An Introduction to the Schramm-Loewner Evolution Tom Alberts C o u(r)a n (t) Institute March 14, 2008
Outline Lattice models whose time evolution is not Markovian. Conformal invariance of their scaling limits. Their common Domain Markov Property. The SLE scaling limit
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Simple Random Walk Here we think of simple random walk as a law on paths (without a time parameterization) in some grid domain with mesh size δ. All paths start at some point z and stop when they hit the boundary of the domain. Refer to this law as SRW δ (z, D), where D is the domain. Important question: does the law SRW δ (z, D) converge to something as δ 0? In this case the limiting law is the Brownian law on paths. Refer to this law as µ(z, D).
Conformal Invariance of Brownian Motion Suppose f : D D is a conformal map, with D and D simply connected. Let B t be a complex Brownian motion. Then f(b t ), modulo a time-change, is also a Brownian motion. Easily proved by Ito s Lemma. Consequently the measure µ(z, D) is conformally invariant: f µ(z,d) = µ (f(z), f(d))
Conformal Invariance of a Lattice Model conformally map approximate the region
Models with Conformal Invariance but no Markov Property There are very simple lattice models that have (or are conjectured to have) conformally invariant scaling limits, but whose time evolution is not Markovian. Schramm-Loewner Evolution was introduced by [Sch00] as a possible candidate for the scaling limit. Has proven to be extremely successful, and is proven to be the scaling limit of at least 4 different models. Is conjectured to be the scaling limit of many more. Here I describe two different lattice models that have an SLE scaling limit: loop erased random walk and the percolation exploration process.
Loop Erased Random Walk Very simple model to understand. Derived from simple random walk by a loop erasing operation. First sample a simple random walk path, and then walk through the path and chronologically erase the loops in the order they are encountered.
Loop Erased Random Walk
Loop Erased Random Walk
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Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
Loop Erased Random Walk
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Loop Erased Random Walk The loop erasing procedure induces a new law on paths from z to D, on a grid with mesh size δ. Denote this new law by LERW δ (z,d). If a scaling limit exists, then one expects it to be conformally invariant, i.e. f LERW 0+ (z,d) = LERW 0+ (f(z), f(d)) In fact, the scaling limit should correspond to loop erased Brownian motion. However, there is no canonical way to loop erase a Brownian path, so the scaling limit can t be as simple as that.
Loop Erased Random Walk The time evolution of loop erased random walk is not Markovian. The curve can t intersect itself, so the future path is always highly correlated with its past. Still, there is a Domain Markov Property.
Domain Markov Property of LERW First we fix a boundary point y D, and consider the law of LERW conditioned to exit D at y. Call this law LERW δ (z,y,d) y z For LERW δ (z, y,d), it makes sense to reverse the direction of the path so that it goes from y to z.
Domain Markov Property of LERW Now suppose we ve observed the first step of the loop erased random walk. What is the law of the remaining path? y y 1 z Expect it to be, and it turns out that it is, LERW δ (z,y 1,D\[y, y 1 ]).
Domain Markov Property of LERW Can easily generalize to having observed the first k steps of the loop erased walk. Law of remaining path is LERW δ (z, y k,d\[y, y k ]). y y k z
Domain Markov Property of LERW Constrast the Domain Markov Property with the one for SRW. For SRW, having observed part of the path, the remainder has the law of a SRW in the same domain D, started from where the observed path left off. For LERW, having observed part of the path, the remainder has the law of a LERW in the domain D\observed path, started from where the observed path left off. Thus the LERW process is somehow Markovian in the evolution of the domain.
Percolation Exploration Process
Percolation Exploration Process
Percolation Exploration Process
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Percolation Exploration Process Again the curve can t intersect itself, so the future path is highly correlated with the past. Evolution of the curve is not Markovian in time. There is still a Domain Markov Property in the model. It is still expected that the scaling limit is conformally invariant (if it exists).
Domain Markov Property of Percolation Exploration Process Suppose we ve observed the exploration curve after a certain number of steps.
Domain Markov Property of Percolation Exploration Process Suppose we ve observed the exploration curve after a certain number of steps.
Domain Markov Property of Percolation Exploration Process Suppose we ve observed the exploration curve after a certain number of steps.
Domain Markov Property of Percolation Exploration Process The law of the remaining path is that of a percolation exploration process in the original domain less the slit created by the observed curve. This is the same Domain Markov Property that LERW has.
