We're here at the Random McCoy Well today for another special announcement. We have created what is called Fuel of the Feud. Josh Martin is a local business. We're here to support this local business. We think using the water out of the well is a great way to honor our heritage. You know with the Half of McCoy Feud you have violence and vengeance and death and a lot of negative things. In the last 20 years I have in reunions the families have tried to take that negative and do something positive with it, show the unity of the families and this is just another way of doing it. Fuel of the Feud as he said is made out of the water here from the well at the McCoy site. It's a West Virginia corn and a Kentucky rye mix. The well keeps a constant 18 to 24 feet of water on a constant basis. We have the water tested that passed the test. Naturally when we take it to the distillery it goes through the process again of being purified. You know with Bob's well, you get all the minerals and stuff, you know there's no chlorine in the water, it's a high quality water, you know it's like our well's limestone, Bob's the sandstone, it's the nice filtration as it comes into the well. It's something that I've always wanted to do, it adds to tourism. It's got a nice little twist to it. He always had this dream of using the water from the well to make a quality authentic heritage product, a moonshine that was done the way we used to do it. This is not the first moonshine made out of this well, I promise. I was driving down the road and I thought what was things with the Feud and everything else that kept things going and everybody knows that alcohol kind of fuels bad tensions towards people and everything else. I was always told a jar of moonshine, it had three lovens or three fightings and it was your choice, so I guess they chose the fighting part. It helped, it helped to have a little liquid courage when you had to go out and face the enemy, so it's not the cause of the Feud but I'm sure it helped. The half-hills were great moonshiners, they were the big moonshiners back in the day of the Feud. We used the water from a source that had made moonshine in the past, we all know that and now we're doing it in the future and trying to you know give everybody a taste of our area and our heritage basically. It's about as original as a recipe as you can get.


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Thank you very much. So this morning we've heard the DMA speak about the E-sync model and the E-sync model is one of the paradigmatic models in statistical mechanics and there's beautiful work on the E-sync model not just in two dimensions but in various other settings as well. What I will be now be talking about is another paradigmatic model in statistical physics and that's called percolation. And who of you have heard about percolation seen a course of it or okay who has not? That's pretty good. I will still give the introduction so don't worry but at least I know that there's something that I can rely on. Now percolation is best known when you do it on an infinite lattice. So what is the idea? You have a lattice, you have edges, I'll just stick to a bond percolation so you have lots of edges and you remove them independently with fixed probability. That's the simplest setting. And then you can imagine that when you erase almost all the edges then basically nothing remains so in particular you're not going to have an infinite component after erasing all the edges at random. But when you keep lots of edges then actually you will have an infinite component remaining and often this infinite component is unique. Now you can sort of feel that this process is monotone. If you keep on removing more and more your clusters are going to become smaller and smaller or if you're adding more and more edges your clusters are going to become bigger and bigger and therefore there has to be some critical value at which you see something peculiar happening namely above it you will see an infinite component and below it you will not. So that's percolation on an infinite graph. So what you should think about is zd, z2, z3, z4 or whatever. What you have is that there exists a unique pc which of course depends on the graph and below it you're not going to have an infinite component above it you're going to have an infinite component and close to it is where all the fun happens. So that's basically what Dima was saying today but then for the easing model where you also have a critical value and sort of everything that is interesting in the easing model is happening very close by this critical point. Okay so above infinite component below none. I will be talking about a slightly different setting and this is the setting in which our graph is not infinite but it's finite and then this this beautiful picture of this uniquely identifiable phase transition collapses completely. So it's not at all clear how you should define what the critical value is because you know the cluster is going to be finite at every point of p even when p is 1 because you start with the finite graph. So what my main message will be today is that in many settings you can actually identify what critical behavior means but you have to be a bit more careful and also this critical value is not uniquely defined. There's a large range of values for which the behavior is roughly the same that's the main message and we'll discuss this on high-dimensional torii. The main example of course is the hypercube which is an example of a finite graph and of which it's not entirely clear how many of its edges you should remove in order to get something that is somewhat connected but not completely connected. So this is work with Asaf Nachmias which was published in January this year. It was completed in 2012. Let that also be a lesson for PhD students. It can take a long time before your paper finally appears and it's building on joint work with lots of people including a former PhD student of mine, Markus Heidenreich and there's also some other PhD student coming up later later on. My postdoc, boss Gordon Slade, Christian Borges and Jennifer Chase whom I've met when I was a postdoc at Microsoft Research and Joel Spencer. All right so what are we talking about? Well first of all we're talking about the hypercube and the hypercube is very easy to imagine in three dimensions where just the corners of the cube and you connect the corners to one another but if you're thinking about a hypercube in 76 dimensions then it's a bit more difficult to imagine and actually our dimension is going to go to infinity because we want to have large graphs, right? So how do you define this? Well the vertices are just collections of zeros and ones. Think of these as the corners of a high-dimensional cube and you connect two corners with one another when they're actually differing by precisely one coordinate. If you think about what a cube is in three dimensions this is precisely what it is, okay? But this definition holds in any dimension. Now this is a finite graph, the volume is 2 to the power n, that's the number of elements in it and the degree of every vertex is n, that's the number of neighbors that every vertex has, okay? So n on the one hand is the dimension but n also describes the size of the graph as well as the degree of every vertex in the graph and it's a very nice graph because it's completely transitive, every vertex looks the same, etc etc. So percolation is described here, we look at bonds and we make them independently, there or not there, occupied or vacant and it's occupied with some probability p that we will later have to choose appropriately and vacant with a probability 1 minus p, independently of everything else. So I think of the vacant edges as being completely removed from my graph, that means that I have a random subgraph and I'm only allowed to walk along the edges that are remaining, okay? Now this model was first studied by Erich and Spencer, two very big names in random graph theory in 1979 and what they looked at was the connectivity transition. So how large should p be in order for this, for percolation on this graph to be completely connected so that even along the edges that are kept you can still walk from any vertex to any other vertex and what you see is that p actually has to be pretty big, it's basically one half, right? So let me make a little table here about what we know about the critical value because that's what this talk is all going to be about, I mean how to quantify critical values and here we're thinking about the hypercube which I'll denote by t2n, the two-dimensional, oh sorry, the hyper, the torus with side length 2 in dimension n, the reason for introducing this is that later we will generalize this and then we know that the critical value for connectivity is roughly a half and that's Erich and Spencer 79, okay. Now what we would like to study and that's actually what Erich and Spencer proposed in their paper is the connectivity, not the connectivity threshold but the occurrence of a large cluster in this graph when p varies and we will see that p has to be much smaller for that. Yeah. Is there any headband argument why it must be a one half? Yes there is. So what you should look at is the number of isolated points so they're two to the power n points and the probability for a point to be isolated is one half to the power n so the expected number of isolated points at this point is precisely one and this is a slight generalization and then you see that this mean is roughly correct it's a possible approximation. Good question. All right. Now one of the things we see is that the hypercube it's a high-dimensional graph where vertices have high degree right but it has a lot of geometry that actually makes it difficult so rather than studying the hypercube you could also study the complete graph which is sort of somewhat similar it has size little n and all the edges are there so there's no geometry whatsoever every two vertices are a neighbor and now I can do precisely the same thing namely keep each edge with probability p independently of all the other ones and then see what what happens to the connectivity structure of this graph. Now this is a very old problem this is called the Erdos-Renyi random graph sometimes called the binomial Erdos-Renyi random graph was actually invented by Gilbert around the same time that Erdos-Renyi started working on the problem and what Erdos-Renyi did was display a double jump phenomenon and the double jump is that if you take p to be of this form where bear in mind this epsilon could also be negative think of epsilon as being relatively small but in the first case fixed if epsilon is negative then the largest connected component is basically logarithmic in the size quite small dust whereas if epsilon is positive then actually it contains a positive proportion of the number of vertices so it's huge it's like the giant component in infinite graphs right the giant component also contains a positive proportion of the vertices and here that's said the same thing is true so it basically says that the the critical value should be somewhere around 1 over n now what you