The 2D Gaussian Free Field Interface Oded Schramm http://research.microsoft.com/»schramm Co-author: Scott She±eld Plan 1. Overview: motivation, de¯nition, random processes, properties... 2. Harmonic explorer convergence 3. Gaussian free ¯eld (a) Basic properties of GFF and DGFF (b) Interfaces (level sets) (c) Convergence to SLE(4) (d) Extensions 1 Gaussian Free Field ² Generalizes Brownian motion to case where time is d-dimensional ² Satis¯es Markov property ² In 2D is conformally invariant ² In 2D diverges logarithmically (is a distribution) 2 1D Gaussian random variables Recall: A standard 1D Gaussian has distribution given by the density £ ¤ exp(¡x2=2) P X 2 [x; x + dx] = p dx: 2¼ 3 Gaussian random variables in Rn If hx; yi is an inner product in Rn, then µ ¶ ¡ hx; xi (2¼)¡n=2 exp 2 is the density of an associated multidimensional Gaussian. This is the same as taking Xn zj ej j=1 where fejg is an orthonormal basis and fzjg are independent 1D Gaussians. 4 Gaussian RV in Hilbert space In (separable) Hilbert space H, we may take X1 h = zj ej: j=1 But note that this is not an element of H. If we ¯x x 2 H, then hh; xi is a random variable with variance hx; xi. However, there are some random x for which hh; xi is unde¯ned. 5 Gaussian Free Field De¯nition If D is a domain in Rd, we let H be the completion of the smooth compactly supported functions on D under the inner product Z Z hf; gir = rf ¢ rg = f ¢g: D D The corresponding standard Gaussian h of H is the GFF with zero boundary values on D. When D = [0; 1), h is standard Brownian motion. R D h g is a one dimensional Gaussian with variance ZZ ­ ¡1 ¡1 ® ¡1 ¢ g; ¢ g r = g ¢ ¢ g = g(x) G(x; y) g(y): 6 Relation to Stat-Phys Rick Kenyon: GFF is the scaling limit of domino tiling height function. Kenyon asked: Is the double-domino contour scaling limit SLE(4)? 7 Relation to Stat-Phys Rick Kenyon: GFF is the scaling limit of domino tiling height function. Kenyon asked: Is the double-domino contour scaling limit SLE(4)? 8 Relation to Stat-Phys Rick Kenyon: GFF is the scaling limit of domino tiling height function. Kenyon asked: Is the double-domino contour scaling limit SLE(4)? 9 Relation to Stat-Phys Rick Kenyon: GFF is the scaling limit of domino tiling height function. Kenyon asked: Is the double-domino contour scaling limit SLE(4)? 10 Relation to Stat-Phys Rick Kenyon: GFF is the scaling limit of domino tiling height function. Kenyon asked: Is the double-domino contour scaling limit SLE(4)? 11 Level curves? The level curves (a.k.a. interfaces) of a continuous function f are the connected components of f ¡1(0). When D ½ R2, h diverges logarithmically. How can the level curves be de¯ned? 12 Discrete GFF Fix a (¯nite) connected graph G = (V; E), and let V@ ½ V be nonempty. Fix some F : V@ ! R, and set HF = ff : V ! R : fjV@ = F g: On HF , we take the inner product X hf; gir = (f(x) ¡ f(y))(g(x) ¡ g(y)) = rf ¢ rg: [x;y]2E The DGFF is the corresponding gaussian random element of HF . Its 2 probability density is proportional to exp(¡khkr=2). 13 Properties of DGFF ² E[h(v)] is the discrete harmonic extension of F to V . ² var h(v) = G(v; v)= deg(v). ² E[h(v) h(u)] ¡ E[h(v)] E[h(u)] = G(v; u)= deg(u). ² Markov property. 14 Proof for E[h(v)] This follows once we show that h0 := h¡F^ is the DGFF with zero boundary values if F^ is the harmonic extension of F . The probability density of h is proportional to ³ ´ 2 ^ 2 ^ 2 exp(¡khkr=2) = exp ¡kF kr=2 ¡ hF ; h0ir ¡ kh0kr=2 : Consequently, it is enough to show that hF^ ; h0ir = 0. This holds because F^ is harmonic and h0 has zero boundary values: X X (F^(x) ¡ F^(y))(h0(x) ¡ h0(y)) = deg(x) ¢F^(x) h0(x) = 0: [x;y]2E x2V 15 DGFF interface ¡a +a ¡a +a ¡a +a Theorem (Schramm-She±eld). For appropriate choice of a, the interface of the [discrete] Gaussian free ¯eld [scaling limit] is SLE(4). 