>>: Before we start, we're going to have a few words from the director of the
MSR Redmond lab, Rico Malvar.
>> Rico Malvar: Thank you, Yuval. I'm actually surprised to see all of you here.
This is a much bigger event than I thought it would be. But I'm gladly surprised to
see that this is the case. I remember when we had a sad event about Oded's
passing. It's -- I mean, we all felt that very much. And very soon after, when we
had an event here at MSR then about celebrating his life, and you've all
mentioned we should also have an event to bring the community together.
And from day one I felt that was a great idea, and we supported the idea very
much. But we carefully kept our expectations low. So I'm very pleased to see all
of you here. Oded was a great member of our community here unlike most of
you I didn't have the pleasure to interact with him that much, just a few
conversations here and there. But since you've all mentioned stories I'll share
one little bit one. We'll just add a little bit to what you already know much about
him.
I was also impressed of course like most of you know him now how smart he is,
how quickly he would pick up any particular conversation and add to it. And a
couple of things that were interesting in particular that showed that, it's when we
have these group reviews here in Microsoft. So every year the research groups
have to present a little bit formally what they've been doing and all of that. And
I'm there, and so I watch the presentation.
And Jennifer at the time was the manager of the group. And they were
presenting very nice work. And I ->>: And Christian.
>> Rico Malvar: I'm sorry. And Christian.
>>: Only mention the lab one. [laughter].
>> Rico Malvar: And now I mean you guys now driving the whole lab, which is
wonderful, which shows that our investment in these areas and related areas
keeps growing. And Oded had an influence on that, for sure.
But the interesting thing on that particular presentation is that he was actually not
presenting and one of the post-docs was presenting about their work, and I
remember asking a couple of questions about the work. And for some reason I
think the person -- I'm not going to mention the name and all that, misunderstood
the question and had thought it was a big criticism or something. And you could
sense in the person how uneasy he was feeling.
And Oded kind of got the whole thing, got what I wanted to know about the
question, and then very calmly interceded and made some commented about
that. And then I said oh, okay, that makes sense. And then he said a few words
to the person not really to me, saying oh, this was great because you were able
to show this, this, and that, and at the same time answered my question.
So basically in one statement he cleared my concerns, he cleared that person's
concerns, and everything was back alive and everything was super happy. Then
I saw wow, this is great work and the personal is like, this is out of the way. And
that's just one example of how he was always on top. Even if he was there in the
little corner and you might even think he was not paying attention, on the contrary
he was completely on top and rescued that person.
And I think that is one aspect that shows how good I believe he was and you've
all mentioned that to me many times in terms of his connections to people,
collaboration, and that's just one more example of that.
Anyways, I just wanted to share one little story with you. That's one I have.
Once again, thanks to Yuval. Congratulations to putting this event together.
Thanks to all of you for coming. This is a great event. And I hope you have a
very good second day. All right.
[applause].
>>: So now it's my pleasure to announce the talk by Omer Angel from UBC,
limits on limits of planar maps.
>> Omer Angel: Thank you. And it's a pleasure to be here. It's been a very nice
visit so far. So well, let me just get right to it. So this is a planar map, and this is
since this is going to be an introductory talk, so even though there are some big
experts on this in the room, I'll give almost all the definitions.
So essentially it's just a plain old graph, and it's embedded in the plane, but we
don't real care about how it's embedded, so if you distort the plane with
embedded graph, we consider it is the same planar map. So we only are
interested in the combinatorics of what is connected to what and the order in
which things are connected, but not in the specific embedding.
So okay, so now, we are interested in to start with in two different types, so
there's triangulations or maps in the sphere and then there are maps in the disk,
but -- and one more comment that's in here, so frequently we are interested in
not in the maps, just in the graphs but also in the rooted maps. So in the rooted
map we just have marked edge which just is a -- which we think of as the root.
This is mostly for technicalisms of eliminating symmetries, but still it's a
convenient assumption. So this is an example of a particular map. Here it's a
triangulation in the sense that all faces are triangles, but as you can imagine
there are many other variations so that the triangulations, all faces have
triangles, quadrangulations have four-sided faces. You can consider K regular
cases where the graph is -- where every vertex has degree K or we consider
several other restrictions. You could allow mixtures of those. You can have a
connectivity constraints. You might want it to be free connected. You can
consider variations where things also colored and the colors have some
interactions, but that's getting a bit far.
