Random, conformally invariant scaling limits in two dimensions

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Random, conformally invariant scaling limits in two dimensions
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Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models. Only recently have some of these predictions become accessible to mathematical proof. One of the new developments is the discovery of a one-parameter family of random curves called stochastic Loewner evolution or SLE. The SLE curves appear as limits of interfaces or paths occurring in a variety of statistical physics models as the mesh of the grid on which the model is defined tends to zero. The main purpose of this article is to list a collection of open problems. Some of the open problems indicate aspects of the physics knowledge that have not yet been und erstood mathematically. Other problems are questions about the nature of the SLE curves themselves. Before we present the open problems, the definition of SLE will be motivated and explained, and a brief sketch of recent results will be presented.
Keywords statistical physics conformal invariance stochastic Loewner evolutions percolation
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plan of the talking this is my plan it would like to look at random systems such as the 1 you see on the sly and this 1 has a very simple description all that makes a gonzo independently a White of with probability 1 half and we're going to
study such systems and the way we're gonna learned information about that Is it by studying random curves that's the Pyrenees systems such as the here and Oh accommodation going to be the prime example but not the only 1 so that this is called coalition and the 2 closely related talks they're actually very closely related talks to mention at the talker bystanders me an off day yesterday in ended in burners talk later today in some things which I will make for lack of time you you can probably learned from and the Nunavut of and let's start with something rather simple and let's discuss said when they mention Bonnie in motion and 1 way to think of one-dimensional body in motion is of simple and no more curiously a simple and walkway is the Timex In the Y axis the spatial coordinates In the M in each at a specific time units on an integer lattice and their walk with probability 1 have goes up or down regardless of what has happened in the past and you get such a graph that these this shape rectangle steps are not drawn With the height difference equal to that and time difference because we're going to take a limited as the and as the time and space scales and get there Brownian motion and you have to scale that time and space differently if you skiers faced by Delta Air Force scare time by the square of this is some a definition and show on bond in motion here are some properties Brownian motion hands and it is a continuous With probability 1 and it satisfies the mark of property which means that if you want to know if you're a time she'd anyone predict something about the future and not with certainty but with some reliability well the only information that relevant for you that's in the past as your current position and a emotions satisfies Runyon scaling Delta time the Times the Brownian motion is the the same in law has the same distribution centers and the body in motion with time scale by Delta squared and let's go on onto 2 dimensions most of the talk will be entirely two-dimensional when you take care to Barney's Martian you have to to an actual ways to define it since we defined by emotion as the limit of their simple walk you can also define to their initial abound in motion is enamored of simple and walk on the square footage but turned everyday you can take 2 independent bunny emotions 1 in there x coordinate wonder y coordinate and therefore gave them a 2 dimensional cooked and these 2 2 definitions of equivalent but definitions suggest that Brownian motion world actually what both definitions used the coordinates the X and Y coordinates about a nice property of two-dimensional body motion and neither mission abound in motion is that while emotional effect doesn't depend on the coordinate system only depends on there In a product I'm so yeah namely its alteration and its into mentions it has a higher form of symmetry its conformity so that means the following and you have viewed the made in you have a basis point and you start running motion from the the based pointer stop when you're the evaluating and you consider an analytic all Eicon formal map to another the main you start In the other the main body motion from the image of the based pointer From at me put it this way around the image of this but in motion in this the main is the same as a Brownian
motion starting at the image of the based pointer except that the time kilometers Asian changes and so you should keep this in mind because later on we will discuss various conformal invariant different models and Iowa Eduardo Lewis and I will not say what conformal invariant means I'm but you should think of something something in this in this legs on where maybe I should say it did a simple fact is that just that they hit think point on the boundary from formally but disk on formally invents means means much more means that all distribution of the car Is conformity OK let's say move on to a more difficult a topic which is percolation so you have a measure of Hanks the Ghana's this is the most successful 2 dimensional commission modeled by now and if you doing an annuity collection he checks on his white open with probability p independent thinker and then well so far just as a system of independent did but once you start studying connected components of white all-black black wages and then you you doing accommodation and various alternative models was a large variety of alternative models of accommodation and one-note model is you start you work on a grid and you flip a coin for each edge to decide if you keep it afloat away now and that's percolation now we will discuss critical percolation turns out and that's pretty easy to see I want not pull over to you that there is some number between 0 and 1 c we have this
