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>>: Good afternoon, everyone. So very happy to welcome Ivan Corwin coming
to us from Microsoft Research in New England. The first strong fellow. And for
all of us who are still stuck in the Gaussian University Class, he's going to take us
beyond that.
Ivan Corwin: Thanks, (inaudible). Thanks for having me here.
I just started around two and a half weeks ago in Cambridge, Boston, and having
a great time and very happy to be the first strong fellow, and I hope there are
many more. You guys all know this is a good thing to remember.
So I'm interested in talking about beyond the Gaussian Universality Class.
Perhaps it's a lofty title, but let me say that it's not. You know, the Gaussian
distribution arguably is one of the most important societal contributions that math
has made in the past 200 years. Over the course of 200 years, people have
really developed this Gaussian Universality Class, they are able to compute all
sorts of things and they're able to prove all sorts of limit theorems.
And over the course of time, people realize that Gaussian is not all
encompassing. There are plussonian (phonetic) statistics. There extreme value
statistics. And a theme of the last, you know, few decades has been a realization
that even these are not sufficient to describe suitably interesting situations that
really occur naturally.
So what I'll be focusing in are models in which there are highly nonlinear
functions of inherent noise. And due to this highly nonlinear function you're
looking at, you end up with non-Gaussian statistics. And it turns out in fact that
despite a variety of different models that I'll show you, you actually end up with
the same statistics. This is an interesting phenomenon. In some cases we
understand why, and in other cases we really don't.
So this will break into two parts. But first, I'm going to focus on a very particular
class of models. And these are models for random growth interfaces in one plus
one dimension, which means that we have a growing curve and it's -- you're
looking at the statistics of how it grows in time.
And, you know, I'll explain a very simple model and then move into two more
interesting models. I'll focus on this third one, this corner growth.
What you're really interested in studying here is the fluctuations. Just like any
sort of central limit theorem, you're interested in understanding the fluctuations of
your growing interface. And it turns out that there are actually two universality
classes, depending on the parameters of the model. The main one of interest is
this KPZ universality class, and it describes the type of statistics and the scalings
of the statistics. And actually underlying this whole picture is a continuum object
at the interface between these two universality classes, and that's this stochastic
PDE called the KPZ equation.
So this will be the first part of the talk. I'll really explain to you in some depth how
this works for growth models.
Now, you could just take that as being a single thing. And this is its own
interesting subject, but it turns out that there are many other fairly disparate
subjects which also give rise to the same sort of statistics and the same sort of
scalings. And in some case, we really understand why and in other cases it's a
real mystery. So I'll fill up the last part by giving you five different examples.
And if the content is not interesting enough, the tools are quite interesting in their
own right. So it really involves a lot of interesting math.
So if somebody were to ask you to come up with the simplest growth model,
you're trying to -- you're a physicist and you want to really understand the way
that surfaces grow. And they ask you to come up with it.
Perhaps the first iteration which you would quickly throw out for reasons we'll see
is something called the random deposition model. And this model is given as
follows:
You have an initial surface. We can take it to be flat. And in every column, you
have an independent fau-saunt (phonetic) process of blocks that fall. The blocks
fall independently in each column after exponential waiting times.
And, well, we can record these for instance in terms of these w x,i. This will
represent the time it takes in column X for the I block to fall. And the object that
we're interested in studying is the interface. So the height above a position X at
time T, which here would be four.
So this is of course a very trivial model because of the fact that everything can
mean coded in terms of sums of these wx,is. So these are independent
identically disputed random variables. And if you want to know the probability of
the height is less than some value, it's equivalent to that the sum of these is
bigger than some other value.
So there's this easy property. And then from that, you can immediately compute
law of large numbers. So you say that the interface grows roughly linearly with
speed one, and you can actually compute just with this standard central limit
theorem and that if you subtract off the asymptotic growth rate and divide three
by T to the one-half, you converge to normal random variables. And each one of
these is independent for every location X just by construction because the
processes that were -- you know, there was no spatial correlation.
Okay. Now, what was the point of -- what was the point of spending two minutes
explaining to you the first thing that you learn in probability is because I want to
emphasize what I mean by Gaussian Universality Class.
So what I really am getting at here is I'm looking at a class in which the
fluctuations of your interface or of your variables scale like T to the one-half. This
is the very classical Gaussian scaling, and they have a Gaussian central limit
theorem. The fluctuations are given by Gaussian statistics.
On top of that, and the thing I really think is pertinent here, is that there's no
spatial correlation. So you have an exponent of zero. Your next-door neighbors
are independent.
And in general, when I talk about a universality class, this is what I'm going to be
keying in on. This sort of information about the scale of the fluctuations, the
spatial correlation, and then the actual probability distributions, not just one-point
distributions but how multipoint distributions are correlated.
Now, why would you call this a universality class? Well, you know, we know that
changing the distribution of these wx,is has no effect up to centering and scaling
on the properties of this system asymptotically.
Now, this is a horrible growth model from a physical perspective because, well, it
doesn't really capture any of the physical behavior that people will observe. If
you look at a bacterial growth colony or any sort of growth model, this is not what
happens. And in simulations in actual laboratory experiments, people see that
these exponents are all wrong. In particular, you have a lot more spatial
correlation.
So in this second iteration, if you want to introduce spatial correlation, a very
simple way of doing it is to go to what's called the ballistic deposition or the sticky
block model. And now you're moving toward some more physical model and the
rule here is that, again, you have blocks which are falling independent in each
column according to independent Fau-saunt (phonetic) processes.
Now, the change is that a block will stick to the first surface it comes in contact
with. It just falls straight down, and it sticks to the first surface it comes to. So
what you see is you get this sort of overhangs and here's another example of an
overhang. But again, we're really just interested in understanding what the top
interface is.
Now, this small change, we don't know how to actually prove. Even a law of
large numbers for the -- this should grow linearly. Should have a certain speed.