also know is that at the point 1 over n there's something special happening and there the cluster sizes are of the order n to the power two-thirds so you have anomalous behavior intricate scaling relations critical exponents okay this is interesting so basically what we see is that for the complete graph we get that pc is roughly 1 over n but there's a little bit more because you can study for which values of p you basically have the same scaling as for p precisely 1 over n okay you can imagine that if you change p just a little bit then you're going to add very few edges so actually you're not going to change the critical components too much if you change it by too much then all of a sudden the whole thing is going to clump together and you're going to have something which is which is a lot bigger the precise amount that is allowed is something of the order a constant over n to the power one-third that you can multiply your probability with that's the same thing as saying that you can add a little bit which is lambda divided by n to the power four-thirds i have a reason for writing it this way okay so this is sometimes called the critical window now what we see in this critical window and there's lots of beautiful results here i'm just scratching the surface what we know is that the largest connected component there is of the order n to the power two-thirds but actually you have many components that are that large in fact if you multiply by n to the power minus two-thirds so you renormalize then actually this vector will converge in any topology that you would like and the limits will be proper random variables so you have really beautiful intricate critical behavior with funny critical exponents here i would like to mention two the n to the power two-thirds so you have a two-third there and you have a one-third there okay so that's precisely the kind of results that i'm after but then in more general settings now actually you can show that these these the scaling window really is a unique scaling window if you take your p to be slightly different then actually you're you're sort of subcritical or supercritical in the sense that only in this regime your critical exponents your critical components are random even after taking the scaling limit and the first and the second are of similar order of magnitude so if you approach this this critical value one over n more slowly than like n to the power n to the power minus one-third then you'll actually either be supercritical when the thing is is positive or subcritical when the thing is negative okay and that can be quantified in the following sense that if epsilon is much larger than n to the power minus one-third and p is one plus epsilon over n then the critical x the the largest connected component is basically two times epsilon times n which is substantial and that actually is concentrated and the second largest component is much smaller so there you have a unique largest component whereas in the subcritical regime same setting but now epsilon much smaller than minus n to the power minus one-third so you're away from the critical window but on the left hand side then actually your your largest connected component is roughly of the order one over epsilon squared times the log of epsilon cube times n and it turns out that the second largest is equally big and the third largest is also equally big and the fourth largest is also equally big so you basically have a conglomeration of many clusters that basically have the same order of magnitude in the subcritical regime and they're really much smaller than what happens here in the supercritical regime so somehow what you should think about is that you have the scaling window where you see critical behavior this is a rough proxy for subcritical and supercritical behavior but actually it connects up nicely and continuously as soon as you're outside of the scaling window you're either subcritical or supercritical that's the picture now we would like to establish such a picture in more general settings in particular for the hypercube all right now of course this this question of Erdogan-Spencer was picked up sometime later the first paper on it is 1982 and in the paper of 1982 by Aytai Komlos and Szemeredi they basically did something very similar to what Erdogan and Renyi did for the for the complete graph namely you take p to be a constant over the degree so 1 plus epsilon divided by n or a little n you should really think of as being the degree and then you see different behavior for the settings where epsilon is strictly positive but fixed and epsilon strictly negative and fixed but here the change is rather dramatic because if epsilon is positive then the size of the giant component is basically 2 to the power n which is rather substantial whereas it's it is going to be much smaller here that's not so much visible here yet but in the next result by Bologbash Kohaiakawa and Butchak which is 10 years later they really managed to show something about how small this object here becomes but these are pretty detailed estimates where they're showing that if epsilon is sufficiently negative the largest connected component is basically 2 over epsilon square times the log of 2 to the power n whereas if epsilon is sufficiently positive something like 1 over n then actually it's roughly epsilon times 2 to the power n so bear in mind these bounds are polynomial in n whereas these this number here is exponential so you basically go from something that's polynomial here to something that is exponential here that's a pretty big jump so you could say well we aren't really close to the critical value yet but then it immediately raises the question again what is critical all right so let me write down what these what we these results are so itai Komlos and Semmeradi 82 they basically say that pc is 1 over n times 1 plus small over 1 Bologbash Kohaiakawa and Butchak in 92 they say that pc is 1 over n times 1 plus capital O of check well something polynomial i think the worst thing is log n squared over the square root of n but on the other side it's it's slightly different so it's small but it's not that small and what you see is that you really jump enormously by going from one side to the other side so i would say that these results are still in the barely sub and barely supercritical regime but of course the question is how to define what is critical and in their paper they actually raised the question whether the critical value is equal to 1 over n minus 1 but that may look a bit weird but it's not so strange the reason why you have this is that on any graph any transitive graph let's say an infinite transitive graph you can show that pc is always greater than or equal to 1 over the degree minus 1 there's a branching process comparison that you can do all right you have in total the degree number of edges that you might use but in order to get to a vertex you have to eat up one so there's only the degree minus one left and you know you better be able to continue otherwise you're going to be dead right so this you should really think of as one over the degree minus one so that that raises a question the question is is this true now we were told of this story when i was doing a visit at Microsoft Research by Joel Spencer and he said you know with the results on high-level acceleration that have been established at that point in time can you say something a bit more precise and in fact he came to us with a question you know if you take p to be 1 over n plus something over n squared say can you identify what that something should be now we started thinking about this as a as a mathematical physics problem and we were trying to figure out what should be critical behavior so we basically didn't really answer the question of Joel at that point in time but we rather started thinking about what critical should mean in this setting so let's look again here what is special here well here the largest connected component is n to the part two-thirds but that's a random variable it's very difficult to say pc should satisfy that the largest connected component is n to the part two-thirds because it's random so rather we take this criterion expected cluster size is the number of vertices raised to the power of third expected cluster size is is a continuous function of p so you could take that as a definition then of course you still have to prove that it's a proper definition of the critical value but this is what was done in a series of papers with borges chase myself uh Gordon Slate and uh Joel Spencer sometimes these papers are referred to as the bitches papers for obvious reasons um meaning the acronym um and what we do was that we just take the expected cluster size as a function of p it's a continuous function it will grow from zero for p is equal to zero to the volume of the graph two to the power n for p is one and we just take the point where this expected cluster size is equal to two to the power n over three of course this seems like a definition that I just pull out of my hat and that may not make any sense to you right so what I'll argue in the remainder of the talk that this is actually a relevant definition and it's probably the right definition all right so that means that the challenge is that we have to prove that the the critical value that is defined in this way really is a critical value and one way of doing that is by investigating the the structure of connected components above and below this critical value all right so this is the first result and this you should really think of in terms of what it was for the for the Irish Renewable Engram which really serves as the inspiration for our results so let me go through it slowly I take p to be the pc that that was defined here this pc and I multiply by one plus epsilon and either epsilon can be positive or negative and I'm trying to figure out which epsilon will give me something critical now in the first result you should think of this as being the subcritical case I'm taking my epsilon to be negative and much smaller than the inverse of the cube root of the graph so that actually sounds that I'm in this regime but then with a lambda that slowly tends to infinity and here I should really think of this as corresponding to the volume of the graph not the degree whereas this I should think of the degree of the graph okay and what we see then is behavior that is very similar to what what we see on the on the Irish Renewable Engram in particular the maximum cluster size is of the order 2 over epsilon squared times the log of epsilon cubes times 2 to the power n and this logarithmic term really is substantial because that's precisely what this is saying and there should be an absolute value there by the way which I didn't write down okay so the left hand side is it is there a log of something you see no there isn't so in in these results that actually is not good the lower bound is not good but this was later improved by Asaf Naghmias and Tim Hulshof and they can prove that that the same thing is true but then with a constant times this not with a constant 2 still an improvement but it's it's not there yet but that you