is parameter which we can decide the probability for each extra can't be white we can play with us and there's some specific value that is course shot and that's called piece of cedar critical value of heat and in the end it can be defined in various different ways has been defined in various different ways but the simplest definition of medically is by saying that if he is larger than that value you would have an infinite white component and if it is smaller than this value than the probability for the existence of an infinite white component is going to be 0 and this is a good a model for phase transition because the large-scale behavior of the system changes drastically when P pushes beyond be and so what do I mean by that I mean if slightly smaller than PC you look at a very large picture then almost nothing is connected the White legions of small islands although small lakes and inside a vast area of blank and the change in this situation vs when he is above said this is very easy it's harder to determine what B. and set for sample collection models we will probably never know what these years for this but ticket model that I've described he found it has we know well Harris formed 1 half of the year showed that he was on hand who is now infinite and this means that peace is at least 1 half in 20 years later tests and showed that an easy actually is 1 hand because about about one-half you do have an infinite cluster in the the 1 I have here is a value which is easy to get this is because your head at 1 one-half year have symmetry of white and black they behave in the same end if you believe my story about 1 completely dominating the other when you're off awful PC then you should believe that one-half has To be the critical value but moving it it was a significant achievement and based on an earlier important result of where was the same 1 Welch OK cell Hamas's theorem tells us that at 1 and we don't have an infinite cluster and that is the case now what happens if you you condition on this where event that the cost over the origin is going to be launch so if we refuse sample percolation you know that with probability 1 the cluster of the origin is going to to be fine so typically expected to have maybe size 5 6 0 X on something like that but you can't and when you're simulations a few times and you can wait until you get a simulation which has many thanks God and the cost of the origin and then and this is roughly what the cluster looks like if you wait till you get a sample with a thousand fixing on all and so you can go on and you can ask yourself How long will I have to wait till I see a cluster of a thousand Mexico how how would likely does the probability of having them a large cluster of how fast as it decays as the notion of large increases in whether such questions are governed by a what's known as critical exponents and physicist such as the knees now it's comedy for addicted and the values of these exponents in particular the probability that the origin is in a cluster of Diana at least on behaves like not to the minus 5 over 48 and with some no Local elections as art tends to infinity and if you asked a similar question but require that connections be inside its inside the high half plane then they exponent you get minus the evidence that of my bible for years in the air these were conjectures and the arguments typically as daughter conformal invariant and many outdoor authorities the way the physicists like to assume conformal advance they they look at state and the probability that 2 takes gongs and all in the same cluster and em they I believe that those gonna be a limit that is the measure of fines after you we scared of folk quickly and then they say well this limit should have nice con formerly invest poplar trees and make similar assumptions and then once they have a few assumptions and place they can whoops of machinery to to calculate the value and so could formally invariants plays an important role now that also would pictures which I want more refined than just saying in next opponents annex but it just gives you 1 number which is very important because it tells you know what these events how do they know how likely they are and so forth but as useful and as it is don't put and things to know about for example comedy had the following addiction if you look at the rectangle and you take a measure fine in U.S. What is the ability to have a connection in white makes advanced from the left of the line in the limit is the measure of finds is this quantity approaches this expression right here this is the Gamma function after 1 is the would geometric function in a is the
cost is the cost ratio of these 4 points after you may the rectangle conformity to the upper leg so this is an a reasonably complicated formula Leonard also found a nice of form for this formula conformal invariant cities is as far as conjectures go with you you want to assume it then naturally because otherwise you can you can't get even started and Carlson's form of God is formulas the following you take care triangle and you want to connect the bottom managed to his segment on on top on the white bandage and that edge length of this triangle is wanted in the length of this segment is the probability to have this connection converges to X is the measure goes to 0 and this is somewhat simpler than and colonies performed well just so that we don't get carried away with percolation I want to mention and a few other models because percolation Though it's essential see if theme it's not the only topic I want to discuss in here I can use this picture to describe 3 different models