We don't know how to compute that, let alone compute its fluctuations. So from
the beginning, we're kind of in trouble. Okay.
But what are the -- I want to emphasize that despite that, we can still make
conjectures about that. So what are three pertinent features of this model? And
I want to emphasize these because this is what is going to come up when we
look at these other growth models. And in particular, this continuum growth
model.
So there are three features. So one which is obvious, which is being illustrated
right here, is that if you have these sorts of big valleys, they get filled in very
quickly. So in a sense, there's a nice mechanism for smoothing out very rough
areas. So the first I'm say is there's a smoothing mechanism.
And the second one is maybe not quite as apparent. The second is that there is
a slope-dependent, though rotationally or kind of positive/negative slope
independent, dependence of the growth rate.
So let me illustrate this pictorially.
So if you have an interface which looks like this, has a high slope, and a block
falls, it's going to stick, and the height will increase by something like three.
Whereas if you had a perfectly flat interface and a block fell, I would only
increase by one.
But of course it didn't matter if it was way or if the slope was this way. You still
increase. Okay.
So I'll say this second is slope-dependent, maybe rotationally invariant growth
rate.
And finally there is some sort of space/time independence or space/time noise.
Okay. And this is just modeled by the fact that you have here these independent
Fau-sant (phonetic) processes in each column.
So these are really the three features of this model. And because of these three
features, the behavior of this model is expected to be the same as the behavior
of the model which I'm going to focus on, which is that of the corner growth
model.
So immediately -- so this is the third model, and this is what I'll talk mostly about.
So this model, you see immediately that actually I've changed the geometry.
Previously, I was looking at growth off of a flat surface, and now I'm looking at
growth in a corner. And the effect of this is it should not change the asymptotic
scale of the fluctuations. However, it should change the type of statistics
associated to these distributions. So there is detection of initial data in the -- in
terms of statistics but not in terms of scale.
So how does this model behave? Well, initially, let's say you start out with an
empty corner and then the rule is that every time you have a local minimum or a
small valley, you can fill it in or you can invert it with a box and you do this so at
rate Q. And simultaneously, every time that you have a box or a local hill, you
remove that at rate P. So here we have three valleys and two hills, and
simultaneously, they're all competing to be acted upon.
And for normalization, we take Q plus P equals one, and Q minus P is the
asymmetry of the model. We call that gamma, and we take it between zero and
one.
So in this case, you can see Q is going to be generally larger than P, which
means there's a net drift upwards. If Q is one, then you would only grow as
opposed to some -- something being removed.
So the height function again is recording this interface. Initially it just gives
[inaudible] value of X, and the question here, or the first order question is what's
called hydrodynamic theory. It's a question of what is the limit shape? If you run
this for a very long time, how do the boxes grow? What do they look roughly
like? And then the real question that we're interested in is studying again: What
are the fluctuations of the interface around its expected growth shape and in
what scale do they live and what are the statistics behind them. How would you
describe that.
And as I said, the conjecture is that it shouldn't have the same scale as this
ballistic deposition model, but different statistics based on the geometry being
used here.
So let me show you a simulation of this. This is on Patrick Ferrari's website.
This is in the case where gamma is equal to one. So there's only growth. So
initially, you have nothing. And then you see the red line is representing the
asymptotic growth shape, which you can see is a [inaudible] line and it goes
according to this -- according to this wedge. And then I'll run [inaudible] time of
thousand.
So what you see is -- well, it generally stays above this, so that says something
about the centering of the limiting statistic. And then you see that it actually
fluctuates on a very small scale. But if you look at area between maximum and
minimum, it's pretty big. Here's a minimum; here's a maximum.
So if you did some sort of -- you keep repeating this experiment and you want to
understand how big the fluctuations generally are here versus how correlated
they are transversely, you see that they're roughly fluctuating up and down on the
order of ten blocks and the spatial correlation, say, between maximum to
minimum is something like over a hundred. This is numerology, but what are
these numbers? Ten and a hundred, a thousand. So the fluctuations are like a
thousand to the one-third, and then they're spatially correlated like a thousand to
the 2000. Okay.
This is not math. This is just some numbers. But there is math here.
So what is the theorem? Well, if you look at the height function, and you fix the
asymmetry to be gamma, so this is the drift upwards, then Johansson proved in
the case of gamma equals one is the one that I was illustrating, and then
Tracy-Widom more recently for any positive gamma, that if you look at the height
function, now, in order to make all things equal, you need to -- if you have less of
a drift, you need to run for longer. So you need to run for time T over gamma.
And let's just focus right above the origin.
So the asymptotic growth shape tells you what to subtract, and it turns out to be
something like a half T, and then what you show is that you divide by T to the
one-third, then this probability distribution that's bigger than minus S converges
to something called the Tracy-Widom GUE distribution. And it's a distribution
function, which I'll tell you a little bit more about it later. It first arose in the study
of random matrix theory for the statistics of the largest [inaudible] value of a
Gaussian [inaudible].
Now, this is a growth model. Those are Gaussian random matrixes. And why in
the world would they be the same in statistics? And people still really don't have
a good explanation for this.
Now, what I want to emphasize is this type of scaling, this is a highly
non-Gaussian sort of statistic. In a usual Gaussian statistic, you would expect
fluctuations like T to the one-half. And you would expect Gaussian.
What's happening here is you have this characteristic T to the one-third
fluctuation scale. Now, I didn't say anything yet about the correlation between -in the spatial direction, but it turns out at least in the case of gamma equals one,
so Trey, Hopper and Spun (phonetic) show that if you look at two different points
which are, say, you know, zero, and then T to the two-thirds difference apart,
then their joint distribution when properly normalized converges to a nontrivial
random variable, two-dimensional random variable.
So this shows that the correlation is ordered T to the two-thirds. And in
particular, you have this characteristic ratio of exponents. So if you look at the
time exponent, which is like T to the one over -- T to the three-thirds, let's say.
And then space is like T to the two-thirds and fluctuations are like T to the
one-third.
