Fields Medalist: Wendelin Werner

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Fields Medalist: Wendelin Werner
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Lecture of Wendelin Werner, Fields medallist 2006.
Fields Medal Process (computing) Direction (geometry) Position operator Wärmestrahlung
Point (geometry) Area Finitismus Potenz <Mathematik> Multiplication sign Sheaf (mathematics) Algebraic structure Basis <Mathematik> Perspective (visual) Brownian motion Theory Number Formal power series Many-sorted logic Identical particles Sinc function
Statistical hypothesis testing Randomization Group action State of matter Multiplication sign Archaeological field survey Mereology Perspective (visual) Dimensional analysis Plane (geometry) Roundness (object) Many-sorted logic Different (Kate Ryan album) Cuboid Renormalization Descriptive statistics Representation theory Physical system Area Potenz <Mathematik> Theory of relativity Gamma function Physicalism Infinity Category of being Order (biology) Renormalization group Conformal field theory Resultant Point (geometry) Functional (mathematics) Transformation (genetics) Number Kritischer Exponent Latent heat Centralizer and normalizer Kritischer Punkt <Mathematik> Energy level Analytic continuation Stochastic process Scaling (geometry) Percolation Physical law Model theory Exponentiation Algebraic structure Vapor Loop (music) Spring (hydrology) Physicist Gravitation Object (grammar)
Complex (psychology) Randomization State of matter Correspondence (mathematics) Multiplication sign Range (statistics) Water vapor Function (mathematics) Mereology Grothendieck topology Mathematics Plane (geometry) Roundness (object) Many-sorted logic Lattice (group) Different (Kate Ryan album) Forest Matrix (mathematics) Cluster sampling Position operator Descriptive statistics Physical system Predictability Area Injektivität Theory of relativity Cohen's kappa Power series Moment (mathematics) Infinity Price index Time domain Connected space Category of being Bethe lattice Order (biology) Configuration space Right angle Arithmetic progression Point (geometry) Metre Functional (mathematics) Finitismus Vapor barrier Sequel Connectivity (graph theory) Resampling (statistics) Distance Theory Number Power (physics) Latent heat Population density Cross-correlation Kritischer Punkt <Mathematik> Interface (chemistry) Boundary value problem Theorem Condition number Stochastic process Scaling (geometry) Percolation Matching (graph theory) Fields Medal Model theory Physical law Commutator Counting Algebraic structure Division (mathematics) Incidence algebra Line (geometry) Evolute Limit (category theory) Invariant (mathematics) Physicist Object (grammar) Pressure Marginal distribution Family
Trajectory Group action Randomization State of matter Interior (topology) Multiplication sign 1 (number) Set (mathematics) Parameter (computer programming) Mereology Dimensional analysis Group representation Derivation (linguistics) Mathematics Plane (geometry) Roundness (object) Spherical cap Many-sorted logic Different (Kate Ryan album) Forest Cluster sampling Physical system Curve 3 (number) Measurement Time domain Sequence Connected space Category of being Symmetry (physics) Uniformer Raum Conformal field theory Configuration space Bounded variation Resultant Point (geometry) Vapor barrier Supremum Divisor Transformation (genetics) Harmonic function Translation (relic) Similarity (geometry) Graph coloring Rule of inference Number Kritischer Exponent Interface (chemistry) Boundary value problem Analytic continuation Condition number Module (mathematics) Shift operator Multiplication Scaling (geometry) Percolation Model theory Physical law Independence (probability theory) Limit (category theory) Formal power series Invariant (mathematics) Loop (music) Game theory Object (grammar) Family
Slide rule Group action Finitismus Link (knot theory) Interior (topology) Multiplication sign Correspondence (mathematics) 1 (number) Archaeological field survey Student's t-test Parameter (computer programming) Dimensional analysis Mathematics Natural number Interface (chemistry) Boundary value problem Analytic continuation Cluster sampling Descriptive statistics Perimeter Condition number Area Theory of relativity Percolation Scaling (geometry) Fields Medal Forcing (mathematics) Model theory Algebraic structure Measurement Time domain Connected space Formal power series Category of being Arithmetic mean Loop (music) Physicist Order (biology) Object (grammar) Mathematician Family
are good afternoon earned by it is a really great pleasure for media tradition bad burner for the final field Medal lecture the receivers Ph.D. in 1993 anniversary party peace whether the directions
job Francois Le Gougne since 1997 is that body sued in said from 2001 to 2006 he was member ideas Due to the rest of France currently also holds a position at a column about subsidiarity party if prices include the role of surprised European Medical Society thermal prize shockable lives let fries and the pulley a price
on a lighter note in tritium is also 1 of the few people I know who has a finite identical you're numbers and Kevin Bacon numbers subbing for people look up later 3 in both cases on personal perspective I would it was about 10 years ago that I 1st I was giving a lecture and we are all about intersection experts are right section 1 case in