should really think of that as being the order of magnitude so this corresponds to the cluster sizes on the Irish Renewable Engram in the subcritical regime the barely subcritical regime now on the other hand I can also look at what happens within this critical window where bear in mind we should take this epsilon to be of the order 1 over the volume of the graph to the power one-third so in this case that's 2 to the power minus n over 3 and then what we see is that the maximum component is of size 2 to the power 2 and over 3 both in terms of a lower bound and in terms of an upper bound that's precisely the same as it was on the Irish Renewable Engram where you should think of this 2 to the power minus n over 3 as being the volume of the graph which is little n for the Irish Renewable Engram okay yes you don't write anything about high or P in this window why is it is it clear that this is the window it remains of the same order no you don't write anything about the mean where the mean size of the cluster oh it's it's it's 2 to the power 2 and over 3 yeah right yeah the lord the expectation of c max can also be bounded from above and below by constant times 2 to the power 2 and over 3 yeah yeah what is the omega oh so this basically says that i do this for a regime of little lambda it's actually a capital lambda where where this lambda in absolute value is bounded by some constant that's what it means oh this is a large constant right so you should think in you should think that c max multiplied by 2 to the power 2 and over 3 converges in distribution to a proper random variable that has no atoms at 0 and infinity now we can't quite prove that but this sort of looks like it right it means that if i multiply this by 2 to the power 2 and over 3 then it's a tight sequence of random variables because the probability that will be larger than some large value apps omega will be small and also one over it will be tight because the probability that one over it is larger than omega is also pretty small yeah so it's a tightness statement good question any other questions all right so i started talking about the the hypercube but as it turns out there's nothing special about the hypercube but i also think about other high-dimensional tori i mean in the on the hypercube you take zero one and erase it to the power n but what would happen if you would take zero one two to the power or if you would take a large cube and raise it to a fixed power then you get high-dimensional percolation on a torus we are in high-dimensions because for the hypercube we take the dimension to infinity so okay now we actually address that question as well our results are are less sharp there but we basically do the same thing so we have a torus here the width is denoted by little r and the dimension is denoted by little d and then we do precisely the same thing as what we do on the hypercube namely we say that the expected cluster size is equal to the volume to the power of third ignore labdat just take it one actually labdat plays an important role in the proof which is why it occurs here but for all practical purposes you can forget about it and then we need some peculiar condition which sort of seems to come falling out of the sky and if we have that that assumption then actually the results that I was referring to here with appropriate modifications because the volume of the graph is changing carry through now this condition looks very weird it's called a triangle condition but it's not that weird because we actually know that if you do percolation on zd the infinite zd then this triangle condition plays a very important role there it actually quantifies when the percolation phase transition is mean field meaning when the percolation phase transition on zd is very similar to the percolation phase transition on a tree which does not have any geometry or when it's the same as the phase transition of a branching process and if you want to include geometry when it's the same as the critical behavior of a branching random walk you know so this condition has appeared before but then on infinite graphs it was invented by Eisenman and Newman and it has become one of the key properties that signals high dimensionality of percolation on a graph and this is an adaptation to finite graphs because on a finite graph of course this this triangle will always be finite but we want it to be uniformly finite that's what it says now we actually do know this condition after a lot of work for percolation on the nearest neighbor torus for the dimension large but fixed think of d being 25 and r tending to infinity then you would believe that percolation on this large torus should be very close to critical percolation on zd and that's what we'll see later on so that's precisely this setting now you have to be careful here the reason is that i'm actually quantifying that this pc satisfies this funny equation right now if i think about percolation on a high dimensional torus which is extremely big i would believe that the critical value ought to be the same roughly the same very close to the critical value in the infinite lattice that's certainly not obvious from these statements it is true but it requires quite some proofs all right now this is a result that only works on the hypercube it's just a result with asaf nachmias photo taken at obrovolvach and it really shows that the the critical window is what we believe it to be from the earlier two theorems because in these earlier two theorems what we were showing is that if you're on the left of the critical window then actually clusters are very small if you're within the critical window then clusters are of size volume to the power two-thirds what we are missing is the statement that above the critical window the cluster is concentrated and it's large and that's what this says it says that above the scaling window the largest connected component is roughly a constant times epsilon times 2 to the power n it's identical as what it was for the error 20 random graph and the second largest component is smaller so the giant component is unique as you would hope it would be and we can also compute the expected cluster size okay so this can be interpreted by saying that the percolation phase transition on the hypercube and on the complete graph are very alike now there's another interpretation that you could give here that is actually quite relevant and that plays an important role in the study of the error 20 random graph and that is a connection to branching processes you can really think of this two epsilon here as being the probability of a Poisson branching process to survive and that probability is the same as somehow taking a vertex uniformly at random in your complete graph and asking its cluster to be large whatever large means that's also that probability is also going to be two epsilon okay and the remarkable statement that you see here is that almost all the vertices that have a large cluster are actually in the same connected component they're all connected to one another that's the statement that's a bit of a surprise is that statement clear okay now we're still left with this question which we haven't which I didn't even write down so there was a question whether pc is actually equal to 1 over n minus 1 and that was a good reason for this so if we think of somehow the tree approximation to the hypercube we would just build a tree in which every vertex has a degree precisely n and the critical value on that tree would be precisely 1 over n minus 1 okay but it turns out that this is not the critical value and it's not even close why not well as it turns out there is something like an asymptotic expansion for the critical value in terms of inverse powers of 1 over n you should think of n as being extremely large right so bear in mind that if I would want to think about 1 over n minus 1 well that's the same thing as 1 over n plus 1 over n squared plus 1 over n cubed plus 1 over n to the power 4 etc adding that sum that geometric series is indefinitely will give me 1 over n minus 1 so this also has this same feature and as it turns out this asymptotic expansion is correct but it turns out that the coefficients are slightly different the first two are the same also both one but the third is seven halves rather than one so that means that the critical value on the hypercube is a little bit bigger than what you would believe it to be on a tree with the same degree that's what it says and that's actually not so surprising because if you think about a hypercube um it really isn't the tree because there's lots of cycles of size four lots of little squares and if you were to approximate it by a tree if you would go two sides of a square you would think of these points as being different but they really are the same so on the tree you're over counting this point by a factor two now of course it's not that likely that you have these but you are going to have many of those so somehow you're if you if you equate this to be the critical value you're underestimating what it really is and this size that this is indeed the case now we cannot compute all these other exponents there are lots of them here we know that they're rational we know that they exist but we don't know what they are we can compute the first three that was for us relevant because we see that the third is different from the one of one over n minus one i'm guessing that the fourth one is 16 but i don't have a proof for this and the proof would probably be very long all right so let me spend a few words on the proof not that that many the proof is fairly complicated the the paper with asaf in the journal of the european mathematical society is 90 pages and that was sort of the six or the seven in a in a row the other ones are somewhat shorter but they're also quite long it's a difficult story it's difficult to make these things precise so how do we do these subcritical proofs well we do this immediately for several graphs at the same time not just for the hypercube but also for these high-dimensional tori and and in fact the proof also works for the complete graph so we take a relatively abstract approach and the abstract approach is very similar to the abstract approach that is used in high dimensions if you want to investigate critical percolation let's say in z25 there we know that the critical behavior is also very much alike the critical behavior of a of a branching process and the tool of the trade there is some is a is a combinatorial identity it's an expansion technique which is called the lace expansion and the lace expansion is a difficult technique but it can be understood and at the end of the slides i will refer to a survey on the topic where we basically do the lace expansion in all detail at an expository level and also describe all the the progress that has been made on high-dimensional percolation including hypercube percolation but also some other topics that's what i would like to say about this proof i think that's enough now i would like to spend a few minutes on the supercritical setting which bear in mind we were only doing for the hypercube in fact the results apply a bit