And the interesting thing is just as random walks under mild conditions belong to
the same universality class, it's believed that all of these structures will even
belong to the same universality class, meaning that if you take such a random
cluster, such a random structure you take it and you take a very large one, it
looks more or less the same at least in the large scale as if you took any other
distribution.
There's one more way of thinking about this. So this picture is shamelessly
stolen from Duplantier's talk a couple of days ago and some of the other pictures
are also taken from other talks, so they might look familiar to some of you.
So there is another way of thinking about it, which is in terms of the random
surface, and this is part of the people were interested in this. So what you see
here is you can think of it either as a triangulation or quadrangulation. Let's think
of it as a triangulation for a moment. And you can think of this as having some
triangles which are just glued together. So this is -- so the analogy between the
random walk which is sort of as the triangles walk, so there's a story that these
surfaces like a drunken quilt maker, so he has a lot of pieces, so the triangles are
squares and he just takes one and glues it to well to the previous one and sews
it, and each time he takes another piece and sews it and eventually you get
some surface, but instead of being flat and comfortable, you get something that
looks like this.
So these are some of the interesting questions that we have about this. So we
have this random structure, and we want to know what -- what does it look like?
You take one with a million vertices, what does it look like, is it similar to the
Euclidean lattice in some sense, is it hyperbolic? So that's one question.
Then there are questions about the scaling limit. So you have this -- so you have
this structure you can try to rescale it, try to see if you get some limiting object
like brownian motion.
And then there are some interesting questions about conformal properties. So
you have a random surface. If you think of it not just as a graph but as I
mentioned as triangle so square that are sewn together, then you can think of it
as a surface, and you can try to say -- to ask questions about it's conformal
properties. So there's many other questions, and I'm not going to talk about all of
this.
So this is a simulation of a large triangulation done by Duplantier. So it didn't like
planar because it's embedded not in the plane but it's embedded in three
dimension. So you can think of it as a sort of a surface. And you can see
various phenomena. It certainly doesn't look very regular like a sphere. You see
that you have these areas where you have clumps of a lot of points clumped
together.
So this is another picture again from Duplantier's talk. The same objects are not
exactly the same but the same structures. So you have a triangulation, and it's
embedded in three dimensions as a surface.
So okay. So it's fairly clear that it didn't quite look nice and smooth but how -- but
exactly how rough it is is a more interesting question.
>>: Could you clarify the embedded [inaudible] or not?
>> Omer Angel: So it's a two dimensional surface. Topologically it's a sphere.
>>: Right.
>> Omer Angel: So ->>: But not all random spheres presumably are identical ->> Omer Angel: Well, so this is a discrete one. This is a finite thing. So there's
no problem.
>>: Then I'm wondering the [inaudible].
>> Omer Angel: It's not an isomorphic embedding.
>>: I see. Well, that's what I'm wondering. So how is it drawn?
>> Omer Angel: So usually the way to do these things you try to do to find some
energy minimization, so you try to -- you try to keep the lengths similar but you
can't, so that's the way both of these pictures I believe were generated.
Okay. So a very brief overview of the history of this topic. So there's several
directions from which people came to these, so physically people thought of
these -- it's called two dimensional quantum graft in physics. And the idea is to
take some average of various quantities over surfaces instead of over paths
which is done with Feynman integrals. And particularly interesting aspect is the
KPZ formula. So this is a relation between critical exponents of critical models in
Euclidean lattices in two dimensions and the corresponding models on these
random graphs.
So just as you can do percolation you can think of self-avoiding work or using
model and the two dimensional lattice you can also perform these models on
these random graphs, you can compute exponents, and the nice thing is these
random graphs in many cases the exponents are much easier to compute and
the KPZ relation allows us to go from the exponents in this quantum gravity
setting to the exponents in the lattice so there are some works by Bertrand
Duplantier and others relating that were used to compute the exponents in
Euclidean lattice along these lines.
