loewner equation | Eventually Almost Everywhere

I couldn’t resist breaking the order of my revision notes in order that the title might be self-referential. Anyway, it’s the night before my exam on Conformal Invariance and Randomness, and I’m practising writing this in case of an essay question about the Gaussian Free Field and its relation to the SLE objects discussed in the course.

What is a Gaussian Free Field?

The most natural definition is too technical for this context. Instead, recall that we could very informally consider a Poisson random measure to have the form of a series of Poisson random variables placed at each point in the domain, weighted infinitissimely so that the integrals over an area give a Poisson random variable with mean proportional to the measure of the area, and so that different areas are independent. Here we do a similar thing only for infinitesimal centred Gaussians. We have to specify the covariance structure.

We define the Green’s function on a domain D, which has a resonance with PDE theory, by:

$G_D(x,y)=\lim_{\epsilon\rightarrow 0}\mathbb{E}[\text{time spent in }B(y,\epsilon)\text{ by BM started at }x\text{ stopped at }T_D]$

We want the covariance structure of the hypothetical infinitesimal Gaussians to be given by $\mathbb{E}(g(x)g(y))=G_D(x,y)$ . So formally, we define $(\Gamma(A),A\subset D)$ for open A, by $(\Gamma(A_1),\ldots,\Gamma(A_n))$ a centred Gaussian RV with covariance $\mathbb{E}(\Gamma(A_1)\Gamma(A_2))=\int_{A_1\times A_2}dxdyG_D(x,y)$ .

The good news is that we have a nice expression $G_U(0,x)=\log\frac{1}{|x|}$ , and the Green’s functions are conformally invariant in the sense that $G_{\phi(D)}(\phi(x),\phi(y))=G_D(x,y)$ , following directly for conformality of Brownian Motion.

The bad news is that the existence is not clear. The motivation for this is the following though. We have a so-called excursion measure for BMs in a domain D. There isn’t time to discuss this now: it is infinite, and invariant under translations of the boundary (assuming the boundary is $\mathbb{R}\subset \bar{\mathbb{H}}$ , which is fine after taking a conformal map). Then take a Poisson Point Process on the set of Brownian excursions with this measure. Now define a function f on the boundary of the domain dD, and define $\Gamma_f(A)$ to be the sum of the values of f at the starting point of BMs in the PPP passing through A, weighted by the time spent in A. We have a universality relation given by the central limit theorem: if we define h to be (in a point limit) the expected value of this variable, and we take n independent copies, we have:

$\frac{1}{\sqrt{n}}\left['\Gamma_f^1(A)+\ldots+\Gamma_f^n(A)-n\int_Ah(x)dx\right]\rightarrow\Gamma(A)$

where this limiting random variable is Gaussian.

For now though, we assume existence without full proof.

SLE_4

We consider chordal SLE_k, which has the form of a curve $\gamma[0,\infty]$ from 0 to $\infty$ in H. The g_t the regularising function as normal, consider $\tilde{X}_t=X_t-W_t:=g_t(x)-\sqrt{\kappa}\beta_t$ for some fixed x. We are interested in the evolution of the function arg x. Note that conditional on the (almost sure for K<=4) event that x does not lie on the curve, arg x will converge either to 0 or pi almost surely, depending on whether the curve passes to the left or the right (respectively) of x.

By Loewner’s DE for the upper half-plane and Ito’s formula:

$d\bar{X}_t=\sqrt{\kappa}d\beta_t,\quad d\log\bar{X}_t=(2-\frac{\kappa}{2})\frac{dt}{\bar{X}_t^2}+\frac{\sqrt{\kappa}}{\bar{X}_t}d\beta_t$

So, when K=4, the dt terms vanish, which gives that log X is a local martingale, and so

$d\theta_t=\Im(\frac{2}{\bar{X}_t}d\beta_t$

is a true martingale since it is bounded. Note that

$\theta_t=\mathbb{E}[\pi1(x\text{ on right of }\gamma)|\mathcal{F}_t]$

Note that also:

$\mathbb{P}(\text{BM started at }x\text{ hits }\gamma[0,t]\cup\mathbb{R}\text{ to the right of }\gamma(t)|\gamma[0,t])=\frac{\theta_t}{\pi}$ also.

SLE_4 and the Gaussian Free Field on H

We will show that this chordal SLE_4 induces a conformal Markov type of property in Gaussian Free Fields constructed on the slit-domain. Precisely, we will show that if $\Gamma_T$ is a GFF on $H_T=\mathbb{H}\backslash\gamma[0,T]$ , then $\Gamma_T+ch_T(\cdot)=\Gamma_0+ch_0(\cdot)$ , where c is a constant to be determined, and $h_t(x)=\theta_t(x)$ in keeping with the lecturer’s notation!