So this is a very different scaling than the Gaussian Universality Class, and you
also have this GUE statistic.
Now, all of this, this happens for every positive gamma. .001, .0001, as T goes
to infinity. But on the other hand, you need to ask what happens when gamma
equals zero. And when gamma equals zero, of course this theorem can't be true.
Can't divide by zero. So from the start, it's wrong.
What happens is something very different in characteristic. So gamma equals
zero corresponds to the symmetric case. And in the symmetric case, you're
growing and dying just equally likely.
Now there is -- there is a difference. Because you're in this wedge geometry,
there is actually something like a hard wall repulsing you. There's nothing you
can remove from the wedge. So there still is a tendency to move upwards on
account of this. It's similar to a random lock with a reflex of the origin. Even if it's
symmetric, it will still drift to the right like T to the one-half.
And so in this case of gamma equals zero, you have a law of large numbers. It
lives on a scale T to the one-half. And if you center by that, then you find
fluctuations around that order of T to the one-quarter, which is fairly small. And
they are Gaussian. However, they also have a very nontrivial correlation scale.
So it's a nontrivial Gaussian process. And this is what's called the Edward
[inaudible] class.
So now there's a little bit of a question here. I'll ask three questions. This is the
first of it: Is that you have for every positive gamma one behavior, and then you
have for gamma equals zero another behavior. And they sit right next to each
other. And the scalings are totally different, and the statistics are totally different.
So what happens at the interface? What happens between them? How does
this work?
And there's a general mantra, and we'll see that it holds true in a few cases in
this talk, that at these sorts of interfaces, you expect that by scaling into them,
you will find the universal scaling options. And that is what this [inaudible]
equation turns out to be. But -- wait a second. Sorry.
So before I go on, let me give you just a drop of the history. So what was known
about this KPZ universality, people in physics had been very interested in
studying growth models and scalings of these growth models. And what had
been determined -- and I'll tell you just a drop of how. What was determined was
that the exponents should be one-third and 2-thirds. This was determined by
very black magic. Nonrigorous mathematics, but it's still -- they had guesses.
What came as a complete surprise was this GUE distribution. People had no
idea that random matrix theory would rear its head in this set.
Now, so how are these exponents computed in physics? I'm not going to go into
the exact method, but what it relied on was this work of Kardar, Parisi and Zhang.
So Kardar, Parisi and Zhang in 1986 had in mind that there are all of these
growth models that people had been studying. And in particular, the growth
models involved these three characteristic properties. So they were interested in
models with smoothing, slope-dependent, rotationally and varying growth rate
and space/time wipeouts or some sort of space/time noise.
And the question was: How can you determine the scaling properties of these
models? But people didn't know how, so what they proposed is, well, we might
as well choose the nicest one amongst them, and in general, for analytic
purposes, discrete things are sometimes harder than continuum things. So they
proposed the following continuous model as something of the typical, the
prototypical growth model.
So this is what's now called the Kardar, Parisi and Zhang or the KPZ stochastic
PPD. And what it says is that you have again a height function. Now it's a
continuous height function of time and space. And it changes in time according
to three factors: Smoothing term, a -- I call it a lateral growth term, but it's the
same as this slope-dependent term. Smoothing is always just given by a
[inaudible]. This is the general recipe.
If you want something that depends on the slope but doesn't depend on whether
it's positive or negative, you could take the affluent value. This isn't particularly
nice, you know, analytically, so they took the square of the [inaudible].
And then if you wanted to be random, you put in some space white noise. And
this is how they derive this equation. And then using earlier work of for
(inaudible) Nelson and Stevens on the stochastic bur gone equation and dynamic
renormalization group methods, they made these one-third/two-third predictions.
Now, recall, what does it mean for a model to be in the KPZ [inaudible]? It says
that it grows roughly like something deterministic, plus something that fluctuates
like T to the one-third and has a certain type of characteristic fluctuation
distribution.
So now the second and third question come, which is I have given you two
growth models. One is a very concrete corner growth model. On the other hand,
I've given you this very continuous Kardar-Parisi-Zhang equation. How are they
related? How does one scale to the other?
And the third question is -- the first time I heard it, it was laughable. And of
course it's true. Must be true -- is the KPZ equation in the KPZ universality
class? And, you know, just by construction it needs to be because it's the same
word. But actually, this is a very nontrivial question because just because you
write something down and do some black magic, this doesn't mean that at all that
it would be in the same universality class.
Of course people have this in mind. You know, that's where the name comes
from. They say something is within the KPZ universality class if it has the same
scaling behavior of this equation.
But there is a fundamental problem in all of these questions, which is that from
the get-go, this equation makes no sense. So let me explain why.
If you're interested in this equation, you need to somehow make sense of this
nonlinearity. Now, what the solution -- if you remove the nonlinearity, so you try
to think of it just as a perturbation of the additive stochastic heat equation. You
can convince yourself and prove fairly easily that the additive stochastic heat
equation, if you look at the spatial regularity, so at a fixed time you look at what
the function looks like in space, it will look something like a Brownian motion. So
we'll have [inaudible] continuity, just under a half.
So now you try to kind of think of this as a perturbation, so you put this back in.
But you can always take the derivative of something, which is [inaudible] less
than a half, but you get a distribution and then you can't square a distribution. So
you really don't know from the start how to make sense of this square here.
So turns out that there is a good way to make sense of it. And by good, I mean
there is a physically relevant way, a way in which this -- what I'm going to give
you actually comes about as the scaling limit of real growth models. And it has
the correct scaling [inaudible]. And what you do is you actually make sense of it
in terms of another equation, which is called the stochastic heat equation with
multiplicative noise.
So it's the following: You have the heat equation, and then you have this sort of
space/time/white noise times the value. So you keep inputting more or removing
more information based on Z times this white noise.
And now, this is a well defined stochastic PDE. You can say exactly what it
means to solve it. You can prove the existence of solutions for every -- for
almost every white noise you can go through. And we're going to be focusing on
the fundamental solution of this equation, which is the delta function at time zero.
