The 2D Gaussian Free Field Interface

Oded Schramm http://research.microsoft.com/∼schramm

Co-author: Scott Sheffield Plan

1. Overview: motivation, definition, random processes, properties...

2. Harmonic explorer convergence

3. Gaussian free field (a) Basic properties of GFF and DGFF (b) Interfaces (level sets) (c) Convergence to SLE(4) (d) Extensions

1 Gaussian Free Field

• Generalizes Brownian motion to case where time is d-dimensional

• Satisfies Markov property

• In 2D is conformally invariant

• 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 ∈ [x, x + dx] = √ dx. 2π

3 Gaussian random variables in Rn

If hx, yi is an inner product in Rn, then µ ¶ − hx, xi (2π)−n/2 exp 2 is the density of an associated multidimensional Gaussian.

This is the same as taking Xn zj ej j=1 where {ej} is an orthonormal basis and {zj} are independent 1D Gaussians.

4 Gaussian RV in

In (separable) Hilbert space H, we may take

X∞ h = zj ej. j=1

But note that this is not an element of H.

If we fix x ∈ H, then hh, xi is a random variable with hx, xi. However, there are some random x for which hh, xi is undefined.

5 Gaussian Free Field Definition

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, gi∇ = ∇f · ∇g = f ∆g. D D The corresponding standard Gaussian h of H is the GFF with zero boundary values on D. When D = [0, ∞), h is standard Brownian motion. R D h g is a one dimensional Gaussian with variance ZZ ­ −1 −1 ® −1 ∆ g, ∆ g ∇ = g · ∆ g = g(x) G(x, y) g(y).

6 Relation to Stat-Phys

Rick Kenyon: GFF is the scaling limit of 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 ⊂ R2, h diverges logarithmically. How can the level curves be defined?

12 Discrete GFF

Fix a (finite) connected graph G = (V,E), and let V∂ ⊂ V be nonempty.

Fix some F : V∂ → R, and set

HF = {f : V → R : f|V∂ = F }.

On HF , we take the inner product X hf, gi∇ = (f(x) − f(y))(g(x) − g(y)) = ∇f · ∇g. [x,y]∈E

The DGFF is the corresponding gaussian random element of HF . Its 2 probability density is proportional to exp(−khk∇/2).

13 Properties of DGFF

• E[h(v)] is the discrete harmonic extension of F to V .

• var h(v) = G(v, v)/ deg(v).

• E[h(v) h(u)] − E[h(v)] E[h(u)] = G(v, u)/ deg(u).

• Markov property.

14 Proof for E[h(v)]

This follows once we show that h0 := h−Fˆ is the DGFF with zero boundary values if Fˆ is the harmonic extension of F . The probability density of h is proportional to ³ ´ 2 ˆ 2 ˆ 2 exp(−khk∇/2) = exp −kF k∇/2 − hF , h0i∇ − kh0k∇/2 .

Consequently, it is enough to show that hFˆ , h0i∇ = 0. This holds because Fˆ is harmonic and h0 has zero boundary values: X X (Fˆ(x) − Fˆ(y))(h0(x) − h0(y)) = deg(x) ∆Fˆ(x) h0(x) = 0. [x,y]∈E x∈V

15 DGFF interface

−a +a

−a +a

−a +a

Theorem (Schramm-Sheffield). For appropriate choice of a, the interface of the [discrete] Gaussian free field [scaling limit] is SLE(4).

16 Stochastic Loewner Evolution

SLE as a limit (pictures)

Definition of SLE

Convergence proof strategy

17 Loop-erased / 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, ∞) → H is a non self-crossing path starting at γ(0) ∈ R, and gt is the Riemann map from the unbounded component of H \ γ[0, t] to H normalized at ∞ by

gt(z) = z + o(1), z → ∞, then with the capacity time parameterization

dg (z) 2 t = , dt gt(z) − Wt where Wt = gt(γ(t)).

25 (chordal) SLE definition

Defined in upper half plane. Transported to other domains by conformal map.

Let κ > 0 (a parameter).

Let Wt = B(κ 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 φ : D → H be a normalized conformal map, and parameterize γ by capacity of φ ◦ γ.

Let Wt be the Loewner driving term for φ ◦ 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 sufficient 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 \ 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 effectively a and on the left side of γ is effectively −a.

31 Since harmonic measure is conformally invariant, and the harmonic measure of [s, ∞) in H from z is 1 − arg(z − s)/π, we get

£ ¯ ¤ 2 a ³ ¡ ¢ ´ E h(u) ¯ X ≈ a − arg g φ(u) − W . t π t t

Therefore, Mt := arg(zt − Wt) is as close as we wish to a martingale, where zt := gt(φ(u)).

32 One step at a time

Fix δ > 0 small and mesh much smaller. Define t0 = 0 and

2 tj+1 := inf{t > tj : |t − tj| > δ , or |Wt − Wtj | = δ}.

Set ∆f = ftj+1 − ftj . Taylor expansion of ∆M up to first order in ∆t and up to second order in ∆W gives

Im(z) ∆M = ∆W + Im((¯z − W )−2) ((∆W )2 − 4 ∆t) + O(δ3). |z − W |2

£ ¯ ¤ ¯ 3 We have E ∆M Xtj = O(δ ).

33 Im(z) ∆M = ∆W + Im((¯z − W )−2) ((∆W )2 − 4 ∆t) + O(δ3). |z − W |2

Taking two distinct u gives £ ¤ £ ¤ £ ¤ E ∆W = O(δ3), E (∆W )2 = 4 E ∆t + O(δ3).

34 Are we done?

We actually have ¯ £ ¤ 3 E ∆W ¯ Xt = O(δ ), ¯ ¯ £ 2 ¤ £ ¤ 3 E (∆W ) ¯ Xt = 4 E ∆t ¯ Xt + O(δ ).

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 Hausdorff 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 |h| 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 kfk∇ is bounded.

If f is reasonably smooth and large f >> 0 at some v, the probability that a specific 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. If that does not touch the pink path, the average height of a collection of vertices near the path is outside the reasonable range.

39 Interface for continuum GFF

Theorem (S-Sheffield). There is a natural coupling of discrete and continuous GFF. Under this coupling, the limit (in probability) of the discrete GFF is a function of the continuum GFF.

40 Extensions - by Scott Sheffield

At least for κ ∈ [0, 4], the SLE(κ) paths may be found in the GFF. Instead of “solving” the equation h(γ(t)) = 0 the corresponding equation is

h(γ(t)) = c winding(γt), or, equivalently, dγ = exp(i c h) . ds

41