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\title{Bishop Notes\\January 23, 2006}
\author{Gabriel C. Drummond-Cole}
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The book is ``Conformally Invariant Processes in the Plane,'' by Greg Lauer.  It's AMS Mathematical Surveys #114.  This will be about Brownian motion and SLE, either the Schram-Louner Evolutions or Stochastic Louner Equation.

I'm not going to worry so much about going through the proofs.  The requisites are complex and real analysis, which I'm qualified to do, and probability, which I don't have a background on.  So we can go through just the statements.  I think most of the material is pretty basic, but we should just go through to get the shapes, maybe going back later to get the details on our own.

Brownian motion is a conformally invariant process in the plane.  The easiest way to think about this is discretely, with a random walk on a lattice in the plane.  It can easily repeat positions.  After $n$ steps there are $4^n$ paths.  There's a probability on them that gives each equal measure.  Say I draw a rectangle over here.  What fraction of thousand step paths end in here?  That's a simple question we can ask about these things.  Brownian motion is the continuous version of these.  Make the grid smaller, and in the limit you get a continuous random path.  You can ask whot the chance is that you're in a measurable set during this interval, at this time, at a time in the Cantor set, ever,  when am I on this curve with dimension $4/3.$  You can write papers and papers.  There's a huge literature about this.

It's certainly a stochastic process.  He should have put this in the title but because he does probability he thought it was obvious.  Conformal means, put this in the disk, and map this holomorphically, then you go to a path over here.  You get a random family of paths, but it may not be Brownian motion, but it still is.  You start with a family paths with a probability measure, and you get a new one.  So you hove $C[0,\infty)\to \mathbb{R}^2.$ and you have a measure.  Brownian motion tells you you have a measure with certain properties.  It's radially symmetric, it's independent (starts fresh at every time).  The conformal image has the same properties, with a proviso.  Time changes.  Say you blow it up, you can make time speed up or slow down.  

For a general conformal map, the time dilation is proportional to the derivative of the map $\phi.$  What's the chance that I hit this piece of boundary, maybe $.1.$  Then look at the preimage on the Riemann sphere.  If it's conformally invariant, you can tell by rotational invariance, there's only one rotationally invariant measure on the circle, Lesbegue measure.  This measure on the boundary is called harmonic measure.   $0\le \omega(2,E,\Omega)\le 1.$  This is an important measure.

That's an example of a conformally invariant process in the plane.  This book isn't about Brownian motion, it's about SLE.  Let me tell you some questions about Brownian motion.   I have two or three papers about Brownian motion, a paper in Annals of Statistics, you don't have to know anything to write about it.  Here's a picture of Brownian motion, in two dimensions.  Suppose this was for time one, $B[0,1].$  A few things I can tell you about this, suppose I call this $T,$ the trace, the points that get visited.  The area is zero, this is not obvious.  On the other hand, eventually it is dense.  The Hausdorff dimension is two.  In this picture it looks pretty thick in some areas, that's a trick of the printer.

Here are some questions.  Brownian motion is continuous so this is a compact set of zero measure. There are all these complementary components.  Let's look at the unbounded component, look at its boundary.  Now Brownian motion is a fractally looking object, so is the frontier.  This is a subset of the Brownian motion.  This may not be trivial to prove.  This is quite a bit smaller.  There's a conjecture that the dimension is $4/3$ by Mandelbrot.  Any of the other components should also have dimension four thirds.  This was open, and then four of us proved it was dimension greater than or equal to $1+\epsilon.$  We had grids, but we decomposed the plane using fractal tiles.  There was no straight line anywhere.  That was some progress, but in fact it's a theorem now, it was proven using SLEs.  This theorem is stated around page 187.  I want to understand the proof of this theorem.  This is Lauer-Schram-Werner.  He doesn't reference it, let me just check.  Yes, 2002.  These results are published in the last five years.  They do some other stuff as well, like with cut points.  A cut point is a point you can remove from the trace and disconnect.  There is no intersection between the time before and after $t.$  The cut points have dimension $3/4.$  I don't know why they are reciprocals, but in particular they are totally disconnected.  Pioneer points have dimension $7/4,$ meaning that they were frontier points at some time.  These are points on the outer boundary at some time, then this number becomes $7/4.$

In some sense if we didn't have an intrinsic reason to be interested in SLE, these examples give reasons to be interested in SLE.  I want to get to this for personal reasons.

So far no details, total hand-waving.  Here are things that were totally inaccessible ten years ago and now are solved.  How would you ever believe that these numbers could be solved at all.  Computing these have to do with computing intersection exponents.  They have to estimate.  You have the disk of radius one, and two Brownian motions starting at say $\pm \epsilon.$  They might hit each other and might not.  If $\epsilon$ is small it's likely they'll cross.  If $\epsilon$ is $1/2,$ it's likely they'll escape before crossing each other.  There's an exponent that goes down like a power.  It's like $\epsilon^eta.$  There are limit theorems from ten years ago.  It was known that there was a critical exponent that could get you this.  That's why the paper is called Brownian intersection exponents.  I hope that this makes it seems more reasonable.  It's really computing the probability of some low probability event.  You could do some numerical things, estimate things, and we were pretty sure about this.  This was from the physicists, I guess, rather, and I never understood it.