which was the brought the but the never brought conjecture on about Bradley motion when deadline was in the audience in a couple couple weeks later receiving e-mail with a very nice idea out or quotes the problem from the start of an e-mail start collaboration less a long time and I've seen many many of its nice idea I don't like to mention a couple of them today what about the couple years later asked lawyers as big as mentioned this morning struck by that he was talking with their dead trauma he realized that Odets beautiful sullied might be keyed Bharatiya section exponents and at that point guard two-person collaboration on this I became a three-person collaboration sadly I had 2 outstanding collaborators with giving nice ideas back and forth secondly I'd like to certain the general idea although a lot of work with problem driven by these particular problems steadily many types of of wonders actors said Becker said that sort of looked said not only we solving these problems that were asking but it back structures that it be done but that's only in Brownian motion really good perhaps the key understanding Brink formal appeal theory on a rigorous basis and now that's an active studied art area being pursued by many people with that I'm happy to that 1st how how a thank you make much of you as you might imagine this week has been a very special week for me and enjoyable want uh I hope I'm not too tired to deliver reasonable took a today and that since it's the 1st time I get
the check a chance to speak and of course many people to think from trust notorious seems to Peter G. supervisor but I'll do that on personal film a level that would be too long to do now but of course I like to our very happy about that I'm speaking in and author with that plenary later morning in that Greg was able to be here too also moved and chair of the session because a lot of what I'm going to say I'm not maybe part of the reason that I'm standing here in front of you is that there to be nice people so the topic of my talk is random planner loops and con former restrictions and added the survey because of course this is supposed to be more of an introductory lecture in order that gives some flavor of some of the ideas and some of the results that a mutated in the last during last year's end this talk is of course not unrelated Robert closely related to their store this morning and also that's will store yesterday but I will I'm going to assume that you all went to these tonight's lecture uh I think I'll try to make it a self-contained today but maybe with some offshore additional at some point on some issues with what mostly what said this morning but outside to take another perspective which is more to try to look at described description and understanding of continuous random structures in the plane using the courses discrete models of the guidelines but really focusing on properties of the continuous off so I think it's fair to stop With some background about her motivation and history in the physics community because the not subject of researcher 1 of the main motivation comes from physics and physicists have been giving a lot of input to these subjects before we actually started to look at them and actually I'd like to thank also take the opportunity to thank uh to thank moon's Michael Eisner that other physicist who also made the effort to come to match the offer and said that you stay here we have a nice some problem for you that we no also know how to solve but which as a nice problem for you edition because the way we approach them is probably not the way want should do it mathematically and then when we came up with some ideas they also accepted and didn't just say well knew it before so they acknowledge the fact that these where it was really new input so I like to stress so I'm going to say a couple of very general statements and to start with a with the danger is that this general so that you don't see what I mean that in general when you 1 about physics you learned that well the laws of physics something when you're repeated twice the same experiment you get twice the same results that's the result the most of them signals of physics and it has been observed long ago actually experimentally that on microscopic scale when you are exactly point at which we physical estates transition point that means the point where say for instance you can imagine the temperature which liquid become vapor or those competition between 2 possible states in the system that when the exactly that point this
spring previously domestic doesn't hold any more because you see some features that become random on macroscopic scale so you repeat the same experiment twice a you and you will get different answers and part of the story to understand some of these macroscopic random teachers or complex systems that use your microscopic scale and a closely related question is that well How would you describe what face transition point is well usually it means that some of the physically fit Mr. quantities that you observe on a macroscopic scale when your away from the a critical point go to 0 or go to infinity and it has been observed that of this quantities often obeyed the volatile behavior when your approach the critical points so that some quantity to domestic quantities soaring basically it's like to see some our gap and this exponent gamma critical exponent of this model for this of so she have a description of the 1st a random objects at a critical point and then you you have hedonistic behavior or Mr. Mr. points are quantities near the critical point and Of course well of of course but it has been observed also that used to argued that these 2 phenomena are very closely related with each other so physicist have came up with her ready is have came up with the number of food