what is this picture you know you have a white here and on the white and you should think of the White who it is a graft probe it's about all a bolograph but you should think of attempt by a grid square grid and the whites is the Spanish victory in that it if you notice the white maize he has now cyclists and it spends every verdicts in the book withdrew it and so if you have that then by wanted you can think about choosing and Tree from that and you can say well so I want to be unbiased I'm going to choose the spending Chile uniformly among all spanning trees and that will at 1st I was kind of suspected it skeptical about then this model why would the uniform measure being a natural thing to study well it turns out this uniforms spending has many nice connections to different types of questions such as electricity and random walks and so forth so it it is very natural infected take a uniform measure so all is the uniforms spending to the big and is also a uniform spending 3 of the duel of the original graph the bank is just essentially to complement of the wife and family In the Black Cove between them is what's called the piano associated with the uniforms spanning tree and it's also known as the Hamiltonian from the Manhattan land now those so here we have a two-dimensional system we have 1 Cove associated with it and in fact there's another nice of associated system if you fix 2 vertices say white but this season you fix them before you see the jury menus sample the chili and those just 1 path in the maze connecting the the 2 vertices because it's a train that that and what's known as the EU In a walk they you and walkers model and invented by Greg although and if it is intimately connected to their uniforms and drink Hey wet and so the was a belief that the uniforms spending In the poets and walk a lot to be conformal invariant and end the has some partial progress in that direction with Kenyan that some properties of the uniforms spending to renew poets and off formerly in the skating limit in here an example for property which if you take it here is the main thing you fix they points in the bound and you take the uniforms standing of the grid the restricted to this domain then near the points you can take season you go in and join them out by the paths that joined them in the drain and you will have the kind of the a Y shape topological white to and spending these opponents and there will be a special meeting points this led me meeting point and if you you can't do this thing in another and you can I'm assuming the demands of simply connected and you you can map wonder made in conformity with the other domains and what a Kenyan as food is that the distribution of this town where the meeting point is conformal invariant namely the image of the measure here the long the point is going to meet the of the red points you provided the play basis points a map of the fleet based on that that is a good a good bit of farm West it doesn't quite conformal invents of this whole system for example you don't yet know that the these bats are actually have the same distribution in both picture OK to study book relation it turns out that that we want to look at an interface let me down and the demo year-round showing this interface so you start and stop them by declaring that you want the boundary a hexagon still be there to change color Colosio voyage in here this is what's called it a devotion boundary conditions and then you can start an interface between their blue and yellow and you want to know how this interface continues so you look at what this X adorns gonna do them In this taxing on you you flip the corn you see its yet Soviet tonight and then you look at the next exit monitor and blue you turn left in the next 6 to go on and next Texaco now sometimes you you that 1 exit on you the rest all of which slipped 1 exit said on and we could that made flee steps of the of the interface because the interface when it hit takes a gantlet that have been college fault it doesn't need to sample a you 1 if you continue and so on and so on and so on and if you continue like that you build this interface which has no choice but to a go out to infinity we
should think of it as an interface connecting the 2 boundaries switching points which is 0 and infinity Hey and so now we're going to take care of but so far I haven't done really any just told you about now we're going to full slides roughly we're going to to actually do some math here and I'm going to try to what so their intuition for and why you want to consider an 11 hours equation with Burundi in motion is that and driving the whole you don't need to know all any of these words were going to see and this is awfully their way this actually a villager motivational villager motivation was in the context of the random walk with that the pictures so we take our we take our chrome and buzz that we are we only look at a lot of it which stopped when it reaches size Epsilon and I'm not going to be found very clear at this point what size means we will see later what size me and then we believe in conformity invariant from now we will task of justifying conformal invariance we will discuss later we believe come forward in Brentwood we've sampled the part of the wall of this interface and would like to know what the West looks like whether the traumas that this he's here is very complicated it's kind of fractal shit we want to simplify but we believe in conformity variants and see the boundary this the main is after you remove the path is a the by to Collins the widen the black here and they change in exactly 1 point which is the tip of the cook and that fall we can simplify the domain which in May Apple vote by the conformal man we use of England's you are telling us that the complement of this region that we've examined those already is simply connected therefore we can method over to the upper half playing again it when