I will report on some. Result that we are so together with Rob Maurice at the impact least in that than in the early in Paris. She's a student of Christina. You know I mean within the bigger project that I will be able maybe to explain only towards the end OK so. So let me introduce them or they'll actually there are two more this connected with the name of the at the great the if is a physicist we introduce this model in the ninety's so both models. Say spin zero one mother's on Z. to. Subvert this is I read there in fact that in fact that correspond to zero or green in the picture are Alfie one or red and. There is a what what is called in the Update family of they do at the modeller So in this case the update family does that to some of a given vertex V So I did that to subset of the north west neighbor of the OK and then and there lament of this update families call in that date rule so on the element could be for example the north and south neighbors or the north and west out of the South and West OK. So the first model is their bootstrap a collation model and it's called the B.P.M. do at their booth separate Collishaw mother and goes as follows so at time zero you spread out infection namea vertices that are a zero. All over the lathis with by tossing a coin so the probability that vertex is in fact of this Q We call it Q. and peace which is one minus. You can then. So this is the only randomness in them or they'll be starting configuration and so then they're following maps a time T. one Earth event this is. Then that day three on which is completely infected they become infected for example in this case this guy is Elfie But there's the north and west neighbors which are infected and then becomes infected itself OK And then you eat it OK so that's standard blood separate collation OK. So you first of all is there more not on process so infection can only growing time. And this is a picture that pulls me. Give to me so here you see that saw so this was some low Q. Q Is there then city having initial infection and so. Most of the vertices out of nothing fact there is a like a little cloud them in fact in them and then here there is in initial infraction that is localized here then then and there are these and then the booster mapping grows and in facts like something that looks like a growing corn OK So we will see that also later on and so the basic question of Bill sort of put a collision that the standard one so the first question is where there with almost surely the whole that this will and that being infected and the answer is yes this problem was solved by a robot to show man in the ninety's. Very very fast yeah OK maybe this guy will have to wait there very long time and then we get maybe a wave of infection coming from somewhere else. Yeah yeah there is that the reaction but remember that the infection is spread out the time zero in there. And then tical e I The ways are they do they will always be patches I mean faction in the at this in a I've been wearing the sensor so the more their them much more delicate question is how long does it take saying mean are in media and. How many steps I required to in fact the origin. So the answer is this. Scaling Experimental of rock square over of Q. or over Q.. S Q goes to zero so Q. is the dense initial density of infection. So this scaling was found by Tom and for in one thousand five and for a basically twenty year it was open up the question where there were it was possible to pin down exactly the constant here and this problem was solved by a. Problem are a simple Smith in two thousand and seventeen. Actually they found a they connect Konstam for him very closely related constant quantity. And essentially it may be a year before this same group. In the sense solved the same problem they may find in this case the scaling for the infection time of the origin in two dimension for basically any. Collision model with the general update family. That there's the problem with the that almost surely there lattice will be eventually be infected OK so far so this is like air so physicists vote called that is universality result OK And then so given an update family base on the edge some just geometry of their plate family. They were able to pin down exactly they're in for a scaling of the infection that OK So this is their moves that up it Coalition did were out of the books for a pre-condition mother then the second model that we can see their. Motivation come from physics of glasses there is a twenty million dollars grant by the Simons from mission for a project called Cracking the glass than Zeeshan OK in very big lead this project is that related to this to this problem OK so this this we called the K. C.M. and you can think of the of the model as a stock casting version of the booster percolation so. So we create one every vertex tosses a coin so it's kind of OK so. If the vertex as an update rule which at that the moment is completely infected OK then the base state of the vertex is updated according to their result of the contact says so of the contrasts namely becomes the contacts was one then becomes healthy otherwise is infected if there was no in fact in. Updating the rule for V. that was in fact that at that moment then there then the proposed update is the refuse and you start thinking OK So implanted this is not a monotone process because infection can heal back OK. So. Just give you two general facts about this so this is a matter of process or if we run it on say on there are and then on the final square. Is a continuous time Markov chain So first of all is a reversible with respect to their. In the only product measure OK It's very simple to verify and it has a positive spectral Gapper find that the relaxation time this was proved years ago by these people. An. OK It belongs to a certain universality class that that will be able to explain only in the last light OK but is called critical one group that