invented and clever ideas in order to describe the these of problems in these questions and the 1st of which as we developed by Jr it's not true intimidate you just the dance on the names of visits that have contributed to these issues and of course I omitting a lot eventually the thoughts ideas that of renormalization group and that basically is so of course is a very simplified explanation but explain roughly gives you convincing heuristic to the fact that a different models all different gasses all different experiments will give rise to the same explode insult the same around behavior at a critical point and the is basically to say that his random microscopic Yediot intrepid fixed point off some renormalization function that basically you divide system in a large system into smaller boxes and boxes this random system areas created you put them together to create a large systems and what you argues that at this point the system is a fixed point of the because become skating variant it will be a fixed point of this transformation so he items that are specific to two-dimensional systems and a soap conformal field theories and also what is called on gas techniques all quantum gravity our ideas or that sort of based on an allergy is based on explicit computations based on but many different facts that provides mathematical tools in all of that enabled resisting predicts the value of these critical exponents of many different systems in a case where the dimension is too so we are looking at planners and the exponents of the models are classified according to what is called the central charge of the mold that each model or just look at has a specific central charge and time central Charles refers to the fact that some of these mathematical tools that I didn't behind the call for fury seeing you have to do with the representation theory of summer into dimensionally art and the but item is explanations of the fire the precise relations that there exist between the behavior of this round of behavior at a critical point ended hedonistic behavior near the critical and I think it's fair to say that apart from the last item which has been treated in the mid eighties by Harry tests and that's 1 of the many things that Harry test and all these the questions so treated the case of grid of percolation into dimensions where he made sense of the scaling relations and explained that if you understand the ran the behavior of the system at the critical point you also understand the behavior of the system near the critic
and it's sad to say that pop-up that these 3 even those can't fall a theory was related to her very rigorous mathematics the relations between these deepen reversed that matters and the actual question look that was question the a question mark OK so I'm going to repeat very quickly so that you have 1 modeling minor chose to repeat the moment has been introduced but you may be seeing twice despite before but but well I guess that the simplest model to explain that you have in mind something specific it's not general further specific wanted Mullis Coal amateur percolation and the ideas the following you have honeycomb lattice of you tell you playing Texas and you just going he checks donors to be back with probably would probably P and white with quality 41 line is clear and the state of each of the next dance is going to be independent of each other and the question of looking at posting just doing Conte tossing so you have to explain what questions looking at ways so the pressure to look at it you are interested in the connectivity properties of the picture you obtain when you draw when you do this Malaysian you'll see what the movement and keep plays the role of the temperature Archibald that's a promise to be allowed to play and that it that Polish connectivity properties the face charges of Booker appears to occur he sequel one-half and went so the ideas when he is larger than one-half then you have 1 infinity of majority of black and you get 1 black connected components call the infinite plaster and that hazardous if you want a positive intensity or positive density is achieved the rights where of Goldstein of which is hedonistic function of went larger and when he is more equal to one-half you have no incident that connected components only small Idaho and that he got one-half which is a critical points so these random what we would expect to be the random future you see clusters at any scale and their shape over the shape of islands appeared to be random so that's critical populations are so should just think of it as a great division scream and you try to take them is great Red television stream the connectivity properties you realize that well it's not the trivia question because I will train detect If there's a letter right what crossing in this in this picture and actually this is some of the more keeper of question and the fact that this is what the way randomness is organizing and put together in order to create a macroscopic eventually left to right crossing very soft and this is what a percolation clusters of looks like in the previous pictures so that's why I and I surrounded and out about already but forget about bound for the moment so this is what I want closer in the previous picture shows you that when you look at your television screen you'll see clusters offsides compatible to the screen you look at but you will not see infinite OK so are predictions by physicists that were I now mathematical theorems and I'm not going to explain you the root of that these prediction these theorems now because I would repeat