Now, how is this related to the KPZ equation? Well, at the first level, if you
formally define H to be minus the log of Z, this is called the Hopf-Cole transform.
And assuming this makes sense, which is to say assuming that Z is positive,
which you can actually show that it is almost surely positive for this type of initial
[inaudible].
>>: [Inaudible].
Ivan Corwin: As opposed to just plus log. No, it's just the issue of choice of
signs. You could have put a plus here of course. White noise is symmetric.
>>: [Inaudible]?
Ivan Corwin: You know, guess in the context of PDEs, this is -- if people want to
solve these types of equations, the transformation of burgers types equations
to -- yeah. It's the minus log solution to KPZ.
So if you formally compute, you plug this definition, you define HT minus log of Z,
you plug this right into the KPZ equation, you check, you use the chain rule. You
check that it satisfies it. Okay. This is maybe not the best. This is just a formal
reason.
As we'll see, actually if you interpret the KPZ equation as minus log of Z, then
you can show that this H comes about as the scaling limit of real growth models.
So from a physical perspective, you want continuum objects to really explain
discrete objects. And this explains the discrete objects that you're interested in.
Okay.
So now that we have this definition in our hands, we can actually go about
answering these three questions that I asked. So that are the answers? This is
joint work with Didi Amir, Jeremy Quastel, and myself.
So first I ask, you have this very nice corner growth model. Very simple. You
either grow, you fill in boxes, or you move boxes in a certain asymmetry.
How is this related to the KPZ equation? And hand in hand with this was this
question of what happens between positive and zero asymmetry? And I said that
there should be some sort of continuum object at the interface.
So what do we do? Well, if you look, you remember there's this very
characteristic scaling of the KPZ universality class. Time scales like three and
three-thirds, space or time like three, space like two, and fluctuations like one.
So we take time to be like epsilon to the minus three capital T. This is a
macroscopic variable. Space like epsilon minus two X, and then fluctuations on
the [inaudible] epsilon minus one.
This is the usual scaling under which, you know, for instance these results of
Johansson and Tracy and Widom and [inaudible]. These are the results of this
type.
But now what you do is you take the asymmetry, you scale it to zero. And you
scale it to zero like epsilon. And then what you find, well, recall, we only know
what it means to solve the KPZ equation in terms of this hot and told transform.
So really, any convergence results should occur at the level of the stochastic
heat equation. So what we need to do is we define -- and this is very much
similar to Bertini and Giacomin. Bertini and Giacomin, they studied not a corner
type of geometry, but actually a very flat geometry. And it turns out their work
doesn't apply. This is far out of their -- out of kind of the realm of their theorem,
but their approach still applies, and there is a certain logarithmic correction, and
things become a little bit more complicated, but nonetheless, it's similar sort of
approach.
So if you define Z epsilon T, X, this is a process in these macroscopic variables,
and essentially, you know, there are some parameters and I don't want to
emphasize them. Essentially what it is is you look at the height function, you
subtract off the law of large numbers, and you scale it according to this -- these
KPZ scalings. Then -- well, what is the inverse of minus logarithm? It's
exponential of minus. So you look at exponential of minus, the fluctuations, then
there's this log correction, which wasn't expected. And the results -- this is not
the main result, but this is a result you need along the way, is that as epsilon
goes to zero, this process converges as a space/time process to the solution of
the stochastic heat equation with delta function initial data.
And you can see why the delta function comes in and you can see why you need
this logarithmic correction. If you just take the T equals zero distribution, so this
is a wedge, but it's being scaled because of X, the scaling of X, so it becomes a
very narrow wedge. You divide by an epsilon to the minus one, but that only gets
rid of one of these. So you have something like E to the minus, a very narrow
wedge. And that's becoming a delta function if you put in this correct [inaudible].
That's where the initial data comes from.
Now, what this tells you is it tells you -- it kind of answers these questions.
Certainly answers the first question, but doesn't totally answer the second one. It
says that somewhere between the two symmetric and the asymmetric cases, this
KPZ and this Edwards-Wilkinson universality class sits the KPZ equation. But it
doesn't tell you how -- if it's the only thing there or if it fills the space or anything
of that sort.
Now, the -- this question has a more -- there's much more to it. And this brings
us actually to this third question: Is the KPZ universality -- is the KPZ equation in
the KPZ class, and in fact, we can show much more than that. We can give the
solution to the KPZ equation.
So if you take the stochastic heat equation with delta initial data, you run it for
time T, you look at both minus the logarithm of that. That's your random variable.
It depends on the white noise.
What is the distribution of that random variable? Well, when you subtract of a
constant, this constant, its probability distribution is given by something which we
call a crossover distribution, F, T, of S. And that can be written as a contour
integral of a [inaudible] determinant with a -- well, extremely explicit kernel.
And what you should notice here is that for those who have seen this
Tracy-Widom GUE distribution, that's written as a Fredholm determinant of a
kernel. And the kernel is exactly of the form of the kernel here, except this sigma
function, which here is some sort of mollified step function, is replaced by an
actual step function.
So the area kernel is just an integration from zero to infinity of this [inaudible]
functions. And this is a mollified version of that.
Now, I should remark, Sausimoto (phonetic) and Schpun (phonetic), two
mathematical physicists, simultaneously and independently discovered this
formula. They made no attempt of giving a rigorous proof of it, and I'll remark in
a little while what goes into the proof of this. [Inaudible] about that.
But what is the -- you know, what is the outcome? Well, on the one hand, here's
a nonlinear stochastic PDE that we're able to give an exact solution to the
statistics, but from a physical perspective, it also shows a very nice crossover
between asymmetric and symmetric cases of this universality.
So what -- how does this come in? Well, there's an interplay between the
asymmetry and the time. So rather than looking at asymmetry of epsilon, you
look at an asymmetry of two epsilon or three epsilon or four epsilon. You would
still get a scaling limit, and that two, three, or four, that would actually end up
scaling into the time parameter.