we met at over to the Apple half lender would be a special only 2 of them the image of the black curve meets the image of the white and the red view is indicated to show what part of the boundary corresponds to things that an adjacent to the cook In this this meant that ordinary users has has some power series expansion near infinity and we normalize so it fixes infinity and the power series expansion looks like that this is all very easy consequence of the amendment interference in if we think about the continuation of the path he a that's gonna built in distribution about the same as what Buergel path would looked before started it at the tape airway at the Boundary Pointer right here because we believe conformal invariant 2 now we can continue the pussy July we can after we'd we can start here we can build the interface until it has size Epsilon and then Apple will by another man left and then you have too has essentially the same distribution F-1 1 except that it's translated to started seeing that the point TO were black changes to White is not necessarily the same point as black changed to 1 year in cell F Tiller has the same given that we now want to harass the distribution of 1 except translated therefore conjugated by but this translation in this Q denotes equivalents and loss and we can continue inductively if if W denotes there Place where the image of the J map it's because Jerry composition takes the tape to sample a and then we have this composition Jeanne which is a composition of all these deaths and beneath each X has essentially the same law that except conjugated by translation depending on the W Jagr it and this is just metaphor for the compliment of the Cobra examined so far the upper half playing because as we each time with added at peace and we've taken the map that makes it disappear and now each J is going to be very close to their identity because our epsilon is going to be very small therefore we can attempt to think of this composition as if flow either event a composition of discrete steps because if you haven't met the 1st of their identity you can think of it as approximately vector field I'm enjoyed the stand vector field were going to essentially find out what it is we can look again at this map F 1 which has the power series representation of infinity In the canal to simplify things we we want to choose cleverly how we measure the size of the initial segment it turns out that day 1 is monotone In their pants as you stand the 81 increases known for em To make life easier which Tuesday want to be essentially to epsilon therefore we stop the cove once a 1 reaches a size to Epsilon and then easy calculation with with our series and the skating argument implies that F 1 is well they to epsilon C and plus the other islands which are bounded by onto the Rehan I'm not going to do this calculation for you now and then the consequences that FDA plus want is going to be the same thing except conjugated by the translation and WJN therefore has this expression if J. plus 1 is the same thing except you come here have minds WGA and therefore so this is essentially a vector field where this is the vector at the Poynter GQ and therefore instead of thinking of the composition of a new identity Mets weekend and flow according to this vector field this W. is some continues version of the WJC and well what is this continues version of the WGA swelled the W. James heavy stationary independent increments as we go as you go from WGA to WGA Lewis won the distribution of that inflation is going to be the same as W.
wanted W-2 from W-2 W. all the same because and we've remark that the loss of these maps all the same except for the translation once you take away the translation and the is the same and therefore had the WGA Have the saying that it's like a simple random walk except them they're not necessarily have a fixed length but never Random Lake but they have the same Random Lake each time and so this WTU which is limit of these W. James is going to be a process that is has stationary independent and its cement because no problem is symmetric and its continuous that you have to order by some other means but that's not too hard and it follows easily once in a little bit about motion that it's a multiple of bomb motion in this leads leads us to the definition of what their years as Cynthia as the plastic 9 pollution and where we fix the parameter cap and we defined double duty as Burundi emotional time kaput in this body motion is one-dimensional and then you solve this differential equations starting at Geneseo of disease was saying this is essentially what was supposed to to be the Con Format for the compliment of the interface to the upper half time and this equation is known as revelers equation is used it to study the about conjecture and them well studied the different mostly used a different a slightly different form of the the equations is what's called called end once you solve this equation you can recover the Panthers itself by taking GQ universe of W what is the basic idea any other ideas that we want to um we want to invoke and differential equations to study and I will local nations now the poker additional include interfaces a fractal object and it's hard to study Frank goes away With differential equations differential equations other better suitable to smooth objects what we have a smooth object at our disposal the complement of the interface that domain and therefore a smooth object so instead of studying devolution of the interface we study the devolution of the complementary and that's amenable to a differential equations and therefore we can not only find out what what we guess it should be but also and then we can do calculations to to calculate properties of the compliment and therefore of the interface well this old discussion wars conditional on having conformal invariance but what we didn't have conformal invariance