So I this say that there are steps in this approach that are not completely
rigorous, so there's some progress but okay. So that's a -- that's about the
physics.
>>: [inaudible].
>> Omer Angel: No, it's a very new slide, but -- and the Duplantier-Sheffield
work is suddenly a big piece of making these things more rigorous. So some
parts of it concerning the relations to the scaling limits are still open, and I'll say a
word about this later.
Okay. So that's the physics motivation. Mathematically, so it's a very natural
object to takes a random planar graph as a focal theorem which was a big
motivation for studying them, and that is a program in the '60s which didn't push
through to prove the focal theorem but a part of it involves enumerating planar
maps and develop some techniques for enumeration, so they came up with the
various formulas, including this one for the number of triangulations of a polygon
with a given number of internal vertices.
So why do we want to talk about polygons? So you node that here I just have an
external face which is not a triangle. So this is a triangulation, but we have one
external face which has a different size and frequently these things turn out to be
useful. In fact, in order to even write down the recursions that you developed,
you need them, so the idea is that if you have some planar structure so you pick
an edge maybe, the root edge or one next to the root edge, and you ask what
happens if you delete the root edge.
So there are two possibilities. One possibility is that you have sort of a bridge
and then you get two, two disconnected maps. The other possibility is that you
delete these so you have some facing which is no longer part of it and you get
smaller map. So this allows one to write recursions for the number of such
objects and then there's some non trivial techniques developed to solve these.
So the precise formula is not so important. What's more important is the
asymptotics.
>>: [inaudible].
>> Omer Angel: Yes?
>>: [inaudible] simulating a random walk?
>> Omer Angel: Yes. Yes. You can use these recursions to simulate.
Essentially because you have -- if you want to have the enumeration it's possible
to simulate.
Okay. So the more important thing is the asymptotics, and what do you note. So
the number of triangulations off a polygon with boundary size A, I mean an
internal vertices, so you have some constant depending on the boundary and
you have an exponential term and in this case 250 gigs over 27 to the N, and
then you have N to the minus 5 halves. And the interesting thing is that so this
constant 256 over 27 is not so interesting. So this is dependent on the model
where you detect triangulations, quadrangulations and so on. And the exponents
of five halves is the universal one. So in every class of maps that were
enumerated you always get the same type of asymptotics, the same exponent,
and this exponent is -- can be seen from arguments as a bigger effect on the
structure of these objects than the exponential term.
Okay. So this is about enumeration. And I have to say more recently there were
newer techniques developed for enumeration bijective techniques so this was
pushed forward quite a lot by Schaeffer and his Coauthors. So idea is that you
have a bijection between the set of planar maps of a given class, say equal
triangulations, so the nicest construct is in the case of quadrangulations so the
duals which are for regular maps. So those also have the simplest formula.
So you have a bijection between those and tree, plane tree, so that the tree will -the children of the average vertex ordered, and the results are some additional
temperature and very roughly you can think of these additional structure as
telling you how to take this tree and glue together different parts of the tree to get
the surface. So I'm not going to say much about this except that this is a -- this
vertex and say turn out to be extremely useful.
Okay. So I was talking about -- I'm going to talk about limits of planar maps, so
there are two types of limits that we can consider. So one is the scaling limit and
one is a discrete limit. So in the discrete limit you try to ask what does the -what does the planar map look like locally. So you take this random planar
graph, this random map, you take a typical vertex and you ask what is it's
environment, what does its neighborhood look like? So you have some
distribution on the neighborhoods.
And in this sense, you can try to take a limit, and you get an infinite, just an
infinite planar map. So the random walk analogy is not so interesting in this
case, it just says that you can talk a random walk of N steps and you can take a
limit and you get a random walk that never stops.
The scaling limit is more analogous to the scaling limit of random walks to
brownian motion so you have this random structure, you can think of it as a
metric space with a graph metric, or you can think of the surface with a glue that
triangles. And you can just rescale these. So you have some random metric
space you can try to rescale it and get some random but [inaudible] metric space.