And the way that it works is that the larger the asymmetry is, epsilon times alpha,
the larger alpha is, that corresponds to the larger T is.
So what you find is actually that as T goes to infinity -- just do some asymptotics
on this equation fairly simple -- that as T goes to infinity, if you rescale by T to the
one-third, when the statistics converge to that of the Tracy-Widom GUE
distribution. So what this tells you is that the long time limit of the KPZ equation
has T to the one-third scaling and is GUE. So the KPZ equation is in the KPZ
universality class.
Now, on the other hand, if you take T to zero, look at the short time which
corresponds to moving towards gamma equals zero, then when you center
correctly, if you scale like T to the one-quarter, the Gaussian -- you convert to a
Gaussian distribution, which is exactly that in Edwards-Wilkinson universality
class.
So the punch line is that the KPZ equation actually represents a transition or a
crossover between these two universality classes. And it's a smooth crossover in
statistics and in scaling. And you can see this directly from this [inaudible].
So how do you prove such a formulae. It comes into two steps. And I'll just very
briefly tell you what they are.
The first step is this result on a previous page. It's the realization that you can
use a discrete model to approximate this continuum model. Then the second
piece of input which you need is some level of solvability. Here's an exact
formula. You need some level of solvability of exact formulas for the corner
growth model. And work of Tracy and Widom, which went into this positive
gamma result of theirs, this provides the starting point for this sort of thing. You
need to be very complicated asymptotics and, you know, all this sort of stuff, but
in the end, you get this formula.
So let me wrap up this first part by giving you just a few open problems. And
then review and then I'll move on to a bunch of examples.
So here I gave you the solution for a very particular type of geometry or time of
initial data. The geometry, you remember, is this narrow wedge type of geometry
or a wedge geometry. But another geometry that you're interested in is
understanding growth off of a flat surface like this ballistic deposition model I
talked about in the beginning. And, well, that would correspond to a particular
type of -- well, I guess height equals zero at time zero. And we don't know how
to solve for the statistics of this or prove that this is in the KPZ universality class.
So this would be one -- the question also computing more than just one point
statistics is another question that we don't know how to do. And this third one is
a little bit more vague.
There is a natural renormalization for all of these models. It's this three, two,
one. So you rescale time like three, space like two, and fluctuations like one.
Take any model in the KPZ universality class. For instance Tasia (phonetic) or
KPZ equation. Rescale it according to a renormalization group operator with
these scaling, and rescale time based in fluctuations in the scale.
It should converge to a fixed point. That fixed point is actually more important in
my opinion than the KPZ -- equation itself because that fixed point is the fixed
point of any model whereas KPZ equation requires this [inaudible] symmetry,
describe and compute the statistics of this model.
Of course we know some stuff about this fixed point. We know it's one point
distribution and we know it's fixed time distribution, but compute the whole
structure of this fixed point. This is an interesting question.
So to review this first part, we looked at a few different growth models, and we
saw that underlying these growth models is really two universality classes. The
main one I focused on is this KPZ one because it involves very interesting
phenomenon. It's much larger in a sense.
And underlying this and going between this is this KPZ equation. We've been
able to actually give an exact solution to that.
Now, what I want do now is I want to fill out this KPZ universality class with a
bunch of other examples. And generally we know that the Gaussian universality
class, there are lots of things that are Gaussian, and they're a good reason,
which is that you can prove under weak hypotheses and fairly weak linear
combinations of roughly independent stuff ends up being Gaussian.
So is there a general theory here for when something will have these random
matrix type distributions? And the answer is as of yet, no. But in certain cases,
we can prove a fair amount.
So I'll talk about these five examples. And within each class of example, there is
some level of universality that should be expected. And what this really means is
that if you have a type of model and there are some natural parameter to tweak,
then tweaking these parameters should really not have an effect on the
asymptotic behavior. So in terms of growth models, if you change the local rules,
as long as it has these three properties, you really shouldn't have much of an
effect on the asymptotic behavior.
And in some cases, you can see why each one is universal in its own right, and
then as to why they're all connected, well, this is a little bit vaguer and there's not
a great answer in general.
So what is the first of these models is that of interacting particle systems. Now
an interacting particle system is essentially a system of particles, that interact.
And the poster child of this, we need to be something called a simple exclusion
process. And that is given by particles sitting on the lattice. So now we kind of -we've transitioned to particle systems. You'll see very quickly it's equivalent to
growth models, actually.
And these particles are trying to articulate independent continuous time random
locks. Jumping to the left at rate P, to the right at rate Q. And the only rule is
that their interaction is that they exclude each other. So if this guy tries to jump
left or right, he can't because this spot is taken. Okay. So it's like a model for
cars where cars decide to go backwards and forwards. It's a very bad model for
cars.
If you have [inaudible] the same way, it's not as bad, but still a pretty bad model
for cars.
Now, if you actually take a basically an integration of the particles, you get a
growth model in the following way. So imagine let's say that you start out with an
infinite number of particles to the left of the origin and an infinite number of poles
and nothing to the right.
Now above every particle you associate a little increment of slope minus one.
And above every pole, you have a slope plus one. So what initially that means is
that you have all minus one and then all plus one a corner. And now every time
that a particle jumps you make the -- either in this case, you know, you have a
minus one and then a plus one. When the particle jumps you need to switch the
two. So that's equivalent to filling in this box or inverting a valley into a hill, and
likewise, when a particle jumps the other direction, you turn the hill into a valley.
So you see actually that this model and the corner growth model are coupled at
the most fundamental level.
But what's nice about this is that it's very easy to now tweak this particle system,
and now here are a bunch of open questions. If you tweak it in any degree,
prove that nothing changes. Okay. And essentially, we don't know for any
tweaking of this model that it stays the same.