more general and i heard somebody speak here about expanders one example for which it applies is expanders with a sufficiently large girth so percolation on an expander with a sufficiently large girth also has this similar qualitative phase transition where you see the giant component emerge above the scaling window and the scaling window has the same shape as on the other screen now if you want to formulate the precise settings under which uh under which the results apply it's three quantities first of all we need that the degree tends to infinity we're talking about high degrees like the d regular graph with d tending to infinity then there is a condition that relates two objects random walk and percolation which are a little bit weird so m is the degree i take m minus one times pc bear in mind that's roughly this first order approximation to the critical value right if it were a tree then this would be precisely the critical value but we know that generally pc is a bit larger than that we raise that to the power this t mix and that should still be roughly one why this comes out this way is not entirely obvious but let me describe why it is true for the hypercube so for the hypercube we know that pc is one over n plus one over n squared plus seven halves over n cubed plus higher order terms right so if i multiply by n minus one i have to multiply this by n minus one and then i get one plus capital o of one over n squared do the arithmetic it's a third order term that actually deviates from this this is precisely one over n minus one so if it would be this then i would get one but it isn't this because this one is a bit larger so that means that i basically have to blow this up by a factor n and that's going to be my error term okay now it turns out that this mixing time the mixing time of random walks is actually pretty well known mixing time of random walk on a hypercube that's of the order n log n but bear in mind we're not looking here at the mixing time of a normal random walk in fact we're looking at the mixing time of something which is called the non back tracking random walk and that's just the same as a regular random walk it's just that you have to keep track of where you were before and just not step back so you don't track back your last step it turns out that also that mixing time is the same that's actually one half year now if i raise this to the power t mix that's the same thing as raising this to the power t mix and then we get one plus well some constant over n squared to the power n log n and that's indeed one plus one bear in mind here it's crucial that i go all the way up to size one over n squared not one over n that would not have been enough if it would have been one plus a constant over n raised to the power n log n i would be out of business okay so indeed this result applies good yes it's believed and this is an open problem so there are lots of open problems in this big survey it's believed to be borel summable but not summable so similar identities hold for zd and there the coefficients have been non rigorously established by the way the the coefficients there are the same rigorously you should replace n by the degree of that graph which is 2d but then the next term is 16 the next term is 105 it sort of starts growing rather uncomfortably so it probably doesn't converge but this is not known it is believed that they're beryl summable and that would mean that if you divide through by n another n factorial or something like that that would be summable so really the idea of thinking of this as being an infinite sum and ignoring the error not a good idea that will probably start becoming negative for larger than one or whatever if if n is too large good question but now there is speculation on the function of which is an asymptotic expansion sorry i didn't get that actually this is still an asymptotic expansion it's an asymptotic expansion but you have to be careful with any other speculation about the nature of this function poles yeah borel summable all right i mean just concrete nature like where it has poles i mean in the complex plane or no idea you're probably the right person to answer these questions and to ask them i'm certainly not so i wanted to give a bit of the ingredients to the proof it's a few slides somehow trying to highlight what the key ingredients are there's four key steps basically in the proof and in the very first step we're trying to analyze how many vertices we have in large clusters where large is not let's say two epsilon times two to the power n but it's pretty substantial yeah so that's what this random variable measures it measures the number of v for which the connected component in which v is the number of vertices that that contains so that's all that the number of vertices to which little v is connected after doing percolation that's what this object is is greater than or equal to some k where k will be appropriately chosen and it will actually be sufficiently large and this means sufficiently large that actually we know that that number for that choice of k not really has the same order of magnitude as what we believe to be the giant component we call the giant component has size two epsilon two to the power n and now we have something that can certainly not be more than the giant component that basically has the same order of magnitude so what we would still like to prove is that all of these vertices or most of these vertices are in the same connected component that was proof one now the second step is sort of this remarkable connection to the mixing time of non-back tracking walk and that turned out to be a very handy technique that allows you to simplify all sorts of nasty sums that we get in our proofs and just replace them by some nasty functions that depend on the spatial coordinate and just replace them by their by their average so bear in mind if I if I think about anything on the on the complete graph you have complete symmetry so if I look at the probability that two is in the cluster of one that's the same as the probability that three is in the cluster of one four is in the cluster of one five is in the cluster there is complete symmetry now in the hard because that's not true because if I take one vertex the other pole is much further away from it than its immediate neighbors so its immediate neighbors are much more likely to be in the same connected component as this vertex than the other pole now somehow on a qualitative level what this shows is that if you take a percolation path that is sufficiently long then the endpoint of that path is somehow completely mixed it could basically be anywhere so rather than fixing this point I could also replace it by the sum over this point and divide through by the total number of vertices that's very convenient if my spatial functions no longer depend on the spatial variable they all become constant certainly life becomes much easier so that's the role that non-backtracking random book plays so it says something like this if I take an r that is sufficiently large then this probability that that zero say is connected to x in a path that is sufficiently large can be uniformly bounded in little x if x if little r is sufficiently large so for example then think about this and I can just replace the little x by summing out the overall possible x and dividing through by v if all of these are the same then of course summing and dividing by the number of summons is the same as any any of its values right but this clearly depends on x whereas this is completely independent of x so that's a nice trick so that's how non-backtracking random book enters the picture and somehow this bound this here you can think of the the the number of occupied percolation paths of length t mix so this is the expected number of occupied percolation paths of length t mix and that's roughly one so on average if I take a vertex there will be roughly one vertex at distance this t mix and t mix is not that large okay good now the the proof will go by a sprinkling argument where we say we have some p p is p is super critical we think of p as consisting of two little parts a slightly smaller value and then we sprinkle extra edges on it so what we would like to do is to investigate what the chance is that somehow clusters are merging together so what I would like to know is an estimate on the number of vertices that will start creating a cluster when I raise p a little bit now for that it's crucial to know how many boundary edges there are between the the clusters of these two vertices because if they're if they're not if they're touching one another and I increase p then I will make some of these edges if they're sufficiently many I will make some of these edges occupied and then all of a sudden my two vertices are connected to one another hey now we have a much larger cluster right so that's the idea but then of course I need to know first that sufficiently many vertices have sufficiently many closed edges between their clusters and their cluster should be large so that's what this statement says so here we say we have two x two two vertices x and y that could be arbitrary arbitrary vertices now I look at the number of edges between the boundaries of their clusters so this says that the path percolation path between x and u has at most l edges the percolation path between y and u prime has at most l edges and then I'm counting these u v's so if this u v was already occupied then actually the two vertices are in the same connected component if they're close if it's closed and I sprinkle a little bit make it occupied then all of a sudden after that change x and y will be in the same connected component okay so this is the this is counting the number of edges and I want that number of edges to be sufficiently large that's what this says don't don't care about the precise formulation of the constant it just says that the number of closed edges between these two vertices are large and then we call such a pair good and what I would like to know is that there's sufficiently many good pairs because the good pairs after sprinkling will start coagulating very quickly and then they will form a giant component so if I know that there's sufficiently many of them I'm in shape for doing a sprinkling argument and this says that so bear in mind I'm already saying that my clusters are large so I cannot have more than two epsilon times two to the power n v is the volume two to the power n squared of these pairs because I have only that many vertices that are in large clusters but what the statement says is that almost all of those pairs are actually good so almost all the pairs of vertices that have a large cluster actually also have many edges closed edges between them that's a pretty nice statement so then this sets us up for sprinkling and sprinkling basically says that we take two p's a p that is slightly smaller than what we have and then we increase it a little bit and we think about percolation as being obtained by first taking the p1 percolation and then sprinkling extra edges on the closed edges which are these ones with the extra p2 and if I do that in the appropriate way