So I'll say a few words, a few quick words about the scaling limits and then I'll talk
about the discrete limits. So one of the first results about scaling limit is this one,
the diameter scaled by N to the one quarter and not N to the one-half as random
walk converges to some distribution. In fact, it's known what the distribution is
and these guys did much more than just the diameter.
Much more recently there's works by Le-Gall and Paulin, so they show that the
scaling limit actually exists and again it was known that in certain other senses
the scaling limit exists before that, but in the stronger sense of random metric
spaces in the global of [inaudible] topology, this was done by Le-Gall.
So if you take this MN, which is this random map of size N, and you take D,
which is the graph distance, now you just rescale the distance by N to the
one-quarter, then this converges asymptotically to some random metric space.
So it's known that this space is topologically a sphere, although it [inaudible] of
dimension four, and the conjecture is that if -- that this limit is essentially the
sphere with a random metric where the metric is related to the smooth metric by
a factor of E to the constant times the Gaussian free field. Yes?
>>: Is this a new result, the end of July it was [inaudible] subsequent [inaudible].
>> Omer Angel: What was the ->>: This [inaudible].
>> Omer Angel: Yeah, it may be for [inaudible]. Yes?
>>: [inaudible].
>> Omer Angel: Huh?
>>: What is gamma?
>> Omer Angel: I believe it's half in this case.
>>: [inaudible].
>> Omer Angel: Square root of [inaudible].
>>: [inaudible].
>> Omer Angel: Eight over three.
>>: [inaudible].
>> Omer Angel: So it's a constant, [laughter] a universal constant. Yes. And I
would say that this conjecture is one of the big missing pieces in making
rigorously support of computing critical exponents using quantum gravity. So we
can -- we know that the KP regulation holds within this random metric space, so
in the metric space with this exponent to the G, but relating this to the limit of the
discrete path is still open.
Yes. So this is a map -- this is a picture of this E to the G and I'd just like to point
out that this also looks like a planar map and I may say some more words about
this at the end if there's time.
But now I'd like to talk -- to move over to discrete limits. So this was some of the
things that are happening with us taking the scaling limits of these objects.
So how do we take the discrete limits and -- I apologize to anyone that heard this
50 times already, this definition, but so we can see that -- so a rooted planar
graph, so previously the root was an edge, now it's a vertex. This is not
particularly important. It can also be a vertex.
So we consider just a -- this planar -- this graph, and in this case it didn't have to
be planar. We just need a root though, which is a marked vertex. And we have
this class G style of rooted graphs, and there's a topology on those graphs where
two graphs are close to each other if large balls around the roots are isomorphic.
So we take the ball of radius say R around the origin in G1 and the ball around
the origin in G2, and we want them to be isomorphic, and if this holds for large
them the graphs are close, so it's easily metrizable, and those of you in the front
rows can see an example here where we have these two graphs so they're up to
radius one they agree, but want to go to radius two they do not agree. So they're
not very close.
Okay. So this is the topology on graphs. And a couple of quick examples. The
limits are you can take a cycle of length N, so when N tends to infinity, this
converges to an infinite line. So if you take any neighborhood of the -- any fixed
size neighborhood of the root, then it looks like a straight line once it is large
number.
You can take an N by N square if you put the root at the corner, then this
converges to a quarter plane, a quarter of the Euclidean lattice. If you put the
root away from the boundary, then it converges to the infinite, to the regular, to
the complete Euclidean lattice.
One last example, if you take a binary tree and you put the root at what is usually
called the root of the binary tree, you take the root up to the depth N, then this
converges to the infinite binary tree.
And exercise for you is what happens if you put the root at the leaves. But -- and
I'll just say that you get a different infinite tree. So this is the topology on graphs.
One we have the topology on graphs, we can talk about random graphs. So if
we have any distribution on a rooted graph, then we can -- any set sequence of
distributions, we can try to consider a weak limit. And okay, so this holds also
without the restriction on the degree, on the bound degree. But there's an easy
fact that if you look at the class of graphs where you have a uniform bound in the
degrees, then this is a compact set in this topology which makes various
arguments much simpler.