So for instance, one can ask about what if the particles jump more than just to
their nearest neighbors? That shouldn't change things much, right? But we don't
know who to prove that, prove that, you know, these two types have jumped. Or,
you know, what might actually be easier, it's generally easier to prove, when you
have a continuum object, it's easier to prove convergence to that continuum
object. So take the asymmetry between the jumps, so you have a certain mean
jump distance. Take that to go zero like epsilon. And do all this rescaling. Prove
you convert to the KPZ equation. Again, we don't know how to do it.
Other perturbations is you can have a -- some sort of location-dependent on
environment-dependent jump rate. For instance, you're not just affected by a
particle to your left and right but by particles a little bit of a distance away from
you. And again, we don't know how to do this.
Weaker forms of exclusion where particles can jump on top of each other, just
don't like doing that, and they're zero range or finite range processes. Each one
of this type of perturbation should not change the asymptotic behavior of this
model. However, as of yet, we don't have the techniques to prove this
universality, so ideas are welcome.
Now, I'm going to -- I'll give you a different way of keeping track of this corner
growth model. And this will move us into the study of polymers, directed
polymers, which is something slightly different now. And this is another model
which is in the KPZ universality class.
So how does this work? Well, I'm going to focus for the moment on gamma
equals one. So remember, we have the corner growth model. And when
gamma is equal to one, you only have growth.
So if you want to keep track of this model, one way of keeping track of it we saw
was to have a height function and keep track of how the height function grows in
time.
Another way is just to keep track of the time it takes to reach different points.
The time that it takes for a point -- for a box to get [inaudible] from time zero
certain time. And this is actually what we define as the last passage time, so L,
X, Y is the time when box XY is grown. This is gamma equals one.
So in terms of -- there's a very nice way of keeping track of this, but if you want to
know the time it took box XY to get grown, so for instance, this box to get grown,
well, we know that this box and this box need to have been grown. It's two
parents essentially have been grown. And once these parents have been grown,
then there's a waiting time. And because everything is independent and it's a
Markov process, the waiting time can be specified actually from the start as just
WIJ being an exponential [inaudible] and variable of Greek one.
So what that actually tells is you that news compute this L(x,y), then you can
compute it in terms of its two previous neighbors plus the additional waiting time
for that box to grow.
Now, because of the connection with the corner growth model, you can write
down and you can use this result if you have constant to write down the exact
scaling properties of this. So it grows roughly, if you look out in the direction XY
times the large variable T, then it behaves like some deterministic time it takes to
get there plus fluctuation. And the fluctuations are T to the one-third and have
this Tracy-Widom GUE characteristic property.
Now, here's another open question. This is a big open question now. Here I
specified these WIJs to be exponentially distributed. What if you changed them
to be anything else? Sufficient moments, let's say at least five or six moments.
There's a little bit of an argument about how many moments you need. Let's say
Bernewly (phonetic) zero, one. 50, 50, zero, one. Prove the same sort of result.
Now [inaudible] there's a big problem. Aside from exponential distribution and
the discrete analog, which is geometric, you don't even know what this term is.
You don't even know what the law of large numbers is. So it's very hard to prove
a fluctuation result when you don't even know what to center around. But
nonetheless, maybe it's possible.
And we'll see that there's some indications that this really should be true, but it's
only in a -- in what's a very high temperature regime. So what do I mean by this?
Well, if we take this recursion and we iterate it, you iterate it until you get this
following form, so L(x,y) can be written in terms of the weights of all of its
predecessors in the following sense. You look at the collection of paths between
the origin -- here is one, one -- and the box that you're interested in. And you
focus only on directed paths which are paths that go up and to the right or up and
to the left. And you take the maximum over all such paths of the random variable
T of pi where T of pi is the sum of the weights along the path.
So you kind of think of these as being pots of gold, and you need to follow the
city street lines and you want to get the most gold. That's what this passage line
is. So it's the maximum over all these paths. Very interesting example.
Now, why is this a polymer -- discrete polymer ground state? Well, what a
polymer is, a directed polymer is a measure on paths. And it's given in terms of a
random environment. And this is the zero temperature version of that model.
So in general, a polymer measure assigns a Boltzmann weight, which is given
here, to a question of paths directed with respect to a disordered environment.
So in particular, what you do is you look at this T of pi, so this is essentially a
discrete path integral. It's the sum of the weights along your path. And you
weigh a path, saying that it's taken proportional to the exponential of beta times
that random variable.
Now, you need to renormalize this [inaudible] to make it a probability distribution
by the partition function which is the sum over paths of these weights.
So what happens for instance is beta goes to -- beta is called the inverse
temperature. So beta going to infinity is like temperature going to zero. When
beta is going to infinity, you find that the longest path is the one that's favored the
most. In a sense, it becomes the only path that's taken. Okay.
Now, you can make this precise. For instance, let's focus on the set of paths
which again go between one, one and X, Y. And then if you look at the free
energy, which is logarithm of Z divided by beta, then as beta goes to infinity, this
actually converges to the -- this last passage [inaudible], L(x,y).
Now, again, the conjecture is that for any beta positive, if you look at this free
energy and you take T to infinity, it will behave like something deterministic plus
something that fluctuates like T to the one-third.
The reason for this -- so why, you might ask, why for any beta. We know it's true
for beta equals infinity. That doesn't mean it's true for beta equals a million.
Well, the belief is because there is this concept of strong and weak -- strong and
weak disorder. And what's been proved is that in these sorts of one plus one
dimensional polymers, for any positive beta, you're in a strong disorder regime,
which means essentially that the path measured is actually affected by the noise.
So it's only when beta equals zero, in which case this just becomes the trivial
uniform measure. It's only in that case where the noise doesn't matter. For
every other -- every other beta, the noise has an effect, and this is where this
conjecture would come from.
Now, this is not known for any positive beta, but what is known is that again, if
beta goes to zero -- and this is totally analogous to this weak scaling of gamma.