Okay, what are SLEs?  They were invented to understand a random simple path.  The Brownian motion is the limit of a random walk.  But Brownian motion crosses itself a lot.  The fact that the cut points have dimension $3/4$ means that.  You can study the set of points hit by Brownian motion twice, or countably many, or uncountably many times.

Suppose I want a random path in the plane that never crosses itself?  What is a random simple path?  How far am I likely to travel in such a path?  What is the Hausdorff dimension of a random simple path, its trace?  How crinkly would it be?  People didn't know what the right definition was.  Basically what Schram came up with, it's a rigorous mathematical definition with some of the right properties, and it's the continuous limit of various discrete processes.  No one knew if you took the limit, what the limit was.  

What are some of the discrete processes?  Those are in chapter zero.  The picture looks different.  Maybe if you had the frontier, with no cut points, then you might get something like a random Jordan curve.  You do, with a particular parameter.

Chapter zero is setup.  He gives a bunch of examples, the limit of this is $SlE_2,$ the limit of this is $SLE_6.$  Later in the book it's unimportant that these are the limits, but the author had probably been thinking about this for a long time.

One thing you can do is go back to thinking about Brownian motion.  There are $4^n$ discrete paths.  You can go off in some direction, with direction $4^{-n}.$  The first thing to do is pick a random nonintersecting path.  That gives you a bunch of nonintersecting paths.  Take the limit as the path shrinks down and scale time appropriately.  No one knows how to scale time.  Take all the choices, throw away the ones that intersect.  Give them equal proportion.  What is the average distance (divided by $n.$)  We don't have a good idea for what is left by throwing these away.  This is computationally difficult to generate in discrete approximation.

With the help of $SLE$ this problem has been solved.  ``While it is only a conjecture that this has limit $SLE_{8/3}$ it is supported by numerical methods.''

Here's a different model, the loop-erased random walk.  Suppose I have a path, I have a loop.  You take the loop out, and then you get a simple path. A loop-erased simple path, you erase the path between $a$ and $b$ if you hit the same point at those times.  That's a nonrigorous intuitive definition.  The problem is whether this stabilizes.  You don't use random walks in the plane, you give them a push to infinity, then the loop erased procedure works.  You do it in the upper half-plane, start on the origin, and you randomly walk, if you're at height $k$ the chance of moving up is $k+1/k$ and the chance of moving down is $k-1/k.$  If you look at random paths they tend to infinity, they tend to cross every horizontal line and never come back.  That procedure does give you simple random paths.  That rigorously converges to $SLE_2.$  They're not the same thing since their limits can be distinguished.

Here's a third method.  You wouldn't think that hexagonal lattices would make much of a difference.  You could take a random walk on a hexagonal lattice, you get the same Brownian motion as the limit.  For some of these other things, it makes a difference for the proof.  In some of these models, there are proofs that go through for one but not the other.  For this, a percolation, you color your edges with two different colors, about half and half.  Now you see what you can connect staying on the yellow.  If there was an infinite component that you could get to staying on the yellow.  Here $1/2$ is a critical exponent.  Above you can, below you can't.  If there's a path to $\infty$ that's a random path, to make things careful instead of deleting loops, you can connect components to the positive and negative line in the half plane.  Then there's an arc going out to $\infty$ that separates the white component and the yellow one.  There should be at least one path like this.  This is a percolation exploration process.  This is like oil seeping through a stone, it flows along the paths.  This is a model for this process.  This is supposed to be $SLE_6.$  

This makes $SLE$ seem pretty interesting, but what is it?  There's something called \wh equations, from the thirties.    You have a path, say, that slits the plane, and then there's a conformal mapping of the disk to the outside of the slit.  You can make the slit shorter and shorter, and get families of conformal maps.  It is $\dot{g}_t(t)=\frac{1}[a_t(z)-u(t)}.$  Depending on where this is coming from, there's a different form, but there's some sort of differential equation.  There's a correspondence between the curve and some data in the differential equation.  If the data and the curve are lovely and smooth there is a correspondence.  I don't know if there's a formula, but you can use the theory of differential equations.

Schram's observation is that for a random curve, stick a random function into the differential equation.  If you plug a one dimensional Brownian motion, you get a random slit, a random nonintersecting curve.  What data you put in gives you the parameter $\kappa.$  For good old Brownian motion I think you get $\kappa=2.$  We'll get to that.  I think all this is is the speed of the Brownian motion, just run at different speeds.

We'll have to review Brownian motion, conformal mappings, and the classical equations.

Any other remarks?

These processes have the same properties of reframing.  A continuous limit is going to be in the half-plane.  Take the slit region, you can flatten it by conformal mapping.  Conformal invariance means that this conformal picture is the same thing as running $SLE$ from the start, the same way conformal paths from the origin preserve Brownian motion.  That's what conformal invariance means.

On Friday I'm not going to lecture, I want to use this time for a lecture ``Isometries of Teichmueller spaces.''  The Thursday lecture will not be too hard to follow the definitions.  Next Monday we'll start with chapter one, we'll do about ten pages a day.  If we focus on the statements, maybe that's enough.  I do want to spend some time on the parts I don't know so much about.  On the other hand I know the author so if I don't understand, I'll call him up.  We'll see how it goes.  As time goes on, I'd like you to do something to contribute.  If you are coming, I'd appreciate if you sign up.  If registration is too low I might have to cancel the class.  Tell me if it's in the bookstore.

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