what it said this morning so here's 1 1st Division I think this Due to nice and the death of the 2 1st probably maybe I'm making up things predicted by most and fodder 48 numbers so the first one has to do with the area near the critical point the intensity decays but he 1 office of power when she approaches 1 house and the 2nd 1 has to do with description of what happens on the random behavior at 1 time the tells you that the probability that there is an open part for begins sites of distance art is decays and so these are typical examples of these predictions that is made that may now accessible another model to keep in mind is that this so-called using mobile but collection has some very specific features because Of the fact that what happens here and there on television screen is independent and others and will come back to their specific features but later and there's a model for the easing model which is basically a model where you're going to similar as the previous 1 way to bias the probability of each configuration depending on the number of commuter that the configuration count how many neighbors disagreeing comedy pairs of all pairs of points they are richer neighbors and such that was black and the other 1 is what you count how many airline you're going to penalize the probability of the configurational according to the number of disagreeing neighbors using all prefers to have a complete you compared to it is ordered system of but she prefers to neighbors that all of the same opinion same and this induces a long range the correlation between what happens at different various OK so 1 of the novelty of all this a new magnetic approach that has started in the very late nineties is that instead of looking at the Croatian functions and I just did gave the example of what you might call a correlation function in the case of a percolation so instead of looking at the behavior when the match of latitude to 0 or probability that you see that far away points or a distance of a given current you are looking I'm going to fall to try to describe the entire the entire around the objects and the actual random picture you see instead of some sort of things that look like a finite dimensional margin of course this is a big simplification I'm not that if so if you want to oversimplify a little bit more you might say well that in complex analysis in general you always have the magic tricks that on the 1 hand you can write an analytic function as a power series and this as a very local future with a X some of the idea that again and looks like analytic thing when you compose a maps to have playing without you right structures and and you have the magic trick that this corresponds in fact actually map from portion of the played to some other portion of the plane and that has some Joe matrix inside so we're going for a 2nd approach if you want here and now I
just try to be very quick on the 2nd side because that was the topic of Odets this morning if you want so 1 of the main ideas is to use all assumed the fact that these novels a large scale orders the critical systems into their mentions behavior following variant wait so what would that mean well you should imagine the following you take a system so I did you get there critical at base system on the very very fine mesh so you all you just go directly for the actual physical system in the in the continuous and you that the system into different domains she wanted to end which are Oracle formally equivalent which means that there's a one-to-one map that preserves bills from the Domaine du warning that went to the demand D so the way to describe random systems is to say that system in the city and you want to have a collection of plaster say that people see you want index by Chase and indeed to have another collection of clusters that you called teaching index by Jay and that these random systems that you want to describe a new budget going to say that these are going to become forming barrier and it basically the law if you take care of the system in your 1st made a map it onto you now what you see in the 1st of May onto the 2nd domain by then you get around the system and a 2nd that had exactly the same role as the system itself in the sick so in this morning's lecturer that describe the status Of these waters proved and what is not approved the ballots to the fact that this skating in of discrete lattice based models are ideal not performing barrier and if you've been to not lecture yesterday you know that there is some I'm going to attach progress going on OK so the goal now is to describe continuous structure that you see as skating limits old gaining limits but that have nice performing various properties in plane so Of course there's is a very easy way to create a performing variants of performing their front structure as well just pick your favorite the main city of behalf half-million Jews any random object in the upper half playing for a new round of structure that the state invariant or in variants of the mistress mission that will give you 1 run structure of half-length and you defined around structure and the other the just by taking become formal image of what you've seen in these behalf the other simply corrected the make and that defines you a family all possible observables or portable systems once for each domain that satisfies falling Barry so got formerly itself is not the very restrictive conditions need to add more condition in order to be able to and pinpoint or describe all possible such systems so the 1st additional I mean 1 possibility was explained by this morning