such that this equality holds then if I look at the union of these two percolations then I'll actually get my original percolation model that's how it's set up so we take a p that is slightly smaller we prove that there's lots of large clusters with lots of nice properties etc etc and then we sprinkle some extra edges on it to make the p the right value and then we see that all of these large clutches clusters emerge I will not go through the details but this is this is where it goes and that actually then completes the proof this argument is very alike what itai komlos and samredi did but they use an isoparametric inequality instead so rather than looking at closed edges they looked at large paths and that is wasteful and therefore you lose a lot so therefore you cannot get to the critical value this argument is a bit more robust all right but I've been saying so far that many of these techniques are true are are usable rather generally in high-damp engines now that's what this result says and that's a result with marcus heidenreich who's now in Munich who was my PhD student graduating in 2011 I think it says that if I look at the critical value for percolation on the full lattice that is what I would call the critical value for percolation and if I compare that to the critical value of on the hypercube as defined by the expected cluster size being the volume to the power one-third and their distance is precisely of the order one over the volume to the power one-third so this says that the critical value of percolation on the infinite lattice is inside my scaling window that was defined in a way that completely did not make use of zd as a whole so this immediately implies that all of the results that have been proving so far also apply to critical percolation on a high torus where critical really means the critical value of zd all right that's one result that I wanted to say about high-dimensional percolation the second is the following which is one of the most novel parts on high-dimensional percolation this is joined with another former PhD student Robert Fitzner and for those of you who know something about percolation there was always this discussion about what is high-dimensional so think about nearest neighbor percolation there was an unpublished document by Hara and Slate that was saying that 19 was enough but there's nothing special about 19 except for the fact that it's the smallest integer that is smaller than 20 so they started doing some analysis and at first it was working for d is equal to 48 they say ah that's not quite good enough and then they got it down to something like 25 and at a certain moment it was clear that they weren't able to go to the appropriate dimension which is 7 because of this triangle condition they weren't able to go all the way down and then they said well you know 19 actually sounds a lot less than 20 so let's go to 19 they worked very hard got it to 19 and then they stopped that that proof was never published there are a hundred notes it's about a stack this big which I've gotten from Takashi Hara that was extremely useful and we decided to take this up again because we were thinking that somehow an analysis that would be slightly improved on the basis of non-backtracking random book might give us get us a bit closer to the upper critical dimension and indeed it does we're not able to go all the way down to dimension 7 and I don't think we will be able to go down there ever with our methodology but we can go to dimension 11 so d greater than 10 and actually that methodology does not only give us control over the critical exponents but it also gives us control over the critical value so our estimates are upper bound on pc are provably 1.3 percent off the real value which I think is pretty okay we don't we have no idea what the critical value is in in 11 dimensions but we can give a bound and the bound is roughly 1 over 2d minus 1 except for 1.3 percent okay I'm not going to explain how this is done I'm running out of time here are a few open problems there are many more and if you want to see a really huge list look at this book which is almost done it's on the basis of a summer school that I've taught at CRM in Morial the PIMCRM summer school 24 hours of lectures on high-dimensional percolation that gave rise to this this book of about 270 pages you can download the pdf from my webpage and there's lots of problems there on boundary conditions on you know name it oriented percolation all sorts of other models so they might not be as big as the questions that Dima was posing this morning but there's certainly quite a few that are quite nice and that might get you going now here are a few open problems that I personally find quite interesting so scud studied the the limiting probability of the largest cluster in subcritical phase of a hypercube percolation so bear in mind this is partially resolved by Nachmias and Hulshof they show that this largest subcritical cluster is of the order one over epsilon square times along but they don't get the right constant so the upper bound is two we believe the two to be sharp but we don't have a proof for that it's a nice problem very much related to this and that's actually what what inspired also this question what is the size of the second largest component in the supercritical regime in the air is renewing graph and it's related to the phase transition of branching processes there's a beautiful theorem which is called duality if I have a supercritical branching process I condition it to die out then actually that's a subcritical branching process with an offspring distribution that you can determine exactly on the air is running on a graph that says something like the following suppose I'm in the supercritical regime in the air is running on a graph now I remove the giant component and have a much smaller graph that much smaller graph is a subcritical air is running on a graph of a smaller size and a P that you can compute explicitly at least asymptotically and you can show that really a subcritical so that explains why the second largest component in the air is running on a graph above the critical value is logarithmic in size because it's the same as a subcritical largest component which is logarithmic right so we don't know that here we only have very weak results on the second largest component we prove that it's smaller than the largest but it should actually be there should be a large gap it should even be true that the second largest component decreases with epsilon we have no idea how to do that very nice problem now of course we have a concentrated giant component that immediately raises the question whether there is also a central limit theorem for it that probably is proving it may be harder um within the critical window we don't know that the the critical clusters are actually truly random that they converge in distribution to proper random variables prove that and prove that the scaling limit then also is the same as the scaling limit of the Erdogan random graph as identified beautifully by David Aldous in the 1997 paper it made me hard extended to percolation on the nearest neighbor tours with a d greater than or equal to 11 we haven't done that it may be possible we don't know whether we can do it whether our our numerical control is sufficient for that here are some of the papers thank you so thank you rimco for this very exciting talk so are there any questions is it possible to do numerical simulations in the critical phase on the hypercube or is the space just too big that's an excellent question bear in mind that the width of the critical window is two to the power minus n over three take n a hundred it's nothing how are you ever going to see the difference i mean it's basically one point and we don't know what that point is we don't know what the critical critical value is so i have no idea how to do this would be very interesting i mean we know that the largest connected component is concentrated above the critical point but you can add for example edges one by one yeah and then well you're gonna you're gonna stop whenever the the largest component is of the size you want volume to the power two thirds yeah you could do that but that's not going to be enough to give you for example a proxy of the scaling limit of the largest connected component because all of a sudden now these are deterministic right you would rather do it the other way around fix a p inside your critical scaling window and then see what the largest connected component looked like i don't know how to do that so it's a good question if anybody has an idea i would certainly be very much interested in this so any other questions how do you compute the coefficients in the asymptotic assumption expansion of the critical yeah that's an excellent question the first answer is the lace expansion the lace expansion gives you an implicit equation for pc that you can iteratively use sort of inductively by looking at the coefficients and doing asymptotics on those and they will crunch out all of these coefficients you could do this arbitrarily far so in terms of the upper bound so bear in mind that for this analysis we only needed the upper bound we also have a completely independent proof of this upper bound with the seven halves that does not rely on the lace expansion so that's technically less demanding and that's just computing computing how many points you are going to be connected with somehow if this expected size is is very large then you know that the expected cluster size is quite large and if the expected cluster size is quite large then you're super critical right because that was the definition of the critical point so that's a bit sort of more calculus alike also again using these non back tracking random walk ideas but that's only an upper bound the upper bound typically is the hardest but still we didn't think about proving the lower bound in the same way or proving higher order terms in this way does that answer your question so maybe a soft question for the and can you tell us a bit more about the your personal story when you you were a postdoc at microsoft and you were a young postdoc and you discovered yeah with all the others these so i was a postdoc at microsoft but that was not when i was started this microsoft research has a very interesting visiting program over the summer so after i was a postdoc in 1998 i came back several times and all of this work was done in a summer visit in 2004 and that was actually quite exciting because for my birthday i just got in the book n is a number or something like that about the life of airdush so i was reading in the evenings in this book about airdush and then i would go to microsoft and there i would see bella bolobos joel spencer you know all the people who were listed in that book would be there that was very exciting yeah so we had a lot of fun trying to discuss this the the the people who are doing airdush when you run a graph like problems they are much more combinatorial than we were at that time so we were more from a mathematical physics perspective so we really had to learn a little bit how to speak each other's language and a large part of the time that we spent there was actually spent on precisely that trying to understand what they were saying and they trying to understand what we were saying so for example i've sometimes heard joel spencer say that this is the lace expansion this is certainly an expansion but this is not the lace expansion so there are some language barriers that you need to come across when you when you do this kind of even interdisciplinary projects within math if you're talking with people from a different community then they they use the same words but all of a sudden it means something completely different and people with experience in biology probably recognize that okay so any other question so if not let's thank Remco again and of course all of the other speakers today