case here I am OK. And so in a child question which is a question which is of great interest for physicists is what is this what is the scaling of the infection time for the for this is the big case here when Q. is Melissa exactly the same question for bootstrap or collision but now you ask what is the first time how big is the first time first time that you see that the infection gets infected OK So this is like a thin time with so so there really is that let's say you pick a vertex OK if the vertex at that moment for the current configuration can be easy in fact a bull by the bootstrap collation you toss a coin if the kind tells you that you should be infected you become infected otherwise you become healthy that. You look at their now and then west then south neighbors if you find a pair that is in fact it. Is like they suppresses you exactly with a different group. OK so there is a general down that says that the average cheatin time is smaller than their relaxation time of the chain are of the process divided by Q.. And then the. Then there is a lower bound in the lower bound is their very nature a one namely well to in fact the already gene within this process should take at least the number of steps that should take for their books or percolation to in fact the origin which is exactly what is a great and here of course work is not actually they mean is the median but. OK so actually so we work in this so-called continuous time setting namely every where text says and in parallel processing clock when the plot when the clock rings passes a current check whether the condition is very Friday and update itself OK or not OK So so there are you know this remains everywhere but never Turing's you know equal OK and that's it. OK so here is there is a help. So they saw the theory and says that this scaling of day meaning fraction time is very different from this scaling for both her pit collisions says' exponential are locked use to the fourth power divided by Q. square so the upper bound and this was obtained by Maurice Christin them myself last last year. And actually was it was OK not for the do at the model specifically but for a very large class of of models. And instead they lowered down this is a paper that we are finishing now with lot of matter shake Christina. Any specific instead to do atom on that OK. Sorry now so without constant OK I mean between two counts OK So so the first come is that it's very different from the divergent exponential log square over Q. in the sense is the exponential of the of the infection time for a pic collation times the log of the infection time for boots or put collection because you get then the lock to the four over two square. And actually I mean for us is important because he's the first example of a critical one the group that modern this universality class for which the operated by one that was proved. In this paper he's actually proved to be sharp. So the rest of the talk I'm trying to explain the A.D.A.'s and the O.E. stick and some you know some map maybe of why. This killing should be this one. So before doing that I would like to compare for a second they do at the model with their more stand up at a collation more then which is the so called two neighbor more then so in that to name one or more there they are they family are just the two subset of the neighbors of the of a vertex so you saw as soon as two of your neighbors at least two of your members are infected you become infected this is their to neighbor stand that puts up a ticklish a mother maybe and we know really a lot about this model. So it. Is exactly exact So for the two neighbor model actually it can be proved that the scaling of the infection time in the bootstrap version in the case C.M.. Version they scaled exactly in the same way as exponential up some constant over Q OK And this is. A paperback Christine than myself where we had. A law correction and then there is a paper with this big group in preparation OK OK where we remove this tension in the UK that's there that's the point so. Then one could ask I mean what's that what's the reason why why did this scaling is completely this killing in case he M. is completely different from this killing collation OK so in the out of the blue separate collation this predicate I mean fraction these various symmetric because you can only look north west and south and you never look east OK And so there is in the symmetry while in there too neighbor model is completely symmetric and so this is this I see my three is first of all is the responsible for a great that divergence. Of the infection time for boots that up in a collision. Because there was exponential over block square over Q. and the lock Square is missing here and for the case here version this as he meant replace in even Mar important role OK that's I'm trying to explain that now OK So first of all one has to understand how infra infection spread propagates in the in the two attributes separate collision. OK so. The group around. When the solve the problem of universality in would separate collision they understood that it key a key step was to. Understand what separate collision when you confine infection on half planes so given a specific direction say you. Suppose that you are in fact all the half plane that is behind you in the direction of you this getting OK then you say that the direction you is stable if by Iranian boats or collision with all the infection behind you is not possible to in fact anything on the other side so this red guy. Remains red then the it is very easy to compute they're stable and stable direction and so there is so the there man aspired to pay over to semicircle is free of. Stable direction except for their zero for the result of their action the zero one that action and instead all this other direction belonging to this other semicircle are stable OK as he made three skeer here and actually does that action here today