So here, we have two universality classes. You have positive -- I'm sorry. You
have strong disorder and weak disorder, positive beta and zero beta. And if you
take beta going to zero and you rescale the other parameters, then again, the
partition function -- let's focus on that for the moment -- this converges to a
nontrivial object. And what is it? It actually converges to the stochastic heat
equation with multiplicative white noise.
In other words, to say that its logarithm -- now there's a question at the minus
side. Its logarithm converges to minus the stochastic and minus the KPZ
equation is up to a little bit of shift. And this observation was made recently by
Alberts, Khanin, and Quastel at math, and actually also a little bit earlier actually
in physics by Calabrese, [inaudible], and Roso, though Alberts, Khanin and
Quastel have a scheme to prove this. Fairly simple actually to prove this in terms
of [inaudible].
So what's really going on is the following. You have this polymer. You're taking
a scaling limit of the polymer, and as beta goes to zero, you're converging this
stochastic heat equation. And the reason for it is the following:
If you look -- if you remember this recursion I gave you here of the last passage
time, there's an analogous recursion for the partition function. The partition
function at time XY is written in terms of its previous [inaudible] except the max
plus goes to plus multiple occasion. This is the tropicalization of the problem.
And that recursion is nothing but a discrete version of the stochastic heat
equation with multiplicative noise. You can kind of work it out and see.
And so what's really going on is this scaling of beta is the scaling which the
discrete white noise converges to continuum white noise. That's where the
[inaudible] all comes from.
Now, why in the world would we expect the KPZ equation should come up or the
stochastic heat equation should come up in the world of polymers? It's because
of the Feynman Kacs formula. And what it says is that if you remember the
stochastic heat equation and multiplicative noise, it was the Laplacian plus a
potential, which is space times white noise times Z. And this is the general form
for Feynman Kacs and this -- if it was a deterministic potential, you would just
have that -- you could write this solution in terms of an expectation over Brownian
motions or -- I've done a little normalization by the heat kernel to make it
Brownian bridges of exponential of the path integral of the Brownian motion or
Brownian bridge through the potential.
Now, what happens here is that of course the potential now is random and it's
very degenerate. And there's a certain type of renormalization. It's called
[inaudible] renormalization, but nonetheless, you can still think of the heat
equation as being the polymer partition function for a continuum polymer in a
space/time/white noise environment.
So the punch line again of this is that if you look at the free energy of this, which
is just the logarithm, our results proves that the free energy minus some terms,
it's probability distribution is given by that of these crossover distributions.
So what this says actually is that in a sort of double scaling limit sense, as beta
goes to zero, that there is universality. So it's this weak form of universality.
Okay. So let me start to wrap up. I have a few more examples, but I'll be brief. I
know that it said till five, but I'm not going to take -- I don't want to go much over
4:30.
So now let me transition to something completely classical. So here we've been
talking about all these polymers. This actually is also a polymer, but something
called Ulam's problem, or random partitions also comes up.
But what you do is you look at a uniformly chosen permutation. You write it in
two line notations. One goes to five. [Inaudible]. And you look at the longest
increasing subsequence of the second row. Okay.
This is a random variable [inaudible] function of the permutation, you think of it
has a random variable. Here it's five. And Baik, Deift, and Johansson showed
that -- I'm choosing normalization. I'm looking at a permutation of length and
squared if it can be anything, but it looks nicer that way.
You look at that random variable, subtract off its law of large numbers, which
turns out to be 2N, divide by N to the one-third, and this converges to the GUE
distribution. This actually was the result that kicked it all off. This was where
people first kind of discovered that these GUE distributions were coming up
elsewhere besides random matrix theory.
And what was underlying this is really this Robinson Schensted correspondence
or Knuth correspondence. Which is a correspondence between permutations
and a certain combinatorial objects partitions which also have interpretation in
terms of representation theory, the sorts.
Turns out actually there's an interesting world of tropical combinatorials in which
there's a tropical RSK correspondence and corresponds to positive temperature
polymers, and you can actually use similar techniques there. Now, so this was
resolved by Ded (phonetic) and Johansson.
Now, let me contrast this to the earlier work of random matrix theory. And you'll
see I copy and pasted a formula.
Now, okay, so this is random permutation, now random matrix theory. And just
because the word random comes in doesn't mean that they should be the same,
but it turns out that they are. At least near the edge.
So let me tell you a little bit about random matrix theory. I would be negligent if I
kept talking about the GUE but never said anything about it.
So the Gaussian unitary ensemble, basically it's an ensemble of Hermetian
matrixes which are given by complex Gaussians off diagonal and real Gaussian
on diagonal. That's all you got to know really.
And what Tracy and Widom discovered was that if you look at the eigenvalues,
these will be random. You look at the largest eigenvalue, and you subtract two
and divide by into the one-third. Then it's probability distribution is given by the -okay. Not going to do that crowd thing. [Inaudible] GUE. Okay. And of course
that's where the name came from.
Now, why in the world would it come up in both contexts or all these other
contexts? There really is still some mystery. Lots of very smart people have
thought about this. [Inaudible] wrote a very nice paper about this in connection to
[inaudible] geometry, but still, all of the connections between growth models and
particle systems are in a sense at the level of counting. Okay.
Now, within random matrix theory itself is KPZ scaling is ubiquitous. It's
universal. And this is perhaps the most universal that people have been able to
prove. So what is known? Just briefly, one slide on it. There's been a lot of
recent work also.
So the GUE distribution sits in two camps. It's both a Wigner and an invariant
ensemble. So your Wigner matrix just means that you can take these Gaussians
on and off the diagonal. You can replace them by some other type of
distribution. Okay.
And the result is -- and this was first shown under some fairly strong assumptions
by Soshnikov using combinatorial methods and then more recently, Erdös, Yau,
Schline (phonetic), and others, Tau and Vu, strengthened all these results,
different methods, that there is universality, which is to say that the actual
properties of these distributions up to the knowledge of the first few moments
really doesn't play a role. There is universality, you get the same statistics of the
edge. And also in the bulk.