personal notice dangerous my stock and trade if the job's too tough for you to handle you got a job for me George Valentine right full details of California on behalf of independent Chevron gas stations and standard stations throughout the West invite you to let George do it friends here is a value that starts paying dividends immediately and keeps on paying in years to come its new rpm motor oil the oil that doubles engine life between major overhauls due to lubrication try it at your independent Chevron gas station or standard station where they say and mean we take better care of your car tonight's story the noose hangs high another adventure of George Valentine dear mr. Valentine I'm jade's but you I'm editor of the local sunny magazine section and come to think of it I've always wanted to run a feature on you that crazy ad but I figured the true revelations of a turkey secret policeman had more family appeal and so I forgot about it anyhow this letter concerns a different matter Valentine I am going to make a report of you you're going to be the author of a serial on the tover emeralds what became of the exotic princess wedding stones did the eminent on ethologist and so fair weather find them or didn't day what was the mysterious curse that brought death to fair weather ten years ago read it the riddle of the two recluses why have the two big brothers refused to cross the threshold of their own gloomy mansion and for that very same ten years what is the mystery of number 37 Ashton Street well well isn't it the greatest story you ever heard well mr. McHugh I noticed you came trotting over here fast enough I tell you it's the best feature we've had for years since the last mommy sued King Turt for alienation of affections what well I always preferred the classic case history of the two headed were wolf now see here both oh come on now McHugh you've been mashing your own corn too long I gather all the purple double talk has something to do with a man named Ansel Fairweather well I looked him up he tied a rope around a chandelier and hung himself at 37 Ashton Street ten years ago ten years ago today in a gloomy brown control your local color he'd been to the Himalayas looking for rare birds once he was probably one himself and he talked about emeralds as strictly for the birds to ones like you well I may have overstressed that aspect a bit just a bit all right forget them look at the facts it took a jury to decide his death was suicide didn't it well there was a very unfair trial of his two cronies yes the Binks brothers and they still live in that same house never stepped out of it since oh we didn't know that I thought not that's the story the rest is all trimmings I'll admit big Jared Binks and little Tom Binks have holed themselves up for ten years in the place where Fairweather committed suicide why it only took two minutes for the jury to exonerate them as murder if that's what you're driving at Fairweather was sick to spawn and out of his mind no no no I'm not trying to say they killed him and to be unfairly accused of his murder might make anyone withdraw from the world a little you think I'm trying to make something out of nothing cook up a story that's a little too well we think you're an expert at it no offensive course not at all and of course you understand Mr. Valentine I would no more seriously think of hiring you even to write your own name then I would jump off the Golden Gate Bridge now that's better now that we get that off our chest what do you really want me for thanks for nine years I've tried to get a little anniversary story on Jared and Tom Binks why they're such reclusive are they really every reporter I ever sent and don't worry they've been clever about it too has been thrown out in his ear to my knowledge the brothers have never been outside that house and no visitor has ever been inside why you get the facts I'll write story you get paid to leave before I see you throw now Brooks here is saddest look you go and check on Fairweather's death would you here's the little one oh and how do you do Sam my name is Valentine how do you do I represent the city fire protection League Department of Fire Inspection Service in the interest of fire prevention I'm sure you won't mind my taking a few minutes of your time for an examination before I what's the matter friend you're not listening well perhaps they have a card here is that girl with you me no I was just asking directions you're not busy young man you're free for the next few hours perhaps well I thank heaven come in come in please do come in I am so glad to see you yes come right in oh Jared Jared where are you we have a visitor isn't that wonderful it's come right in Georgia you sure it's the right somebody's throwing curves now work fast when you angel I'll see you later good luck yes yes hurry Jared this way sir Valentine you say yes yes that's right your Tom Binks oh yes yes quite Tom Binks somehow she have here don't you ever open the shutters oh yes yes forgive the lack of furniture we've never bothered about the hallways and the parlor in them makes such a handy place to store things don't you think we're very informal here brother there's an understatement pile of magazines clothes boxed up books look out this way please I hear what's the matter afraid of a loose chain on the chandelier what well no no not at all yeah sure the chandelier that's what fair weather hung himself from isn't it I thought you said you were from the fire department where he hung himself one of it Mr. Merton what are you doing here young man well I invited him in there there's no reason you might answer the same question friend I thought these guys lived alone yeah the name is Merton Binks boys aren't crazy I am I handle their business affairs Mr. Merton is just on his way out yes you what I mean I've been trying to leave the past three hours now Tom kicks me out oh no no but I think all about that's because you walked in I make one visit a year to this place I spend the rest of the year calming down I don't follow you but I guess that's all right of course you don't who could you don't know it but you've been tapped to take my place holding hands what now now mr. Valentine is interested in our work and I was just going to show him to the house all your son you scare the boogeyman for a while the boogeyman make a good story for you now come on tell the truth what newspaper you from newspaper none but what's the story I don't really care who you are Tom where's my coat oh I'll get it boogeyman that's the story you know it's a haunted house okay buster I'm dumb what's going on so my valuable property but kept in a mess good securities but all gummed up a couple of loony clients that's all look around here nuts lived alone so long they imagine hey what's the matter the light cord mutton look a jerry jerry oh for the love of what well you see another not it's tied in a knot like the others the light cord in the closet here tied in a hangman's not a noose he's found some others around like that well now who's the practical joke of the day that's it none of us ties those knots what is it Tom what is it look look here I didn't do it well of course you didn't and no one did oh Jared this is mr. Valentine I asked him I could hear you from upstairs I could hear I don't think it is good to ask people in there you see young fella I'm gonna card to the club like I have you have finished your business mr. Martin get out Jared now wait I know what happens when people come in but I'm not afraid of any nooses there is nothing for anybody to be afraid of please Jared perhaps you tie the nooses yourself Tom what perhaps you don't know that you do but it's all right there's nothing to be afraid of no no I don't it occurred to me when I saw the shadow on the stairway from this same noose I suppose what shadow stairway and oh no this door was shut oh brother now take it easy Tom look yes you can see it on the wall it's a shadow of a noose this time yeah sure I'm upstairs here only place it could have come from there is nobody else in the house mr.  Valentine broxy everything's funny I was just gonna call you on the world cave then I was gonna take hey Tom brings looks like he fainted or something George for heaven's sake he's crazy they're all crazy the whole place is crazy broxy so am I now listen Angel I was gonna tell you the marquee out here was too thick to be true but I guess I'm wrong I guess I was saved by the bell when you called tell me how everybody sees nooses out here well now I do too yeah Angel there's one made out of rope neatly draped around my neck turn to tonight's adventure of George Valentine in just a moment in the next few seconds I'm going to tell you a short story in sound effects listen that's the artificial wind your car creates when you're driving around 30 miles an hour now that's the sound of tiny particles of road dirt and dust hitting the body of your car and that's the sound of a drop of water on the body of your car after it's been exposed to the Sun's heat for a few hours now those are the three troublemakers in my story wind grime and heat they can literally put an end to your car's finish make it dull and drab instead of slick and shiny the way you want it what to do about these three get dependable auto body protection here's where the hero of the story comes in the car saver at your nearby independent chevron gas station or standard station he has a complete line of protective easy to use body washes cleaners and polishes for your car next time it gets the beauty treatment in these bombing spring days and throughout the year keep your car looking its best with the quality waxes cleaners chrome polishes and finish preservers your car saver stocks ask him about the next time you stop in for chevron supreme gasoline rpm motor oil and car saver service at standard stations and independent chevron gas stations where they say and mean we take better care of your car to tonight's adventure of george valentine the two binks brothers jared and tom two recklessness his friend hung himself ten years ago now the binks brothers are seeing the shadow of a noose light cords and sash cords have been tied with hangman's knots well if your name is george valentine the whole thing looks a little wild to you that is until you yourself get knocked out and come to with a rope around your neck angel i don't