Yeah, okay. So this is -- so this is the idea of this limit. And essentially the way
to think of it so you have this graph and you ask what does the -- what is the
neighborhood of the work look like, you have some distribution in the
neighborhoods, and you can look at these for any finite diameter, any finite
radius for the neighborhood, and then you can take a limit and you get some
limiting distribution in neighborhoods for any given radius.
Knew, in many cases we want to take limits of also graphs which don't have a
root. And in this case, we just take the root uniformly at random. So then in this
case, when we talk about limits of graphs with no marked root, then it just means
that we are interested in the distribution of the neighborhood of typical vertices.
Okay. So one theorem that I'm going to prove is the following. So you take GN,
which is a random planar graph, a random planar triangulation specifically and
with a given size. So just the triangulation of the sphere with N vertices. So this
is a random rooted graph. And the theorem just says that the limit exists and in
fact it also should say that the limit gives you some infinite planar triangulation,
so it gives you some -- okay. Just a infinite planar triangulation.
And there are two parts in the proof of this, so one part the first part is the easy
one to show that the probability of getting any particular environment converges.
So suppose this is the ball of radius R. So you have the root here. And you want
-- so you have some particular ball, and you want to say what's the probability of
seeing this particular ball? What's the probability that we have this particular
environment for the origin? And so we have a ball for this -- it has some
boundary of size M, and we want to say how many ways there are to complete
this to a triangulation of the entire sphere of N vertices. So let's say that we have
K vertices overall in this ball. So it's just given by the number of triangulations.
So once you remove this part, you are left with just a polygon. The rest of the
sphere is just a polygon. And you have external bound face of size M. So you
now think of these as the external face. And the remainder has K -- M minus K
vertices, so you have phi of N minus K and M.
So this is the number of ways of completing this to a triangulation of the sphere.
And we need to divide this by the total number to get the probability. And if you
look at the asymptotics -- I'm sorry, you have an external face of size three for
the total number. And in any case you -- if you look at the asymptotics of phi you
can see that the exponential term cancels out, you are left with something that
depends only on K and M, and in particular the limit exists.
So this is -- so this shows that the limit exists. It doesn't show that it gives us a
probability measure, and so we need some sort of tightness for that. And here
there's a very nice argument. And it might remind you of an argument from all
this talk about considering the -- considering a random walk on the graph and
observing the environment scene from the random walk, and we have the same
effect here. So if you take a random walk on the graph and -- so if you take a
random walk on the graph, so you started the root and you move and now you
consider the map again rerooted at this point, then you get -- then this is
stationary. So at any given time you get the same distribution.
Okay. So now let's look at the maximal degree within a ball of radius R. We
want to show that this is tight and this will -- and this will give us the tightness that
we need.
>>: What is the word --
>> Omer Angel: Tight. So it just mean that the tail decays to zero uniformly in
the size of the whole triangulation. So we have a sequence of distributions, one
for which N and we wanted the probability that MR is greater than T tends to zero
as T tend to infinity, so this holds foreign distribution but we want this to hold to
be uniform in N. Okay. So we want to show that the maximal degree in any
finite radius is tight, so for degree zero just for the degree of the root, this is
known, and you can -- and it's fairly easy, you can compute distribution of the
degree from this sort of arguments.
Okay. Now we [inaudible] by induction, so if the -- so suppose that for radius R
minus one it is tight and now we just observe that the ball of radius R you can
think of it as a union of balls around each of the vertices that are neighbors of the
root. So if in each -- and in each one of those, so, okay, so if we take one
random step, so we replace the root by a random neighbor, and we look in the
ball of radius R minus one of this new neighbor, of this new root, then this has
the same distribution as we started with. So this is the stationality. And this
means that the maximal degree in each of those balls is tight. But we also have
a -- quite a small number of these balls. So the number of these balances is not
bounded, but it is also tight. So this means that MR is the maximum of a tight
number of not independent, but it's a tight number of copies of MR minus one.
And from this it's easy to see that it is tight.
So that's -- okay. So we have -- we finished one proof. That's good. Yeah. I
know. So -- okay. So one more statement to show you how this thing works.
So this limiting result is one -- and I'll skip the proof, but one [inaudible] so the
size of the ball of radius R grows like out to the full. This is closely related to the
R and to the one quarter diameter size scaling and related to the half of
dimension 4 of the limit. So the number of vertices within distance R goes like
out to the full even though it is planar and flat.