16 Stochastic Loewner Evolution SLE as a limit (pictures) De¯nition of SLE Convergence proof strategy 17 Loop-erased random walk / SLE(2) / LSW 2001 18 Self avoiding walk / SLE(8/3)? Half plane SAW (by Tom Kennedy) 19 Ising interface / SLE(3)? (Thanks David B. Wilson) 20 Harmonic explorer / SLE(4) / SS 2003 21 Percolation interface / SLE(6) / Smirnov 2001 22 UST Peano path / SLE(8) / LSW 2001 23 Loewner's theorem 24 Loewner's theorem If γ : [0; 1) ! H is a non self-crossing path starting at γ(0) 2 R, and gt is the Riemann map from the unbounded component of H n γ[0; t] to H normalized at 1 by gt(z) = z + o(1); z ! 1; then with the capacity time parameterization dg (z) 2 t = ; dt gt(z) ¡ Wt where Wt = gt(γ(t)). 25 (chordal) SLE de¯nition De¯ned in upper half plane. Transported to other domains by conformal map. Let · > 0 (a parameter). Let Wt = B(· t), where B is standard 1-D BM, and solve Loewner's ¡1 equation. Take γ(t) = gt (Wt). 26 Proof strategy for DGFF interface 7! SLE(4) Let Á : D ! H be a normalized conformal map, and parameterize γ by capacity of Á ± γ. Let Wt be the Loewner driving term for Á ± γ. Let Xt be the restriction of h to the vertices adjacent to γ[0; t]. ¡a +a 27 £ ¯ ¤ Note E h(u) ¯ Xt is a martingale. Deduce good estimates for ¯ ¯ £ ¤ £ 2 ¤ E Wt+¢t ¡ Wt ¯ Xt and E (Wt+¢t ¡ Wt) ¯ Xt : This is su±cient to determine the limit of W . 28 Evolution of expected height £ ¯ ¤ What is E h(u) ¯ Xt ? Let Vt be the vertices adjacent to γ[0; t] union with the vertices on @D. £ ¯ ¤ E h(u) ¯ Xt is the value at u of the discrete harmonic extension of the restriction of h to Vt. That is, it is the average of h(v) on Vt with respect to the discrete harmonic measure from u. 29 Main Lemma. Let Yt(u) be the harmonic measure from u of the right hand side of γ[0; t] together with @+ with respect to the domain D n γ[0; t]. For an appropriate choice of the constant a, as the mesh goes to 0, £ ¯ ¤ E h(u) ¯ Xt ¡ 2 a (Yt(u) ¡ 1=2) ! 0 in probability and therefore in L2. 30 This basically says that the value of h on the right side of γ is e®ectively a and on the left side of γ is e®ectively ¡a. 31 Since harmonic measure is conformally invariant, and the harmonic measure of [s; 1) in H from z is 1 ¡ arg(z ¡ s)=¼, we get £ ¯ ¤ 2 a ³ ¡ ¢ ´ E h(u) ¯ X ¼ a ¡ arg g Á(u) ¡ W : t ¼ t t Therefore, Mt := arg(zt ¡ Wt) is as close as we wish to a martingale, where zt := gt(Á(u)). 32 One step at a time Fix ± > 0 small and mesh much smaller. De¯ne t0 = 0 and 2 tj+1 := infft > tj : jt ¡ tjj > ± ; or jWt ¡ Wtj j = ±g: Set ¢f = ftj+1 ¡ ftj . Taylor expansion of ¢M up to ¯rst order in ¢t and up to second order in ¢W gives Im(z) ¢M = ¢W + Im((¹z ¡ W )¡2) ((¢W )2 ¡ 4 ¢t) + O(±3): jz ¡ W j2 £ ¯ ¤ ¯ 3 We have E ¢M Xtj = O(± ). 33 Im(z) ¢M = ¢W + Im((¹z ¡ W )¡2) ((¢W )2 ¡ 4 ¢t) + O(±3): jz ¡ W j2 Taking two distinct u gives £ ¤ £ ¤ £ ¤ E ¢W = O(±3); E (¢W )2 = 4 E ¢t + O(±3): 34 Are we done? We actually have ¯ £ ¤ 3 E ¢W ¯ Xt = O(± ); ¯ ¯ £ 2 ¤ £ ¤ 3 E (¢W ) ¯ Xt = 4 E ¢t ¯ Xt + O(± ): It is then easy to show that Wt=4 converges to Brownian motion. Knowing that Wt=4 converges to BM is enough to show that the interface path converges to SLE(4) with respect to the Hausdor® metric. 35 Proof Outline of Main Lemma We want to argue independence of local versus global statistics: What you see in BR is not completely determined by what you see in exterior of B2R. Since average jhj with respect to harmonic measure is an average of mostly nearly independent numbers, we get that it is well- concentrated. 36 Two steps: 1. A priori we have some independence and 2. the probability to connect is essentially a product of interior and exterior. The 2nd step is the harder one. Assume all endpoints are well separated. The probability to connect the RW is about 1= log R. 37 Manipulating the interface If you compare the law of h and h + f, probabilities bounded away from zero are not distorted too much, provided that f has zero boundary values and kfkr is bounded. If f is reasonably smooth and large f >> 0 at some v, the probability that a speci¯c interface of h + f gets near + + v is small. ¡ ¡ f << 0 38 Narrows Unlikly to be a crossing of ¡ between the two green arcs. + ¡ This is proved as follows. Fix a large C > 0. Condition on the connected component of the set of vertices in h¡1([0;C]) and such edges in the bounded region that is adjacent to the interface.