is that to focus on discrete interfaces so here I schematic and that you have been told by my daughter when she left the might of the Tour away handling the pictures on the on the a so you measure that you have a domain with precooked and boundary conditions save you assume that what part of the boundary years red and the other part the largest red and said of black-and-white for obvious reasons and basically if you if you are going to assume that you everything he has about is everything on the right field that don't have the designed them for a Division Street with blue and red forest there and then it's very easy to see that you get 1 single interface that going to separate 1 run the line that is going to seperate the new plaster attacks this part of the ballot to the Red Cross that's the other so this is around the church and the point is that discreet interfaces can be explored and that's the basic the start point of that trial that led to the division of assembly which is returned to explore the system just stopping here and we stopped short this interface because we're looking at supposedly we supposing that nearest neighbor interactions system with nearest neighbor interaction this failure you that once you have to know that this is the way the interface thoughts and you think What is all of what remains to be explored this strange to business screen that still remains to be Broadway but it's still now say percolation or easing and the the remaining may not the main is the Circlewood the slips and with new boundary condition were on this part of the bargain sure everything is here everything is ready and so if you assume former barons in fact once you start exploring the boundaries of this interface you ask OK now what are they continue you to still have used a lot said asking exactly the same question about looking at have to explore in interface between simply corrected the main where you are looking at what parts of the boundaries red part of the boundaries ruined acute interface between these 2 and so it using this pledge from explained this morning that combining so the classical ideas from a complex Alice's name you know fury Wade the basic probabilistic a site you get there is a exists at most a want meter family of such random is that it is in this implicated the main that satisfy both performing property and this Croatia property this idea that you exporter of progressively OK this is of course too short explanation but just to mention that the output of this Of this idea that just that you could make sprawl interfaces progressively leads just actual concrete description of the possible interfaces over for this interfaces for these special precooked boundary conditions and you get a one-family of such random tests call the front of the evolution that's the baby with the warnings lectured the coach from live evolution and this want is usually called Kappa OK he some of properties that what it described as morning so there's not type of some of the most simple run the germs and some of them have
doubled points over the base physician a cap for this number will come out later and capital 6 and a 3rd also very special have some fun out of a very special properties and back that so now the general I want to focus on the go code conformal restriction property so come from restriction is another idea that is complementary if you want to the previous 1 that gives another approach or other characterization is run by silly so actually this ideas has been developed and refined a sequence of papers with Red Dragon dead the result I'm going to represent knowledge paper lying really just a continuation of our knowledge on work and the idea now not to explore the care if you want from me if for which a dynamically but new stores from far away so well How does the curve looked like from far away and 1 way to do you to say this is to say that you're going to compare the the shape of the curve thing to do when its defining 2 different domains so here you have to start having this idea that you're going to a number of course explain this any way on which ought of the understand really wanted that you're going to start playing with variations of the role of the care with respect to were the domain are looking at and that's where he is the DRG grass and into the game but I'm not going to think more about that so he asked his specific questions I give the motivation later see the motivation would be answered if you want and the question is the following you're looking for a measure supported on the set of rooms in the place so it's the set you're looking at is a set of single loops that you could draw in the play and you want this measure dissatisfied property of the people record this tropical forest picture property and this Probert is the following you take any too come forward equivalent domain you don't get the measure restricted to the first and you know that the measure restricted to the 2nd what you want is that if you not gon formerly the measure restricted the 1st one on to the 2nd 1 before map you get exactly without getting factor the measure restricted to the sect right so whatever domain you look at it you want you're going to see the same measure if you if you will after this condition to be true for simply connected the domains only forward Rick we perform restriction that means that whatever simply could add to the menu choose you're going to see the same measure Motorola forward so at 1st you say well such a measure cannot exist as to form condition and some of the 1st 2 items not only are rather easy just consequences