So one particularly interesting question about this, and it's a question that in
complete generality is still open is whether it's recurrent or transient. So you
have a random graph, that's one of the first questions you ask, is it recurrent or
transient, except when it's obvious.
In this case, it's not obvious. So there's the following theorem, due to Benjamini
and Schramm motivated by exactly this question which is the following. So you
take some graph say GN, so you take any finite, any sequence of finite planar
graphs, and there's a condition that the degrees are bounded by M that say the
reason that -- that's the reason for it's not resolved in complete generality. And
now you take a uniform -- a root uniformly at random, so you get a distribution of
random graphs, you take its limit, then the theorem is that the limit is -- so its limit
is a random -- is a random graph and the limit is almost truly recurrent.
So if you take a three-regular map or four-regular map, so any of a large number
of classes of planar maps, then this theorem applies. So you take -- so it didn't
matter if the GN is random. So you take -- so as long as they satisfy the
conditions. So as long as you have bounded degrees, you can get that the limit
is recurrent, okay. So you might say that three-regular graph is the dual of
triangulation and if a graph is rue current, if the planar graphs and the planar dual
is also likely to be recurrent but there are counter examples, and -- and again the
conjecture is that this recurrence holds for any class of planar maps that you
would care to take.
So I'll just finish with a very brief overview of the -- of a -- of how this is proved
and the idea is that -- the idea is to combine this with some of Oded's earlier work
on circle packings which we heard about yesterday. So you take a circle
packing, you have a finite graph. So let's assume for a minute it is triangulation
and this is not really a problem. Then you can consider it circle packing, and you
can consider the limit of the circle packings. So it's possible to take this limit and
in the limit you get a random circle packing in the plane. Which has the same
combinatorial topology, it's a circle packing corresponding to the limit of the
graphs GN. So you might need subsequent here, but that's not a problem.
And then you have two steps. So one step is to show that the limiting packing
has at most one accumulation point, so it's locally finite, so again -- and then the
second step is to use a result of Schramm as saying that if you have a locally
finite triangulation with bounded degrees, a locally finite circle packing with
bounded degrees, then it corresponds to a recurrent graph. So formally say it
uses the notion of extreme length and I think I'm out of time, so I will not try to
describe this. So I guess I'll stop here. Thank you.
[applause].
>>: Questions?
>>: A general comment on a couple of things. And this is almost I find it helpful
to try to say explicitly [inaudible] you consider a phrase like [inaudible] in the
planar graph, well that's a somewhat vague phrase. What exactly does it mean?
And the sort of -- what I find it helpful to say is that there are at least three
different things you can mean by that. [inaudible] mentions it here. That is you
can solve that by taking some very particular model of uniform on all planar
graphs of size N of some particular type and then take local limits and that gives
you these very particular my special models, these special models like brownian
motion or [inaudible] or SLE as a special model. The second thing is the sort of
very [inaudible] result at the end which takes an arbitrary finite planar graphs and
take local weak limits of those, what you get there is a whole class of processes
which are exactly the graph theory analog of stationary processes and it's bizarre
to me that the Benjamini-Schramm hasn't been pursued much further. It's kind of
as if Berkov proved the[inaudible] theorem and then everyone went away and
paid no attention to it. You have this sort of a fascinating class of stationary
processes, you've got one here, but really there are a hundred theorems about
the use [inaudible] interacting product process on these graphs instead of on the
lattice, for example. So that's a large class of things. Again, bring that to my
original saying, what do you mean by the phrase landed in the planar graph. You
can do something entirely different for probabilistic and thus take a [inaudible]
point on the plane and [inaudible]. So that gives you something different still. So
there's all these lines of work and if two of those three ideas Oded was
[inaudible] you know there's this vague idea [inaudible] different things. Okay.
Sorry. I'll shut up. [laughter].
>>: [inaudible]. [laughter].
>>: [inaudible]. [laughter].
>>: One general question you might or might not want to answer. So many
people wanted to be students of Oded but as far as I know, you were the only
one who succeeded [inaudible]. So what was your secret? How did you
[inaudible]. [laughter].