Possible Scaling Limits If a scaling limit exists for either LERW or the percolation exploration process, it should have: * Conformal Invariance * Domain Markov Property If one assumes that there is a law on paths (modulo time parameterization) with these two properties, then the Loewner equation from complex analysis plus some probability theory determines exactly what the law has to be.
Possible Scaling Limits Here I describe the law on paths from one boundary point of the domain to another (as for the percolation exploration process). This law is called chordal SLE. By the conformal invariance property, it s enough to describe the law for a canonical simply connected domain and any two boundary points. The canonical choice is the upper half-plane H, with curves starting at 0 and going to. There is also a law on paths going from a boundary point to an interior point (as for LERW). This is called radial SLE. Here the canonical choice is paths in the unit circle going from 1 to 0.
Loewner Equation Consider a non self crossing curve γ : [0, ) H such that γ(0) = 0,γ( ) =. Then t 0, H\γ[0, t] is a simply connected domain. g t γ([0, t]) 0 U t = g t (γ(t)) Riemann Mapping Theorem says there exists a conformal map g t : H\γ[0,t] H. Loewner Equation describes how the maps g t evolve as the curve γ[0,t] grows.
Loewner Equation Consider a non self crossing curve γ : [0, ) H such that γ(0) = 0,γ( ) =. Then t 0, H\γ[0, t] is a simply connected domain. g t γ([0, t]) 0 U t = g t (γ(t)) t g t (z) = 2 g t (z) U t, g 0 (z) = z
Basic Idea of Loewner Equation Supposing that we know g t, can we use it to find g t+dt?. γ([t, t + dt]) g t γ([0, t]) g t (γ[t, t + dt]) 0 U t g t+dt h t,dt : H\g t (γ[t, t + dt]) H U t+dt Can compute the map h t,dt explicitly.
Basic Idea of Loewner Equation w (w U t ) 2 2dt U t 2dt 0 w w + 2dt h t,dt (w) = U t+dt + (w U t ) 2 + 2dt w + 2 dt w U t g t+dt (z) = h t,dt (g t (z)) 2dt g t (z) + g t (z) U t
Basic Idea of Loewner Equation w (w U t ) 2 2dt U t 2dt 0 w w + 2dt t g t (z) = 2 g t (z) U t, g 0 (z) = z
Technicalities of Loewner Equation First note that g t still has three real degrees of freedom. Choose two as follows: g t ( ) =,g t( ) = 1. Laurent expansion at must look like Set a 0 = 0. g t (z) = z + a 0 + b 1 z + b 2 z 2 +... All other coefficients are now fixed. Two important quantities: U t = g t (γ(t)) and b 1. The b 1 coefficient (which is a function of t) is called the half-plane capacity of γ[0,t], and denoted by a (γ[0,t]).
Technicalities of Loewner Equation First note that g t still has three real degrees of freedom. Choose two as follows: g t ( ) =,g t( ) = 1. Laurent expansion at must look like g t (z) = z + a (γ[0,t]) z + b 2 z 2 +... Set a 0 = 0. All other coefficients are now fixed. Two important quantities: U t = g t (γ(t)) and b 1. The b 1 coefficient (which is a function of t) is called the half-plane capacity of γ[0,t], and denoted by a (γ[0,t]).
Technicalities of Loewner Equation It is easy to check that a (γ[0,t]) is additive, i.e. a (γ[0,t + s]) = a (γ[0,t]) + a (g t (γ[t,t + s])) γ([t, t + s]) g t γ([0, t]) g t (γ[t, t + s]) 0 U t = g t (γ(t)) Can also show that a (γ[0,t]) is continuous and increasing, so can reparameterize the curve so that a (γ[0,t]) = 2t. g t (z) = z + 2t z +...
Using the Loewner Equation to Produce Curves So given the curve γ, the maps g t must satisfy t g t (z) = where U t = g t (γ(t)). 2 g t (z) U t, g 0 (z) = z Can turn the situation around. Given the driving function U t : [0, ) R, the ODE can be solved to determine g t. The g t then determine γ[0,t]. It s not a priori true that any driving function U t will output a curve. A sufficient condition is that U t be Holder continuous with exponent greater than 1/2.