of the definition to push out such a measure that of course it has to be scary variants because some colorful Maoming multiplication of format so should take in this look at what you he does this court twice as you say the same the same measure so the measures stated very if inflation very poor a similar reasons and therefore it must have an answer that's so it's going to be a measure with infinite lasts where this massive going to be supported on both very very small groups and very very large and the thought item as Robert using bird not completely gentle but I ask you to believe that this is not the difficult statement that fact you only have 1 at most 1 measure satisfying we can't restriction so already weak condition is the restrictive and you end up very quickly With the fact that the this is so so restrictive that you cannot anyway have more than 1 measure satisfying and more they which involves a silly is an attack such a measure exists and not only satisfied we come from a restriction that satisfy the stronger control church property so there exists a measure on simple loops in the plane that satisfies the property that whatever domain you look at it you'll see the same measure module former and it turns out that 3 different a priori completely different construction of this measure to say a word about so that that's construction users Brown emotion imagine it take planner ground motion so this is a skating and it if you want delivery of a simple on a very financial the trajectory of a crazy fly say that's just moving around random and the played any condition to be back in the starting point a time want so this creates and of course this is not self avoiding we're looking for a measure remember maybe I'm not insisted on that before we're looking for a measure on groups that are self avoiding upset No . 0 points just things that separate the playing into connected components so a black abroad you have looked like this this is a trajectory of those has been studied this type of property has been studied extensively and it's known since the shift is that plan Browning motion because it's related to that's 1 way to do it led to a la plaster related to harmonic function is going to be invariant in some way on the court for transformations and it doesn't say didn't take too much effort with Greg actually see that measure of on granted loops here you can turn it on the way you of course are going to natural way the state invariant translation very measured on past the playing of the abroad and what you get is the invariant measure on wrong end which is translation invariant and satisfies for Wall in various properties and the very definition of the when you're going to define it is going to tell you that if you vote with just be out about this this is going to be self wanting so each running to define oneself awarding and that the measure under which self-loading defined satisfies me from Forrester now I will not insist on that but if you think about the percolation model the percolation along a large-scale is going to describe very naturally the measure clusters basically each cost of you see in there a bill the continuous of the discrete picture counselors what has must want and then just think this counting measure on classed as defined by the palatial and you get the measure's supporters on the set of Foster's if you imagine that discusses how the scaling that is performing a barrier the independent properties of preparation are going to apply fact that as this measure on on
clusters of percolation clusters that you get here is going to satisfy the same we come from all restriction property and therefore get out the boundaries of again for boosters of the outer boundary of the cost Red Elton was part will satisfy it will be measured on cellphone voiding numbers that satisfies week from former restriction so using without together Elliott you get the fact that outer boundaries of percolation clusters another boundary of the bronze emotions are just as saying the skating now it turns out that with the help of allegiance and more because a silly 1 of these along with foreign that turns out to have a very special property as I mentioned before that makes it possible to define directly the measure as 78 a sudden and to prove not only that this measure satisfies recon formal restrictions but also the stronger version and 1 of the big open questions precisely to prove that this measure on sale for avoiding loops that describing it is in fact a skating image of the if you want I'm cheating slightly the uniform measure self avoiding loops in the plane that's that's 1 way or harm education wanted invariant so what you end up with here is that out about round out the boundary of purplish plaster the skating and that are going to define exactly the same measure the same ran well since the measure but the same round shaky and furthermore you know that's the last 1 satisfies only we from restriction but struggle researcher so the outer boundaries defined in the previous case finals satisfy also this from conform restriction so this is 1 explanation why a silly is useful to solve this man wrote conjecture about the fact that the damage the outer boundaries of a planned rendezvous hero running motion general has dimension for us because I just told you that the outer boundary is exactly the same as a facility that is setting it said you can't perform computations so can compute probabilities that you couldn't computer the In the broader motion picture and then you can deduce From that computation that you get the dimension for its full