>> Omer Angel: [inaudible] connections.
[brief talking over].
>>: Intern, right?
>> Omer Angel: Well, I was already a student of Oded at the time, so that's why
I came here. It started in [inaudible] I believe. Eti is nodding.
>>: Maybe in the [inaudible] session you'll choose to share some more.
Any more professional questions? [laughter].
[applause].
>>: So it's my pleasure to announce now a talk by Michael Freedman who is
going to speak about random triangulations as dynamical variables in quantum
mechanical models. Michael.
>> Michael Freedman: Thank you, Bertrand. I realized I was trying to get
Bertrand to give me the last bit of information I needed to give this talk with the
mike open in the back room, but I got it covered. So the general setting for what
I'll be talking about is Chern-Simons' theory. And I realize that this is the first
approximation a room full of probabilists and your experience with Chern-Simons'
theory may be varied. Now, many of you have brushed up against physics and
may have learned in that way and some of you were educated in physics but
some of you may not have seen it before, so I want to just put it in context. It's
really sort of the child of what's called Chern bay theory, Chern bile, which is the
method of describing twisting of bundles for characteristic classes in terms of
curvature.
The curvature tensor is a two form with coefficients in a lie algebra. And the
characteristic classes are polynominals in this two form, so they're even degree
polynominals. And they give even degree differential forms. And the
Chern-Simons' theory is sort of a boundary variation on the theory of
characteristic classes where an integration is performed to reduce a dimension.
So you sort of subtract one by anti-derivative to get the Chern-Simons' forms,
and if you want a typical formula, you'll see something like ADA plus if it's
[inaudible] there may be a term like this, triple wedge of A. And you can think of
this, if you have a physics background as A as a section in a gauge bundle. It
might be electromagnetic gauge bundle U1 or it might be a U2 gauge bundle or
whatever.
Now, Chern and Simons actually had rather modest ambitions when they wrote
this down. They were trying to solve some very classical problem about
conformal embeddings. This was around 1970. This background is sort of 1940
through 1950s. But it turned out that they stumbled on probably the most well
sided action in the world after kinetic energy.
So, you know, it's amazing the payoff to an abstract mathematical investigation.
So 15 years after they wrote down the Chern-Simons' formula to get a number
associated with a particular connection physicists were integrating over all
connections, doing one of these generalized Feynman path integrals. They were
looking at things -- if this is called the Chern-Simons' action of A, then physicists
in 1985 were looking at things like the integral over all connections of E to the
minus 2 pi I some level K times the Chern-Simons' action of the connection.
And thinking of this as a perturbed Gaussian interval, you can could already see
that there's -- this is sort of a quadratic form in A because there are two As here
with a term which is considered perturbatively.
Now, why is this such a great thing to have written down? Well, there are two
things about it. One is it's all written in terms of forms. The metric doesn't enter
in. So that means there's no metric in the formula, there's no GIJ written
anywhere. So this means it's possibly going to give a topological action when
written up here. So it's going to give something that might be very abstract and
deep and not depend on this geometric structure of space but just on its
topology. That's the first hint that something great has been written here. The
second hint is that there's one derivative here.
And that's a shock. Because in physics you're used to things in the action which
are like one-half NV squared. And this V of course is a velocity, that's a
derivative. When you square it, you see two derivatives in the formula. So you
have two derivatives.
Now, why is one so much better than two in condensed matter physics, some of
the most interesting stuff happens when you take the noise away at low
temperatures. And if you have a translation invariant system, it's always better to
think of it in momentum space than position space because those are
symmetries which diagonalize the Hamiltonian, so you always think of K vector at
a time.
And when you're working -- when you want to see the details underneath all the
thermal noise and you go down to very low temperatures, that means you're
looking at very small K vectors, very small momentum. And differentiation as
everyone knows from [inaudible] analysis in K space differentiation goes over to
multiplication.
So one derivative, if you can see this here, means K, one copy of K, and two
derivatives mean K squared, and K will be much smaller -- sorry, K squared will
be much smaller than K at low temperatures, low energy limit. So if the similar
for some reason has a topological term, however weak, it should come to the
fore at a few millikelvin.
