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# Random, conformally invariant scaling limits in two dimensions

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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
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.
Total <Mathematik>
Extrempunkt
Hausdorff-Dimension
Zeitbereich
Gruppenoperation
t-Test
Gleichungssystem
Bilinearform
Massestrom
Komplex <Algebra>
Computeranimation
Evolutionsstrategie
Exakter Test
Beweistheorie
Auswahlaxiom
Distributionstheorie
Subtraktion
Punkt
Ortsoperator
Hausdorff-Dimension
Gruppenoperation
Formale Potenzreihe
Rechteck
Bilinearform
Äquivalenzklasse
Gesetz <Physik>
Deskriptive Statistik
Einheit <Mathematik>
Symmetrie
Inverser Limes
Minkowski-Metrik
Dimension 2
Umwandlungsenthalpie
Beobachtungsstudie
Zentrische Streckung
Kurve
Graph
Kategorie <Mathematik>
Stochastische Abhängigkeit
Physikalisches System
Biprodukt
Zeitzone
Koalitionstheorie
Brownsche Bewegung
Verbandstheorie
Ganze Zahl
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Koordinaten
Ebene
Resultante
Distributionstheorie
Subtraktion
Physiker
Kritischer Exponent
Punkt
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Rechteck
Zahlenbereich
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Exakter Test
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Lineares Funktional
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Randwert
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Perkolation
Gammafunktion
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Varietät <Mathematik>
Riemannsche Fläche
Distributionstheorie
Einfügungsdämpfung
Abstimmung <Frequenz>
Punkt
Mathematisches Modell
Familie <Mathematik>
Gleichungssystem
Fastring
Gesetz <Physik>
Hamilton-Operator
Massestrom
Richtung
Vektorfeld
Gruppendarstellung
Arithmetischer Ausdruck
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Minimum
Nichtunterscheidbarkeit
Translation <Mathematik>
Analytische Fortsetzung
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Auswahlaxiom
Parametersystem
Dicke
Kategorie <Mathematik>
Güte der Anpassung
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Dreieck
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Topologie
Ausdruck <Logik>
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Abstimmung <Frequenz>
Prozess <Physik>
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Einheit <Mathematik>
Vorzeichen <Mathematik>
Existenzsatz
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Translation <Mathematik>
Schnitt <Graphentheorie>
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Analytische Fortsetzung
Phasenumwandlung
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Parametersystem
Dicke
Extremwert
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Kategorie <Mathematik>
Fläche
Dichte <Stochastik>
Rechnen
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Kugelkappe
Arithmetisches Mittel
Randwert
Konditionszahl
Ablöseblase
Körper <Physik>
Ordnung <Mathematik>
Perkolation
Riemannsche Fläche
Darstellung <Mathematik>
Subtraktion
Glatte Funktion
Hausdorff-Dimension
Physikalismus
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Toter Winkel
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Ausdruck <Logik>
Überlagerung <Mathematik>
Multiplikation
Spannweite <Stochastik>
Knotenmenge
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Einfach zusammenhängender Raum
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Zeitbereich
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Objekt <Kategorie>
Brownsche Bewegung
Offene Menge
Mereologie
Differentialgleichungssystem
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