abound so he you these 2 different the first one is silly gives continues object that allowed to perform computations and you have additional the side of the argument that tells you that anyway evening into different up your different models will give rise to the same object in this case and similar ideas led to to the derivation of all what will grant intersection exposed that had been predicted by the cultures before so OK you continue another consequence years just to mention to show you that this the strong from restriction is not a trivial statement issued you something like we see the outer boundary of his red and has a sudden random shake instead you look at in advance of the outer boundary of this White Island inside the red eyes so you don't get out of here so of course this is very different because you when inside the wrong motion is not the inside of the of the group but it's outside but the previous show the fact that the of the shape of the inside of his wife why is exactly the same and the rule of the shape of the red so you have surprising in there Out of symmetries so you see here that we started with this idea that but you're looking for asking an abstract question about looking for a measure of satisfying property and we end up with the fact that well there's just 1 and is not the wanted said it's a measure of supported on the set of groups that have damaged for and so on however and as it turns out that going that this measure because it's the only 1 satisfying the property is going to be probably is this very natural to related to other teachers and maybe although the boss of mathematics and useful there now I want to spend the remaining time discussing what we call cops with Scott Sheffield and that ongoing work that let me the motivation for that as explained before a silly is going to give you the 1 interface precooked you have prescribed about conditions everything is blew everything is Reggie and you get a lot of this round that is between well you might say Well that's enough the ones we know little of the we have a lot of information about the system and you get a lot of information about some critical exponents and mainly however if you want to describe the skating the entire picture that you are actually seeing you need to continue their relative what happens here and what happens there but once you did draw curfew out here which has monochromatic blue boundary conditions so you cannot really stop the same we can at its array procedure by like starting in U.S. silly somewhere he because now the boundary conditions do not have specific blue and red hot so you try to disco have to figure out something a different and if 1 looks the discrete models if you won't and the properties of the discrete models well How are you going to describe what you see here and in end In view of the main where have to describe a family of loops that correspond to the boundaries of the clusters that showed to see and these loops are going to possess properties that lead to the following definition the continuous case so you're dead this as a schematic picture so you're looking tried to describe what possible laws could be the skating limits of these random collection of other groups that use see so this time here imagine that we're looking at the configuration in our case is going to be a random collection a collection of loops these joint end that is not that I owed disjointed and do not Mr. sold you may see this picture as some sort of showed Kantor said she wanted to fit in material or hoops the idea that you should see some some type of around capital and these new guys can't think of the outermost loop of Foster's sup models and that
the condition going to ask is the following 1st of all we wanted to become forming variants so we want this to be in the making such a way that what you see in any demands of them by taking because former image what you see the and now if I cut something out of this so you got anything take any more than they now you see that you have 2 different type of you have those that interest will stay in the White region and those show up so let's go back the interiors of all the moves that touch and go out Of this white rich so now you see you have cheating slightly of course but never never knew domain which is his complement of all these black groups and a condition Is that June that upset it is not about the rule of of what remains to be explored here the what the troops in the remaining domain is the same as the old what you so in other words and this is a very natural this looks maybe a little surprising a condition of 1st sight but this is the condition you would ask these looked correspondent of indeed throughout most interfaces if you want wanted and that is not so question is what are the possible compilation of loops that she is quite like well where there is a risk of Sheffield that tell you I achieved that again you only have 1 perimeter family of such objects and what happened is that for each of these objects the troops will look like a silly you could do with a certain fixed army to cap full I have described to you what the armed attack was in the middle of silly but you should imagine roughly speaking about for a need dimension deal between the force doesn't Freehoffer there exist exactly 1 of such measures supported on these John collection of groups like this that is supported on collection of room with which our this dimension and you don't have any other such random collection than the ones are described OK you have 3 different construction of these 2 of these the collection and just