know what hit me i tell you george you saw a neck though if i had killed you skipper william all right what did you find well all the ones they've ever found have been in kansas just one or two well you make a lot of sense how it's to be lost somewhere in india i mean someone must have found them yeah only the police don't know who that's right and there's nothing illegal anyway they were found somebody just so angel i love you you're fitting the first piece into the jigsaw only this house out here is such a big barn of a place with so many rooms and so much junk that we that i'll call you back in 15 minutes hello who are you cleaning lady at the booby hatch lady with jokes on i worked in a real booby hatch once come in to bring their week's food for the binks that that's all i'm a long order cook now i must got to eat don't they only say what did you do to tom there nothing he fainted he's all right yeah any a funny one spends all of his time drawing birds looks kind of like a spare what are you doing put him down i know where i'm at i was a nurse once i'm all right no please every time i come to this place something crazy happens um there's a man out back to see you mr binks what's that maple that man who owns the house that fair weather fair weather man who owns any of your business mister don't just go tell Amanda wait why you're losing your job as nurse get out i never trusted anyone mr. valentine i was tried for murder once jerry deny but we hadn't done anything do you know what it's like to be tried sure sure mr banks but i'm only trying to help there's nothing secret about our work all those crates of specimens downstairs birds that's what they are we've collected them from all over the world and i am drawing them jerry deny for 10 years oh we will take much much longer than that we don't work very fast and there are more than seven thousand yes sure i get the idea an answer fair weather was your partner in that work he was an ornithologist i remember so now explain who this other fair weather is who's alive and waiting for you out back a cousin of ansel's he's been trying to get the house maybel said he owned it he has some old deed that ansel gave him he says it's quite valuable property that's why we're moving moving what didn't you know that's why the birds are all created we're being thrown out of the house next week oh why didn't you buy it why'd you let him get it you guys have money don't you or something worth money maybe now see here young man you told me it was some sort of a fire inspector i asked you into the house and all you do is ask questions just like all the rest jerry told me not to talk to people he warned me every time anyone gets it easy take it easy only suppose i go out and talk to the new landlord huh hey all this stuff on the back porch do you suppose it's going to mr fair weather how should i know a couple of blame crack parts i have the house torn down soon as they get out yeah sure go over an inch by inch what's that mr valentine never mind you're a real estate man huh business deal business deal that's all i know i met that lawyer or whatever he is of theirs oh merton sure just saw him down to drug store and he's smart he won't even eat their food he didn't offer to buy the place for them though huh mac a cousin of mine i never met used to live with these bird birds here that's ancient history but a friend of mine discovered this old deed and i found i could get the place so why not so i've got it got a key to i suppose to go in and out anytime 30 rooms three stairs an army could be wandering around trash bro what's eating you anyway just because i won't sell the house to loony and goonie banks well sure they offered to buy it one of it i'm in real estate i got plans for the whole block yeah yeah yeah yeah sure just stop justifying yourself flashlight oh go jump in the lake hey how long do you figure to take to tear this house apart if a man wanted to find some emeralds oh that old story you're just another loony yourself uh well i'm in business friend and i don't have so blame much junk isn't there here ever see a flashlight rigged up like this what's that wire thing found in the trash barrel when you put the button on see the wires twisted around in a circle makes a nice shadow out there in the back wall shadow of a noose jared jared what in the name of jared tom why are you what's happened chandelier the noose he's dead he's dead tom come back here come back here tom oh you crazy idiot who says he's dead huh just as well be i suppose you hung yourself sure oh brother you're not heavy or anything are you sure tom hey anybody okay come on hermit let's get you out of here you need help fast hey you you're the pharmacist yes that's right what do you have get out to that car as fast as you can there's a man been choked hit over the head too what yeah of course uh joe you too come on go on i'll be with you in a minute well hello no well well sit down sit down have a piece of pie no thanks mr merton so he had enough brian around in the booby hatching hermit's hollow once a year is all i can stand sure sure whose taxi is that out there oh waiting for me i got a job back in real life as they call it to worry about see the house from here but nothing happening yet what are you talking about what's the matter come on come on let's wait in the taxi hey what are all those guys doing around that car across the street police will be here in a minute what come on get the kid wait till mac no place jared binks may be dead or alive i don't know he was hanging from the chandelier jared but now look only person strong enough around there to lift him up for it is that woman i guess that mabel get your eyes off of my muscles oh you weren't there i haven't got any muscles unless jared hung himself but that's a fair chance besides i've got a sore neck too i'll look around that rope around he wasn't pulled tight and i don't know what knocked me out but if mabel was telling the truth if she had been a nurse and worked around in asylum well it's one way to quiet somebody in a hurry slap him in the back of the neck healer the hand and what are you guys talking about just keep turned around driver i'd like to know what you're talking about too well why wasn't the rope around me pulled tight why wasn't i killed go ahead watch the house i can see the alien what for oh we're not waiting for mabel to show up that's a police job by now she must be blocks from here so they don't mind it's pretty clear i got the flashlight off her service porch i don't know who else could have been upstairs to throw that shadow of the noose and then duck in a hurry well stop looking so sad mabel's a hired hand that's all a female thug oh brilliant but what i'm not sure of is who she works for hey look there he is there's tom well yeah the little man himself with a suitcase follow right along driver okay yeah i guess jared had all the courage it's a cinch tom couldn't ever face the police again so he's finally running what's that no the guy's walking up there just keep your mind in the front seat driver and that's what the rope was put on me for i guess same reason as the hangman's knots in the shadow of the noose to scare the binks boys only what happened with jared he catch on who was doing it that why he was strung up to put the final friend of the time and make him run oh stop driveling tom hey tom well who is it it's me merton over here tom if there were any emeralds in that house nobody'd ever find them the binks has never talked about them what's been pretty hard on you merton wondering how you can get your hands on you're smart enough to know a gun when you see it i hope hey what if you too shut up oh hello tom hop in well i don't know it's all right tom you can tell me some things like who this guy merton really i told you tom tom get in here get in of course hello merton don't point that gun at me you see merton's been so careful to get himself an alibi taxi already and waited shut up i told you and merton told me he came to see you men once a year to handle your business affairs well i have one of it once a year in april one month after income tax does that make any sense for a business manager all right genius you said enough goodbye well merton give me that suitcase come on a step on it yes no no you get out from behind the wheel and turn around hey what am i i'm leaving you right now get your hands off that get in the suitcase banks it has the emeralds right isn't at the point of the whole business to scare you into running and when you run the one thing you take with you is the emeralds that a man could spend his life searching for in that house that a man could get one at a time you mean once a year and that's who i am i handle a business all right one at a time once a year but never a glimpse of the rest of them well now i've got the whole mess get out of my way you're gonna mess all right buster but look out i'll kill them not in april sweetheart mr valentine yes yes that's all give me the suitcase then tom this guy's safe for the police i want to find how your brother but people will come and they'll bother me there'll be policemen and photographers there's nothing wrong with you having the rest of the emeralds but jarrod and i only use the emeralds to carry on our work now this suitcase you you open it is there anything wrong mr valentine well no not from your point of view friend but in a few minutes i gotta date with miss brooks and a guy named mcqueen at my office i'm gonna find out how they can take a surprise on chevron supreme to give you all eight gasoline qualities you need for dependable all around driving be sure you get not one not two but all eight gasoline qualities area blending mileage power anti knock vapor lock prevention starting warm-up and acceleration ask for chevron supreme at standard stations and independent chevron gas stations where they say and mean we take better care of your car bird feathers you heard me mcqueen skins of birds that's what he carried netsuke uh-huh yeah if tom could take one thing of value from the house that was it after all what are emeralds compared to birds jarrod banks will be all right george good brook see that's fine the police got that woman mabel she had been working for merton birds yeah you see mcqueen merton finally got an itchy palm and ganged up with mabel to try and scare the boys in a running that's the only way they get excited enough to tip where they'd hidden them bird feathers they had over a hundred emeralds and how many did merton sell for them through the years seven what's the matter what are you sure about he wants to know where the rest of them are george but tom and jurid will never tell mr mcqueen we got your story for you what if we didn't find the emeralds it's certainly a better story than the usual vampire scotland yard confessions of a werewolf that you usually printed that so yes yes yes yes yes yes yes but it's a little wild don't you think the noose hangs high hmm see what i mean who will ever believe it tonight's adventure of george valentine has been brought to you by standard oil company of california on behalf of independent chevron gas stations and standard stations throughout the west robert daily has starred as george with virginia greg as brook see let george do it is written by david victor in jackson gillis and directed by don clark bob griffin was heard as tom forest lewis as jared ted osburn as mcqueen larry dobkin as merton noreen gammill as mabel and tim graham as jake the music is composed and presented by eddy dunstetter your announcer john heaston listen again next week same time same station to let george do it let george do it is heard overseas through the worldwide facilities of the arm forces radio service this is the mutual don lee broadcasting system


