it 3 times, but was not able to fully follow the argument. In my understanding, shuffling is the cause, and pathological game is the outcome, so I am not sure what you mean by pathological game causing distance after shuffles to be high. In the second part you argue that the structure that is preserved in the deck after 3-4 shuffles is too complicated and thus effectively imerceptible by an average player (if I understand the argument correctly). I would claim that to be the case only in case a single game is played, in which case any structure can attributed to variance with varying degrees of plausibility. However, in many competitive scenarios a sequence of multiple games is played, and there, minor changes in structure can become apparent. I am more familiar with card clustering, that's why I use it as an example. When magic the gathering decided to host some of their tournaments online a few years back, they noticed a sharp decrease in winrates of their top-rated professional players. While cheating is known to be a factor, it is quite rare it is too easy to detect nowadays. This effect was instead was systematic. What they eventually concluded is that the natural way of shuffling breaks down clusters more than random, resulting in generally higher power of starting hands. While professionals clearly took advantage of this as much as they could within legal bounds, the problem is even simpler than that. They have a natural advantage in playing skill against weaker opponents. However, increase in variance of starting hand power level would dilute that advantage, bringing the game closer to dice rolling than a game of skill. In the end, the game creators decided on a modified random shuffling algorithm to be used online, which (under the hood) randomly generated 2 different hands, and picked the one that is closer to the expected ratio of resources and non-resources.

Oh I see, that is very interesting! This is an indication that the way these players shuffle is somewhat different than the model I presented in the video. In fact, it sounds like what they are doing doesn’t actually count as “shuffling” according to the definition I was working with, since under that definition the distribution of the arrangements needs to approach the uniform distribution, whereas if the players were avoiding clusters then they would be approaching some other distribution. And that distribution should have a high distance from the uniform distribution, since the clusters occur in the uniform distribution. Maybe this also helps address your first concern: the event that clusters occur is one that happens often under the uniform distribution, but rarely under this other distribution, and this gives a witness to the fact that the distance between these distributions is high. It would be very interesting to see exactly what the mechanics of a typical MTG player’s shuffling look like, maybe with some high speed cameras, and to make a more accurate mathematical model for how they shuffle in real life. I would also be curious about a description of the distribution of the arrangements of the deck after they shuffle a few times; for instance, does it converge to some distribution in a reasonable time frame? Or does it sort of bounce around between multiple quasi-stationary distributions? There are lots of questions here, thanks for bringing this to my attention!

Glad you liked it!

In that coupling, each coin individually has the right distribution (ie if you only look at the first coin, you’ll see a p-biased coin, and if you only look at the second coin, you’ll see a q-biased coin). This is the only thing that’s required of a coupling! The point is that the joint behavior of the two random variables can be chosen arbitrarily, as long as each one individually behaves correctly.

Thanks for watching!

Thank you!

Thank you for watching! I’m glad you enjoyed the video and the nails!

Well, according to his paper that I mentioned, it should be about 9

Never thought I'd ever get that answer! Thanks!

I’m glad you found it helpful! In my experience, understanding that distance is one of the more difficult parts about this subject

Yes, it is a Markov chain!

Glad you enjoyed the video!

Thanks! You too...

I’m glad you enjoy them! I am a PhD student studying probability, so I typically make a video about something interesting I’ve learned about within the past year!

Glad to hear it!

Yes there is! And it is very detailed and well-written…

It’s always green because I’m only showing one sample from the uniform coupling

Glad you enjoyed it!

I'm just kidding

4 color theorem applies to connected graphs, while these graphs are never fully connected (except at p=1)

@@nadyanabahi8259, think again, what I want him to color with 4 colors. He colors each connected part in some color. Take each part like that as a «node», and each part it neighbors as an «edge». There you have it, fully connected graph.

It's a clarinet 😉

@@arankahruskova4433 dang and I was so sure lol

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