with a one zero that action one says that as one difficulty I'll explain that in the second. So Suppose for example that we have a. Vertical interval of in fact vertices and next to a column arm non-infected vertices so there are one so. In this equation there is no propagation to their right because none of this verdict of this vertices as an update rule that is satisfied because both the north and south and neighbors. L.T. However if one just one guy of the of the red column is instead in fact it then is you can propagate because the next guy here can become infected and then you can eat or it is OK and then the whole column moves to the right one step OK so you need a little help so the direction one zero is stable but we just they help of one X. three in fact at the guy you can propagate to the right OK so this one is called the difficulty. For difficulties one just one extra guy and you are able to move Of course I mean propagation can also move vertical leak because in the following way that moving through there I mean a a green guy one level up OK So then the all this read this read interval becomes green and then they the green interval is you're on one step vertical E. and then you may continue like that if you sort of meet along the way like a stair of infected best this is OK no progression is possible to the left simply because they're updated rule only looks to the left never looks east. So this is again the same picture so you see that. Here there was a column that was completely infected this. Met some of this little guy helping to there I was able to travel a little bit to there I more and more and then started to meet a like Astaire and was growing OK And then from small it became large. That's the basic mechanism of our infect infection spreads in the do at the mother. OK so. A few words about the upper bound so the upper bound there is. OK So this is the situation with stable and unstable the their action and they have this green semicircle as the opposite semicircle is infinite difficulty you can propagate only with infinitely many L. ping guys to the left. For all Casey M. with this property we prove with Rob and the Christina that the. Meaning fraction. Cannot grow more than exponential lock to the fourth over Q. to the poor Alpha Alpha four do at these wildly OK But actually that is that is much more general than that. OK so I'm not able to explain why so instead I will try to explain why there is a matching lower bound for the duet OK where there is also some agree to a construction that might be of interest OK So a critical notion a critical ingredient for the game is what are called Critical droplet here now as we said them in the vertical interval are being factored witnesses can move to can propagate to their eye with a little help local little help by just one guy OK Now you may ask I mean what is the probability that this. Green column will have a help you know the extra help on the next column OK Well the probability of getting in X. Rapids essentially Q. times the length of the column OK so if you really make the computation you'll discover that if they highlight of their of the column is can stand who log one of it Q. divided by Q. then these probabilities start to be large and math I mean very large ifs if they're constantly this is a part of the be constant and therefore. Is very likely that a car a column of that high will will find help and will be able to propagate OK it's very likely so any such condom will be called a critical droplet and the probability of observing a critical droplet is just Q. to the to the to the to the length OK to the number of vertices and so is exponential of minus see log square over Q OK so actually it's a very small probability so critical droplet. Very small are very rare they're very small probability but once they're there is very likely that they will be able to propagate and while they propagate they can grow as in the picture and then is become even more likely to be help OK and this process turns out that is like an avalanche that's there in the pond so let me give and then released the picture for the infection mechanism of the origin OK so what you expect your You'll see that the origin what you expect to see if you look is West westward OK sorry yes westward there. At some point you will find some critical droplet maybe very far away actually well should be little one over the probability of finding it OK so they will be X. some critical droplet and so now the next key ingredient is this this critical droplet will start to infect. Vertical segment closer to you and so on and so on and at some point it will reach you so the whole point is what's the what's the dynamics of this critical droplet in which the percolation is very simple you start in fact more and more years until you reach the auditing in K.C. M. You don't want to do that because you pay I mean there is a big cost thing creating. In the normal slab rich patch of infection because you pay Q. for I believe extrovert this is exhibit this is of info. Action that you create OK and therefore you want to bring this critical dropper from very fired to next you OK but in the in the in the passive in the most economical way OK So we have to understand what is the most economical way they claim is that these critical droplet moves in the east like way I'm going to explain this in the next life so. For base game some of you will you already know this game but I will repeat for for for the rest so suppose that you as I give un talk cans and this tokens can be placed on the positive integer one to accept that are. Each each each side each vertex can have at most one token. And there and we can place and remove token according to certain rules so there really is there for the following so it talken can always be for free poop or remove from the first vertex so at one OK. For any other vertices. You can remove or add it token there only if there is