The other property is this being an invariant ensemble, which is essentially
saying that the eigenvalues sit in a certain potential. And again, the potential
corresponding for the eigenvalues in GUE is the polynomial [inaudible] squared.
Universality says that as long as it looks something like H squared, just to say a
polynomial with even degree, positive [inaudible] coefficient, you should have the
exact same edge statistics. And this has actually also been strengthened by
Erdös, Yau, and others.
Let me finish off with the last thing. And I just want to give a picture. And you
guys will all know this picture, because I saw it up on the second floor. It's
produced by David Wilson.
Now, this is that of studying nonintersecting random walks and tiling problems.
And we kind of know what we should be looking at now, because I've been
talking about last, longest, top eigenvalue. So what you do is you look at let's
say four random walks. And you look at the uniform distribution on all parts that
start and end in the same points, but never touch in between. Okay.
And you can associate a tiling to this by associating every up one has a little
rhombus that points upwards, every down step has a point down and everything
where there's nothing to do, you put a diamond. And you can erase it. You get
this nice tiling, or you stare at it a little longer, it's a feeling of a corner of a room
with boxes.
Now, we know how to study this. There are various tools because of this
nonintersecting property, but where does the KPZ universality class, where does
this stuff come in? It comes in at the edge. At the top.
So if you look at the trajectory of the top guy, you see in this vicinity it's
fluctuating and it turns out -- and there are results to this extent -- that the top line
as N goes to infinity, N being the side length, fluctuates like N to the one-third,
has GUE has its distribution, and also has correlation of [inaudible] into the
two-thirds. The same one-third, two-thirds scalings.
And this is the distribution of the boundary of this arctic circle. [Inaudible]. And
of course you can ask for lots of other types of universality, tiling other shapes,
nonuniform measure, different types of repulsion of particles, and again, for the
most part, we don't know much about this.
So there are all these questions. And you see that there are all these directions
of universality, and what we've done is kind of uncovered the statistics which
should be universal and in some cases have been shown to be universal and in
other cases have not been shown to be universal.
Now, we know that Gaussian -- so let me conclude. We know that the Gaussian
distribution really comes up for a functional reason. It is in a sense this unique
object which is when you have roughly independent random variables confined in
roughly linear ways, you get it.
Now, all of the models which I've shown you take essentially independent
random variables. You combine them in very nonlinear ways. But kind of
different nonlinear ways. Yet you still end up with the same outcome. So it's a
real mystery as to is there a functional description? You know, I take a black
box, I input random data, I get out -- if I get out GUE, what was the black box
like?
This would be a very interesting question, but we don't have a great description,
you know, something like GUE is the fixed point of a certain [inaudible]
transparent property, any of this sort of -- we don't know how to do that yet.
But nonetheless, we have made a fair amount of progress. Especially in
understanding recently this KPZ equation which really does underlie a large part
of this KPZ universality class. It's very -- these things touch a lot of areas, of
course there is loss of mysteries.
And you know, just kind of a -- an analogy to where we're at, so 200 years ago,
the Gaussian distribution was discovered, and people started to do a lot of work
with that. There were two directions that it went in, right. And one direction was
proving the universality of the Gaussian distribution that not just simple -- you
know, originally was proved for simple symmetric random locks by explicital
asymptotic analysis. This is the work of Du Monde (phonetic).
Now, of course people found nicer ways to prove in and they proved it in much
greater universality and these were the two directions. Computing things without
the Gaussian and Gaussian processes and proving universality and we're very
much still at the very early steps. If we have a few solvable models, then we can
prove in a few cases universality, but we don't have a great description as to why
all of this comes about. And hopefully it won't take 200 years to get that.
But thank you. Thank you very much. [Applause].
>>: Is there any kind of [inaudible] property that characterizes this GUE
distribution, maybe something analogous to the Gaussian maximizing [inaudible]
for invariants? And the [inaudible] question is why do we believe in universality?
I mean, for random matrixes, okay, there's something there. But since your
model isn't coming from random matrixes, should we really believe this?
Ivan Corwin: Why is there something there for random matrix theory [inaudible]
prove that?
Okay. So for your first question, in a sense, I think -- well, no, but there is
something to that extent. So let me tell you just one thing.
So if you look -- so there's work that I've been doing with Alan Hammond on
looking at nonintersecting line ensembles. So if you look at one of these sorts of
ensembles I was just showing you, you look at Brownian bridges, condition
nonintersect, and you take the number of Brownian bridges to infinity and you
look near the edge of kind of the shape, and you look in a scaling window, you
find that the top Brownian bridge is given by an [inaudible] process whose
marginal is the Tracy-Widom GUE distribution. Okay?
Now, in the scaling limit, it -- there's a certain property which is also true in the
finite limit which is that of if you look at the collection of lines, they have a Gibsian
(phonetic) property of Brownian Gibbs property. Which means that if you take
one of the lines and you erase it, you can resample it according to a new
Brownian bridge conditioned not to touch the line above it or below it.
And we in fact believe that if you look at the whole collection of lines in this edge
scaling limit, then it's a -- there's up to some sort of spatial invariant property, this
is a unique set of lines that has this Gibbs property.
So in a sense, the GUE, if this conjecture is true, the GUE is the unique marginal
kind of top line marginal of such of a Gibsian (phonetic) line ensemble. So it's a
little bit of a kind of little bit what you're asking. It's not quite.
As far as universality, why I think it should be universal, you know, by analogy is
one example. There is -- you can see actually if you look at some of -- let's say a
particle system, you can write down the generator of the particle system. And
you can kind of do some formal manipulations and you can see that things kind
of look like they should be going to the KPZ equation.
At some formal level, you can kind of pick out the KPZ equation. So you know,
there are very formal calculations but also simulations.
>>: [Inaudible] here for another two hours?
Ivan Corwin: Yeah. As long as people want to talk.
>>: All right. Thanks a lot. [Applause]