so what do we do and whether it seems just to be a nice noise physics like predictions and not so satisfying for imitation it this Foley and is run of them Jim not with a brilliant idea that well we have if like philatelic disposal which is known to be conforming valiant but still has some some nice questions the tether haven't been hands yet so why don't we applying this method of understanding conformal invariant Frank due to Brownian motion and if you look at bunny emotion itself and in some ways it's very simple however if you look at it How to violative Brownian motion that's not amenable to analysis directly from the definition of onion motion so what you see in this picture is a volunteer emotions started at the your origin and stopped when it hits the but of the unit this into editors that all the places visited by the motion and the blankets the outer boundary of the Brownian motion to is set the point so that you can get to the exterior without I'm hitting any other points before that and there was a conjecture Mandelbaum that out of bounds a Obama much has for dimension for 3rd and world rendered was I a city was delighted to to attack his for men and several related problems about the the Brownian motion and and I think when dealing with their you more about this later this afternoon but let me just say I give 1 of the punch signs and Greg duo and ended in a nite flows and that indeed on the board for conjecture is correct the boundary has been mentioned for floods and and you can extract other information about because for example the set the cut points as house 0 4 dimensions play for and cut points of those points will if you remove them you disconnect the bond in motion In this based on 1st then where this is not how which was written by a poster legally we now know that in fact the outer boundary of this Runyon emotion is essentially the same as the outer boundary of at least 6 and that's what laments that the connection between them what y studying as they can tell us something about money motion at 1st we didn't know that the go actually the same but we knew that the dimensions now well OK so that was winding you but we didn't have to wait to long and then that's not actually 4 of them Toddy's formula in Carlson's for them and in fact the way the simple form of thinking about trying those antics agonal leftist turned out to to be an important observation leading to the vote Bush is quite brilliant but in an amazingly simple ensure dust that infects goddess formula and is co-writer and and for me this it's not talk to conclude that Commission interface skating limit the sank over the showed an opening is a 6 6 years that for Emmett cap of the you a memo of city has a free polemic in this feel allowed the determination of many Of the exponents associated with percolation in particular I'm we showed that the exponent that I discussed in the beginning the probability to connect to distance R & B decays like onto the minus final 48 with some Bordeaux court actions and many others exponents then properties were approved by me often Verna based on earlier work by Justin relating all these exponents to 1 another there less than a point of view only helps determine things at PC at the critical probability and some exponents of relating to offer to call but Keston and linked these with with a critical don't and subsequently and look tourists and and it was both to converge to S still in uniform spending and that's gaining women was proved to be a silly 8 and this was a Norway and here the situation kind of the opposite in this sense the 1st convergence to a city was moved and a consequence of that was conformal invariant rather than the other way around here is a picture of what the new and what looks like and so you see that With With different different values of the parameter Capone give entirely different bosses things and this may be a little bit surprising at 1st that you can change things like How still dimension and properties and just by changing the speed of the Brownian motion it is like to you can think of the bunnies motion as a driver a taxi driver of the taxi driver has won year maybe can still drive so reasonably well but after a few more doing becomes erratic in the consequences on easy to see so does you have chosen Hey but to phase transitions in the behavior of Sydney as it as depending on cap this was approved with Stephana if cap is a must for its a simple asked if it's between 4 188 then it's not a simple matter anymore it kind of swallows parts of the domain as it goes along lastly what they're as 6 pictures look like if you remember we we will see them shortly again and when Campeau is NATO launcher this because is actually space filling it visit every point not necessary that it cannot colossus itself but it contracted therefore this case once you circle some part of the can't go in they'll end that top that 1 difference from Brownian emotional bonding Washington and make it then going the to open out again that's what bond in motion is more difficult to study from from this point of view the no notice that parameter painters precisely what their uniforms spending and that converges to and it's easy to see from the definition that has to converse to a space cover and that's exactly the transition will that a cap where the transition to space filling because does some information about the dimension of that Sydney In this paper with tension effort and we showed that expected box found dimension well expected box mention not don't find this concept because you can probably guess what it means by phone from its name and the expected box mission of the ad is 1 us although and I went kaput between 0 and when campers larger than it bits of course To because it's based and In would cap but is larger than for the outer boundary of the ban is not the same as the act itself because it is it's not a simple plan and there expected