Husband. Yes. Good morning. Good morning. Well, good late morning. Where are we off to? Baby Moon. Ultrasound number. Dooh. Gonna see the baby again and hear the heartbeat. Yeah. Today I am 16 weeks along. Our baby is the size of Batman. I'm just getting an action figure. But the app that I have on my phone showed me that it was a little Batman. So I'm growing Batman. Batman. Does that make me a superhero?   Or a giant oven. I can't do it. I'm a superhero. Now you know.  And a mask. Superhero mask. I don't know that it would go over the glasses very well. I'm not a contacts girl. I can't wear those things. We're in this building. On the second floor. It's called Baby Moon. We come here and it's non-medical, right? Non-medical. I'm gonna tell you the heart rate. Are you excited? I had a pee real bad. Real bad. What's happened? In order to get a proper ultrasound, she has to have a full bladder. No, not for this one. Not for this one? No. That's just for medical. I'm lying to you. Did you know that? That's for medical ultrasounds when they want the measurements on the baby. The last time we were here they needed to. No, that was only because we knew this was so far back. What do I know? I don't know anything. Don't listen to me. You already have an owl, right? Yeah. But we haven't listened to it at home. We'll have to listen to it tonight and show the vlog. Yeah. No, we did show the vlog. Yeah, we did. Yeah, and I showed it to my mom when she came over. It's for the baby. I thought the heartbeat was for us to listen to. Yeah, but it's also for a keepsake for the baby. Okay. When they come out. We have a stubborn baby. She or he did not want to turn on their side so we could get a good profile picture, but it doesn't matter because there's a strong heartbeat and it was a baby in there. Like it looks human-ish. You want to show them the pictures? Oh, sure. I'm not sure if you can be able to tell what it is. This is kind of a side profile. The head is down here at the bottom.  I call it a her because we think it's a girl, but we'll see. Head is tilted a little bit to the side though. So you can't quite get the whole profile, but that's okay. Let's see. We haven't even looked at these yet. That looks about similar to what the last one was a little less clear though. This is, I believe the brain? Yeah, that's the brain because they can see through the skull with the sound waves at this point. Because it's not an ossified skull yet. So it's not hard bone. We can check to make sure the brain is developing. This is a hand.  She kept covering her face. This is the, oh, might be the same hand. No, I think it's the other hand. I don't know. Alex knew what was going on. We don't. Oh, and then here's baby. At first she was on her stomach.  Yeah, that's our baby. Heart rate was 151. Good heart rate. So if you guys are in the Winnipeg area, we come in from Steinback, but this is called Baby Moon. It's in the southeast side of Winnipeg right by St. Fattel Mall. And it's pretty affordable.  You can get just the heartbeat. You can get that, written all the packages a little better than me. And it depends what week you're at. So at week eight to week 15, I believe all that's available is just the heartbeat package, which is what we go for because we don't want to know the gender. But then there's also a mini moon 16 weeks and up. Actually, it might be before that where you can get a video. That's what we'll be doing next time. So the video will actually come from them and it'll be a lot more detailed than what I can show you second hand while I'm sitting there. Yeah. And it'll be a clearer image because when you take a screen and then you film another screen, it gets a little bit blurry. But hopefully next time she or he is on their side and we can get a little picture that knows. That was awesome. Oh, I'm so happy right now. So happy doesn't even describe it. It's a feeling like no other. So if you are expecting and you want that reassurance and just to hear the heartbeat at any time, it's super affordable. Just come to Babymoon here and they'll get you your heartbeat and ultrasound. And then you can relax a little bit because I know anxiety plays a little bit of a role when you, you don't know, you can't feel it. She still can't feel anything. Nope. And we saw the baby and they're just kicking around and just having a party over his face. And that already. She still can't feel it in there. So when you're, especially, I guess, when you've gone through the IVF process like we have, or if you've had any kind of pregnancy or infant loss for peace of mind, you're going to need a little extra peace of mind probably compared to a lot of moms. Yeah, peace of mind. That's the words I was looking for. A little bit of comfort. Peace of mind. And definitely works. You can schedule as many appointments here as you want. As long as they got time to bring you in, you call ahead, you make an appointment. I mean, you could probably make an appointment every day if you really wanted to. They're just the most wonderful ladies there, especially Alex. They are very nice. Yeah. Very good bedside manner. Yes, all of them so far that we've met. Just finished editing this up and I got the approval of the boss.  Well, I think he knew before we even knew. There were a few mornings before we found out he was waking up every morning and laying his head on my stomach, which is something that he never did. Okay, lay down, sir. You're blocking the shot, man. I'm so excited. Lay down. Lay down. I'm not going to cut your toenails again. That's probably why he thinks I'm sitting on the floor again. Oh. Because I cut all their nails today. Chevy, can you lay down? Lay down. We're getting the skin puppy, guys. Skin puppy. It's going to be so good, so cool. I'm your best big brother ever. I messed up my bed.  No, it's because I just vacuumed them. It was way too clean. Chevy, you're such a weirdo. I love it. He's a nice kid. Yeah. So that little baby, you could see it moving around in here. It's facing down in this shot. You can see the legs kicking up behind it a little bit. Ah, she or he is a very active baby. He can't feel it yet though. It's amazing because it's like five inches long. That's cute. That's 5.12 apparently this week. You'd think that you'd be able to feel that, but... You'd think, but no, there's too many layers of insulation, I guess. Amazing. Amazing start to the day. Then we had an amazing rest of the day too. Yeah, we went and we had a baby, a baby filled day. We didn't take the vlog with us because we were just enjoying visiting. We went to my cousin Ashley's and her husband Jared's. Went and saw my two-year-old nephew and we've got a little girl on the way, so she's due next month. Yeah, around the end of next month, so... Five weeks, right? She got six weeks to go. Six weeks. And then after that, we went to my other cousin Braden's and his wife Kylie's and they had twins a few months ago. Four-month-old twins, so we got to play with them and hold them and have dinner together. It was a really good, fun-filled day, but we've done a lot of visiting last two weekends. Yeah, next week can we have another family gathering? Yeah. I don't think I've ever... Well, I've been this social, but not in my 30s. Yeah. Like she said, her cousin has a little two-year-old and he knows my name. Oh, he's the cutest thing in the world. Her other cousin has like the four-month-old twins. I got to practice holding babies and how to pick them up and how to hold them because I've always been the baby of the family. I've never had a baby around. I had nephews. I didn't get... I didn't see them that often. When they were growing up and when they were young, I was in my young adult years and sort of just doing my own thing. And I lived way out in Winnipeg and I wish I would have spent more time with them, but now I'm learning how to pick up babies, hold babies, and this whole dad thing. We got a little bit of light reading from all of them. They all gave us some books to read. A little bit of light reading. That one's the smallest one. I'm pregnant. That's the smallest one. That's for Brett. I never thought I'd hear you say that. What to expect the first year. That's for both of us, I guess. That's light reading for later. It's a little bit of light reading. Once a baby's born, just, you know, just keep yourself busy. And here's one just for me. The expectant father, the ultimate guide for dads to be. Not too bad. This last one is for the mother. What to do when you're expecting. What to expect when you're expecting. No, it's for you too. You said you wanted to read it. Yeah, I can read it too. That one's more specifically for me, but the rest are for both of us. That's so you know what to expect during the labor process and as I get further along and bigger and how annoying I might be sometimes and how much help I might need. Good buddy. Welcome. Are you going to do some light reading too? Yeah? You don't wreck the books with the guys? Lots to learn. Lots to learn. But you know, it's going to be great. It's going to be great. We can't wait. Oh my gosh. I've never been this happy in my entire life. Not ever. Tri-master two is awesome though. For me. For me. Like not everybody has a good time as me, but nausea is gone, morning sickness, vomiting is gone. I have energy again. I'm social. I don't just sit on the couch spinning all the time. She had a pretty rough first first trimester. It was awful. I do not recommend the first trimester, but I do recommend the second. All right. If you can skip the first one, you go ahead and do that. Find a way. Let us know. Yeah, I have energy to clean the house again and visit people. Yeah, it's been wonderful so far. So we'll keep you updated as things progress. Our next big appointment is in two, no beginning of November. Our next appointment is November first and then our big ultrasound, which will be 20 weeks, 20 weeks and two days. That'll be in about a month. It'll be on November 14th. Yeah. So that'll be like the one where, you know, they make sure nothing's wrong with the growth of the baby. They measure them. Well, that kind of stuff, probably more blood tests and stuff. But I've already taken my maternal serum screening or screening serum screening. Anyways, I'm not a doctor. Took that test and that's to make sure that the baby doesn't have anencephaly, spina bifida, Downs, Edward syndrome, things like that. All the really serious stuff. So we'll keep you updated. Thanks for watching this video and tagging along on our exciting journey.