Candidates for the SLE Driving Function To produce random curves, U t obviously has to be random. Should also be continuous. Inputting a random U t induces a law on curves in the upper half plane. Claim: If this law satisfies the conformal invariance and Domain Markov properties, then it must be that U t = σb t for some σ > 0, where B t is a standard Brownian motion.
Candidates for the SLE Driving Function Domain Markov: Given the curve γ[0, t], the law of γ[t, ) is chordal SLE in H\γ[0,t] from γ(t) to. Conformal Invariance: The chordal SLE law in H\γ[0, t] from γ(t) to is the pullback of the chordal SLE law in H from U t to. g t γ([0, t]) γ (s) = g t (γ(t + s)) 0 U t = g t (γ(t)) Hence γ (s) has the law of a chordal SLE curve in H from U t to that is independent of γ[0,t]. But γ (s) is determined by the driving function U t+s U t,s 0, and γ[0,t] by the driving function U s, 0 s t.
Candidates for the SLE Driving Function Then γ (s) independent of γ[0,t] means that the driving function after time t should be independent of the driving function up to time t, i.e. U t is a Markov process. Moreover, γ [0,t] is identical in law to γ[0,t], so U s, 0 s t, should have the same law as U t+s U t, 0 s t. g t γ([0, t]) γ (s) = g t (γ(t + s)) 0 U t = g t (γ(t)) This forces the increments of U t to be i.i.d., and writing du t = b dt + σdb t this is only possible if b and σ are constants.
Candidates for the SLE Driving Function Hence conformal invariance and the Domain Markov property force U t to be Brownian motion with drift. One extra consideration: the law of the SLE curve should be invariant under reflections about the imaginary axis. This means the law of U t should be invariant under negation. Hence b = 0, leaving du t = σdb t. Schramm writes U t = κb t.
SLE Formal definition of chordal SLE(κ): the solution to the ODE t g t (z) = 2 g t (z) κb t, g 0 (z) = z where B t is a standard one-dimensional Brownian motion with B 0 = 0. The curve γ corresponding to the maps g t is called the chordal SLE(κ) curve. Since B t is not Holder-1/2 continuous, it s not a priori true that a curve is produced for SLE(κ). But [RS05] proves that it is true.
The Dependence on κ κ looks like an innocent parameter, but it actually plays a very important role. κ = 0: t g t (z) = 2 g t (z), g 0(z) = z g t (z) = z 2 + 4t 2 t 0
The Dependence on κ As soon as κ > 0 the curve is immediately fractal. As κ increases it becomes more fractal. [Bef07] proves that the Hausdorff dimension of the curve is almost surely min ( 1 + κ 8, 2). There are also phase transitions at κ = 4 and κ = 8.
The Dependence on κ For κ 4, the curve is almost surely simple (i.e. doesn t touch itself). Moreover, it doesn t intersect the real line, i.e. γ R = {0} For a fixed z H, almost surely the curve does not hit z. SLE(2)
The Dependence on κ For 4 < κ < 8, the curve touches itself (but does not cross itself). The curve intersects part of the real line, i.e. γ R R. For a fixed z H, almost surely the curve does not hit z, but does make a loop around z. SLE(6)
The Dependence on κ For κ 8, the curve is space filling. Hits every point on the real line, i.e. γ R = R. For a fixed z H, almost surely the curve does hit z. SLE(κ > 8)
What κ Correspond to LERW and Percolation? For LERW expect κ 4, for percolation expect κ > 4. Turns out that LERW corresponds to κ = 2. Percolation corresponds to κ = 6. To determine this, one has to compute some specific probability or statistic for the lattice model, then compute the same thing for SLE(κ). There s only one value of κ for which these two quantities agree.
Models with an SLE Scaling Limit κ = 2: Loop-Erased Walk κ = 8/3: Self-Avoiding Walk κ = 3: Ising Interface κ = 4: Harmonic Explorer and the Gaussian Free Field interface κ = 6: Percolation Exploration Process κ = 8: Uniform Spanning Tree
References [AS07] Tom Alberts and Scott Sheffield. Hausdorff dimension of the sle curve intersected with the real line. arxiv:0711.4070 [math.pr], 2007. [Bef07] Vincent Beffara. The dimension of the SLE curves. To appear in Ann. Prob., 2007. [RS05] Steffen Rohde and Oded Schramm. Basic properties of SLE. Ann. of Math. (2), 161(2):883 924, 2005. [Sch00] Oded Schramm. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math., 118:221 288, 2000.
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