very quickly describe you 1 of them docket lot but that's just faces the mathematical value so the ideas the following you're going to use the previous con following measure of self warning that described measure you're going to use it and use property in order to construct a more elaborate measure which is With it up right measures which of interacting groups now well you're going to imagine that it's going to rain troops on the screen and you are going to let deranged With intensity given by this measure you and I'm not for those who are probably know about muscle conferences they know what I mean and those who don't know also what mean because that's just learning Israeli troops according to the intensity given by the measure and you had a great time here that's at the beginning you always very small groups because of and then you fill them in and then let it render the the more you have other loops falling and of course all these things are going to overlap presented and they use have leaves falling alone on on the ground OK and the picture and then you continue gross and what happened is after a certain time of course a schematic don't you should think of these groups in groups which have damaged for us and not of the things they would paid with a lot of pain on the on my PC that if well what you see is that at a finite time legs is exactly a finite time they should prove that such at immediately after the finite size or the loops that fall down before that I to who got and form a single connected components so the union of all the lights suddenly going to crystallize In 1 single connected components now this is the way it is the way to define a fractal percolation or which is also still sometimes man broke relations which shoulders just defining random two-dimensional Kantor said that at each scale you're going to look at certain shape since they were either we keep it always throw it away and so that's what we're doing here to do here that if you want a divine around the center said that has of former prime properties because we of group to remove changes which have been the measure you as definition well it turns out that these outermost loop of the clusters you get here for any seized more than this time at which everything crystallizes are going exactly his ceiling Seoul it turns out that you Mediterranean self avoiding groups all wrong whatever you this the according fine according to the measure a seven-time not going to create clusters and you don't get beyond the boundaries of each of the past we are going to correspond to find you so she 1 few like this and have another 1 and that pictures of the is exactly the described for this year why aren't spending a little time describing to this is that In this interpretation the time you're going to let the drain of calling it your from seed because that's precisely what is called a central charge so your very constructive and with bare hands construction of around the geometric objects that have a lot of structure and for which he can now interpret quantities here and not as central charge of some of the additional just at the time you going to do it right so there's as in the previous case of self York description of obvious look of these new boss almost using other means end OK I have to 1 of the is big Alston fields construction which is let very exciting and done by shrine and Sheffield and Sheffie little also partly and here again you have this feature that 2 different up your very different models the continuing going to define the same continues the same objects even tho are priori they're very different just as the outer boundaries of percolation and Alto boundaries of Ron groups the same shape that different objects are going to define the same Sudanese OK so 1 hope of course is that these yearly collection of random interacting groups are going to be fine help you understand better at some of the the mathematics behind all the relation the formal Fiore fury at all OK could or gas and tight links between these 2 we show and should not say so here in the final slide down a cheerful but some people I listed the names of the people active currently active in this area thinking all alone the bulk of such questions you that you can sign your department just not at and off more information about this especially if you want to that mathematician roughly speaking a physicist and I could also list in the most careful Michael across from a recent gage Deason also and actually does want Spanish Long a former student of Greg so if you want more information on the side of the topics the exact date is many surveys lecture notes ICM proceedings and read the book by Greg so you you there's a lot of the introduction the subject and I apologize I realize that wanted to focus really on the self interacting and tried to get my goal was to get you in the oppression of this general feature that you're looking for a continuous objective measure of support continues Object yacht requiring properties natural if you assume that these are the stadium itself the physical models or if they are actually describing physical phenomena and you end up actually with a very restrictive a collection of possible candidates by focusing on long like 1 silly have if you focus on its more global things you get his collection of interacting loops and that just with up the simple argument you get you rather a description of a rich random objects that is and had time to before you probably related to various other pots of mathematics sorry for overtime


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