a token already on that one on the left OK that's there that's the gain rule and if I give you and talk and so the most nature of question is how far can you put. It talk and at the end OK so this this problem was considered by. The back on this NG grammar in two thousand and one and the Maxima is a nice communitarian problem in that you actually this is the easy result of a mark complicate paper and something that you can work it out maybe. At home. Is a mass promise and the maximum distance is to two B.N. minus one OK They message their main message here is the following that the maximum this and grows exponentially in the number of talk ends or vice versa if I want to put the talk in that distance El Al many talk to any luxury to make in in L OK law based who are well that's their that's the message that one has to take OK. Now I want to sort of justify the idea that critical droplets in in this case AM move as acts as a stalker OK So so this is a critical there are plenty. Help on the right hand side therefore it can create and that they're critical droplet here just by booster percolation which correspond to there is a token here I can add and we'll talk and to dare I say OK this is correspond to adding a token to the next birthday to the next type. However a if I have to talk and so two critical droplet next to each other that they drop to the left which is like a token can remove this token here OK can heal most of the vertices does leaving one OK So so this correspond to so add then that remove we're talking exactly as. OK similarly to today's game so let me in now complete the O.E.C.D. OK So we want to in fact the origin and we said OK If a look west worth at this turns one of where they probability of where there are plenty I will find a critical droplet. They can be not are a result of the counties and Graham says that I need to create extra log base two of the distance extra talk namely critical droplets in order to reach the hour it took to put a token which is a critic of droplet at the origin OK. They're far how long do I have to wait until I see that this occurred well you know Barry Raftery you would imagine well that's should be the same fraction time then should be one over the probability that between the origin and the distance and one of it Q two and one of the probability of this critical droplet I see this extra number of critical droplets so if I do a computation this should be just the probability to the number of extra token I'm forgetting all they can be an authorial result or where I put the token OK and if you do their competition explicitly you get this they scaling that that appear in the here in the here OK so. This is very nice R K And the only trouble is to prove that they will pick that this a restrict this is correct OK so. OK so the up and down we did adding in mind exactly this idea. And for much margin there are system for do we now have to prove a lower bound so the lower bound typically goes through the look for the bottleneck strategies of which we will look for a cat set in their Markov chain space OK So we take case. Scale that is. Is like one of it the probability of a critical droplet to a very small power silencers goes like that and then we take a pen glare rape of an decreasing column so we started with lengths and square we end up with legs and and there are a range exactly like that we put zero boundary condition that are around so there is some some playing around with their boundary condition that that they skip and the reason why we take this strange Army trigger is exactly to take care of this boundary condition of them one two and three marring in the details of there. Now the idea now is the following Ok so this is the saw given the configuration of zero when I want to construct their cut set so given the configuration of zero one I want to assign to each column and there are pointing up or down OK and eventually this arrow will really be the talk in. Entering in the east again so. For now I have to explain how do I assign the arrows to there to the column given the configuration of the one OK so. There they must extend the column to the left is the cleric to be up by definition. Then the I scan from left to right until I find if any. A column CIA which if I buy running their moves through a collision process in the column from the first one up to CIA at the end of the page collation problem I see that I see an infected critical droplet on the column C.I. so so I scan from left to right and I see if I check where there is a first column call it CIA such that by your ranting the booster coalition in the in the car up to CIA at the end of the booster process I see a big incredible. Of length a lock you over Q On the last column if yes I say that this column is that and all the previous column are down OK. Then I delete all the infection that was used all day meeny Manning faction that was used took two to create this there are plenty of. These critical droplet in fact a critical drop in on this column. And then they repeat OK so it is a rather rare complicate construction OK but but the but the which in a sense as a guideline for this construction is the following given they seek. A collection of zero one zero mag a circle action of zero one I want to start off saying. Which column every critical droplet are enough I cannot do this really directly because maybe no call our music critical droplet but maybe is almost close to have a critical droplet I have to solve this problem of being almost close in the sense OK. And these are going gives the correct construction so. So at the end they came forget about the the two dimensional zero one configuration that I have a one dimensional configuration of arrows OK And then there is some technical work and this technical words. Boils down to proving that this arrows they'd be a century like I the variable in which the probability of a nap error is essentially the probability