box found they mention of the bundling is given by this formula which was earlier conjectures by Canyon and Vincent the fire a man made quite clever and technical awoke managed to show that this expected box on mission of the ad is in fact that actual house mention the expected box found their mission much simpler to to determine because you just need to calculate expectations and for house dust emissions essentially you need to calculate 2nd moments in the more difficult I think that the corresponding statement for this is still open but probably won't be for long and I want to tell you about yet another model which was relation tourism has been determined this is the Garcia we field what you should think let let's discussed displayed gas and we feel there's also a continuous version which is of the same flavor Longley harder to define should think 1st of graph and then a teacher but they have some value in nature and you want to penalize discrepancies it said it's a model for you can think of it as a model for which has said penalties for discrepancy from from being smoothness so you penalized it should be minus each of you squared for vertices that are close bind so that the total penalties is the exponential of minus the sum of all Neighbors of the square and and this is a nice formula because the distribution of a this probability density is going to be a and Her like Canyon showed that this Gustine fully feel there's a scaling limit of the dominant tightening height function which is another discrete model which I'm not going to describe to you but this was partly motivation to study the guys in free field in is and what you do with a gas flee feel if you want to do it you put the Bush boundary values again you put abounded values plus a few minor staying here and then you can look at the center of edges you can look at that you can Carlisle and vertices warned can't positive another problem so negative and then you can take an interface and this is graphical depiction of this interface Scott Sheffield a nite for that and this interface of the discrete gas field converges to a city fall now for the special value were the interface he above fold the above stops being a simple code when you consider that this this is an actual simulation you could see the go is almost not a simple curve and the shades of blue and red made depicts the heights of the the field now and in order for this statement to all Cho you need to take this the value of a very specific value if you take take a different value then you get convergence to something very closely related which we also wanted the and as an analogous statements about the continued Agustine free field but that Hodel to formulate and those a conjectural so as not purported done much for Orbison this conjecture critical easing model converging U.S. and this is this picture and discuss also shows it with other parameters and things associated with the easing model also converged to listen to you but with a different value of cap this they do John by David Wilson and so the White is dead last and Dorcas minus if assume your familiar with the model if not then I think of this as just another pretty big church hopefully with and if you take a uniform measure on on leftist the dollar self avoiding and stay the upper half Blaine the belief is that that converges to lessen 88 I can't say I can't aboard and followers on the conjecture but supported experimentally by and by simulations by Tom Kennedy in this figure being drawn by Tom Kennedy Concho it to all the other figures In this talk this 1 which you really need to to work hard to produce all the others has very very simple out so it can well time is running short and I'm going to give their quick some owing your list of things were and we know as silly and describes the scaling limit of cricket critical site percolation on the triangle of greater than the value of capped by 6 your worries and Walker values to the piano curve of the value of any gusting free field In the values Warner kicks it is a simple model that is kind of Bill defined to be easy for for this statement and and I'm not and it an inhaler some conjectures regarding other models converging to ascertain whether no other models of critical poker nation which converged to boundaries should converts Characinidae 6 but that's not been for vote the easing model and with different parameters converges to read and 6 and there but we hope to see if both by by us with and they FK cluster boundaries while I haven't defined them but I mention them here because they actually those another programmer took their cue and therefore you have a whole range of Kappa between 4 and 8 and similarly with that when models is self avoiding walk is of course very challenging and the double don't know has relations to the Gaza revealed we mentioned the solidity and those things that destiny does not give it does not give due laying those being there the type analysis of the DLA Nick Collison and call by Hastings Leavitt of that actually was seeds the Sydney definition it appears that paths of the minimal standing Chile or not and described by a Sydney and are probably not conformal invariant this is based on simulations by Whelan and Wilson and a as doesn't apply to many interesting questions and dimension higher than 2 let me mention the focal irrationally high dimensions is almost completely understood due to the work of Holland Slaton are those with the bank a method called the latest expansion which is very different than what I described today and the existence of the poets and walked scaling limiting fully has been pulled recently by God because but gotti hasn't determined what limit is just shown that exist but that's quite quite remarkable achievement well so we have seen things that you can do with silly things that hopefully will be done things that you need to use other methods thank you very much for them