It will suffice to check that for all fixed p with compact support $\Gamma_T(p)+c(h_T(p)-h_0(p))$ is a centred Gaussian with variance $\int dxdyG_H(x,y)p(x)p(y)$ .

First, applying Ito and conformal invariance of the Green’s functions under the maps g_t,

$dG_{H_t}(x,y)=cd[h(x),h(y)]_t$

The details are not particularly illuminating, but exploit the fact that Green’s function on H has a reasonable nice form $\log\left|\frac{x-\bar{y}}{x-y}\right|$ . We are also being extremely lax with constants, but we have plenty of freedom there.

After applying Ito and some (for now unjustified) Fubini:

$dh_t(p)=\left(\int c.p(x)\Im(\frac{1}{\bar{X}_t})dx\right)d\beta_t$

and so as we would have suspected (since h(x) was), this is a local martingale. We now deploy Dubins-Schwarz:

$h_T(p)-h_T(0)\stackrel{d}{=}B_{\sigma(T)}$ for B an independent BM and

$\sigma(T)=\int_0^Tdt(\int c.p(x)\Im(\frac{1}{\tilde{X}_t})dx)^2$

So conditional on $(h_T(p),t\in[0,T])$ , we want to make up the difference to $\Gamma_0$ . Add to $h_T(p)-h_0(p)$ an independent random variable distribution as $N(0,s-\sigma(T))$ , where

$s=\int dxdyp(x)p(y)G(x,y)\quad =:\Gamma_0(p)$

Then

$s-\sigma(T)=\int p(x)p(y)[G(x,y)-c\int_0^Tdt\Im(\frac{1}{X_t})\Im(\frac{1}{Y_t})]dxdy=\int p(x)p(y)G_t(x,y)dxdy$ as desired.

Why is this important?

This is important, or at least interesting, because we can use it to reverse engineer the SLE. Informally, we let $T\rightarrow\infty$ in the previous result. This states that taking a GFF in the domain left by removing the whole of the SLE curve (whatever that means) then adding $\pi$ at points on the left of the curve, which is the limit $\lim_T h_T$ is the same as a normal GFF on the upper half plane added to the argument function. It is reasonable to conjecture that a GFF in a non-connected domain has the same structure as taking independent GFFs in each component, and this gives an interesting invariance condition on GFFs. It can also be observed (Schramm-Sheffield) that SLE_4 arises by reversing the argument – take an appropriate conditioned GFF on H and look for the interface between it being ‘large’ and ‘small’ (Obviously this is a ludicrous simplification). This interface is then, under a suitable limit, SLE_4.

Last time I set up the geometric notions of probability that will be needed to proceed with the course material. Now we consider the deterministic differential equation due to Loewner (1923) which he used to make progress on the Bieberbach Conjecture, but which will also underpin the construction of SLE. This proof is adapted for this specific case from the slightly more general argument in Duren’s Univalent Functions (Section 3.3). Because in that setting the result concerns an infinite domain, readers should beware that though I am using identical notation, in about half the cases, my function are the inverse and my sets the complement of what they are in Duren.

To explain the construction, as with so many things, a picture speaks a thousand words. Unfortunately I have neither the software nor, right now, the time to produce the necessary diagrams, so the following will have to suffice.

Consider a deterministic simple curve in the unit disc, $(\gamma(t): t\in[0,\infty))$ . Removing initial segments of the curve gives the nested simply-connected regions:

$U_t:=\mathbb{U}\backslash \gamma[0,t]$ .

Then define as in the previous post, the unique conformal map

$f_t: U_s\rightarrow \mathbb{U}$ such that $f_t(0)=0, f_t'(0)\in\mathbb{R}^+$ ,

and furthermore set $\xi_t$ to be the image of $\gamma(t)$ under this map. (Note that though the conformal map is not defined on the boundary, it must extend continuously).

$f_t'(0)$ is increasing.

Very informally, this derivative records how much twisting is required at the origin to turn the slit domain into the open disk. Extending the path will demand further twisting. More rigorously, set:

$g_t=f_t^{-1}:\mathbb{U}\rightarrow U_t$ .

Then the $(g_t)$ are injective functions from the unit disc to itself which preserve the origin, so Schwarz’s lemma applies. They are clearly not rotations, so