of a critical droplets so is this exponential minus lock square over Q.. Of it. And and then when asked to prove that if I do a Glauber move because F. F. to compute the captain using like to get inequalities so I have to compute have to compute the effect of a globular move in the original configuration. So I want to prove that the Glauber move essentially. Can modify this the structure of this column of this address. Only if if it occurs in a column such that to build just the column to the left is in the Arrow OK. This is the school OK. And then I also have to prove that a global remove can essentially change only one narrow part time OK and this possible to do that so the conclusion of all this technical work is that these arrows they really act as. East tokens. And then they can't set is inspired by the cat set of the east chain OK I take as a. Set a boundary will be the the cap the boundary of A will be the cats are all manga which can be reached by the old down. By a canonical path using at most a log end of this log a Z. is given by they can be in that area result of true Graham in the counties. Plus there is some technical condition that them hiding and that their act OK now is possible to prove that the probability always almost one. But the probability of the band there is very small. Is mall in there in the right way and then there is your nonstandard cat argument for mark of chain they prove that the relaxation time is larger than then this one to be here and then there is some X. that we see there that transfer this this lower bound not just to their electrician time but to the infection time OK so I'm hiding all that so in order to finish a mentioned work in progress or so maybe this will explain why we look at this new Atom model person though we are not so interested in the specific of the the mother. I wish. OK. So the first. The next step. Is to extend this could this be a construction that this crap for the more the two this so-called briefed more there so that if model added the following so. So you give me some update of family OK these are different from the do at that and then I can compute all this table direction OK this is an easy task and suppose that there this picture product stable direction is therefore in so there is a semicircle that is green except finally three many stable direction and the opposite simu circle as you know as a region like a. Sector which is green except find it in many stable direction but then there are. For example two out there or just one sector that is just a continuum of stable direction OK So this is this model they they show exactly the same feature of dead of the data to put collision will separate collision Amy they tend to propagate in a semantic way more to their eye then to the left because it's very difficult they can only propagate a little bit to the left thanks to this green path but then they are stop because of this red part here and instead they they can propagate will. Basically like to add to the rape so. The next problem is to extend their lower bound to a mother like their. And their even. Bigger step is thaw. Is to prove a conjecture that. So personally we sort of believe it or find it very interesting even to this through with to disprove it so this was a conjecture put forward by a rather Maurice Christian than myself. That farcical allfired route that critical model the behavior they discovered in the correct skinning for their infection time is exponential up relock luck to the fourth power over to to the to Alpha So what is alpha so we imagine the following that if you if you look at there. The geometry of them all then namely the stable and unstable direction they resist semicircle. A green semicircle with Finally many stable direction each with difficulty to remember the difficulties are many extra vertices no need to propagate in that direction so their difficulties less than our favorite where OK an. Is similar picture is five of the holes for the semantics of Mr Coe But nowadays the difficulty here so there is at least one that action that there's this difficulty beginning then twice alpha so this model where call root that Alpha allfired root at the critical models there is a map at a bound on the infection pain that is this one and the big question is whether this this is sharp OK Exactly because of this east gain that is he Demby I mean you know his game for the appropriate critical droplet for his mother OK Thank you. Yes. And also they did this kind of universality result. Proof so fired it only for a bit to then mention the lat this and so the same group is writing a paper for their higher dimension the best chance in the three M. which is as you can imagine a really. Tough task OK but at some point they're so they're just writing as hold a deep days I've been told been OK so. Now on that graph say we did in the next explore that yeah. Yeah yeah so. So this is a good question because actually. So I didn't outline enough I guess or Athol the fact that all this. Through only. The meaning fraction time when they need a shell configuration for the bootstrap it call a shop and for the mark of chain it started from there reversible measure OK So we have basically noise no result if they initial distribution is just event a little bit different from the from the reversible one because. I mean I mean a whole class of poor. That we used to analyze the evolution in parents are in on the regular farms are a gap or. Relaxation Pam and stuff like that I mean they just collapse and that when there's really two really for their true evolution in time OK. So this is this said this I mean this is something really very hard then there's been then only to say to some I mean with with some success only for a specific mother like these more than one they mention or they smile then in a year the mention but in general is really very very ill I mean isn't that their level. Of difficulty in a sense.