Now, what I'm primarily interested in is topological states of matter, sometimes
call them topological phases. And if you want a very quick definition, it's a stated
matter in quantum mechanics or ground state which is degenerate.
So their -- the ground state isn't one dimensional but many dimensional. And
usually this happens for a boring reason like in a magnetic system there might be
a symmetry. If they're a magnet, the spins might point in different directions and
that causes a degeneracy of the ground state but that degeneracy is easily lifted
by a small perturbation. But what characterizes the topological phase it's a
ground state that's stable to local perturbation. You can't lift it. So it's a ground
state that does not have to do a symmetry and instead is stable to perturbation.
Now, you know, so I don't want to get into this, but we want to build a quantum
computer here at Microsoft, and this is how I expect we're going to do it, by
manipulating topological states of matter. And Chern-Simons' theory turns out to
be the main highway to topological phases, our understanding of them. And
there's sort of a weaker path that I want to outline today.
The main highway is the field theory approach, which is precisely integrating this
action, this Chern-Simons' action. And where you land, you land attic examples
of topological phases. You land in these so called spiral phases which have like
a handedness coming from a magnetic field, like right-handed phases. And in
particular, you land in the physical world called fractional quantum hall effect. So
don't worry what this is if you don't know, but it's just some very interesting 2
dimensional low temperature system. And many different lead groups
associated to different gauge bundles and different levels K are believed to be
seen or at least glimpsed in fractional quantum hall systems.
And the degeneracy of the ground state is sort of the computational blackboard
or computational space of the computer that we want to build there.
But there's another way of looking at Chern-Simons' theory which is more
algebraic and starts with the fundamental concept being the representations of
the symmetries, the symmetry algebra of Chern-Simons' theory. And if you take
your representation point theoretic approach, then the primary object of interest
are the particle types, the point like excitations of the theory. So I'll just call them
particles which are usually written like A and B.
And the first thing you want to know when you discuss a theory of particles is
what they might fuse into which in representation language is which copies of A
occur in -- C occur in A tense or B. So you have these fusion conditions. And
then the next most important structure which you start thinking of the algebraic
side is the so called 6J symbol which tells you how you can reorganize fusion
diagrams. It tells you the equivalences between processes.
So the 6J symbol says that if you see a fusion pattern like this with -- think of it
reading from left to right, so 8A and C come in and B and D come out, and they
may be have an intermediate particle I, then a 6J symbol is a tenser which
reexpresses this as the summation over A, B, I, C, D, J of diagrams that recouple
them the other way.
So these coefficients can be calculated from the operator product expansions
associated with the CFT associated with the Chern-Simons' theory and you
should be able to kind of close this diagram, and you should be able to make
other topological phases or models for them by thinking on the quantum algebra
side rather than the Feynman integral side.
There'll be small differences. We expect to get A [inaudible] theories here. So if
we're looking for physical models, we probably don't have a magnetic field
present. What would the physical model we would try to hit? Well, that's right
now big question mark. We're sort of at the stage of working on the math theory
side. We don't really have very strong candidates of what the actual lab system
would be, but we're trying to lay a little bit of foundation. Now, you might say this
looks like field theory and this looks like a lattice model. But I want to make the
point that that lower path is not quite a lattice model. Because if you notice the -not only did the letters change a little bit, which are sort of the states on the
lattice edges on the bonds, but actually the diagram changed. The lattice
changed shape. You know, the left diagram and the right diagram are like a little
piece but of different lattices.
So if you want to build a Hilbert space, the natural cats or basis vectors for the
Hilbert space will be a pair the lattice, I'm calling it a triangulation and a label of it.
So a triangulation and a state.
So the triangulation I don't know, it sort of should be implicitly invisible in the
pictures at the dual structure. I sort of drew the dual thing but in red I could also
draw the triangulation. And you see if you're used to these pictures that I've
done kind of the famous thing to the triangulation on the left to turn in the one on
the right. I did this placket flip where I took this vertical bond and I turned it
horizontal.
So that means we don't have a really fixed lattice which is traditional in
condensed matter physics. That's what most of the literature is. But we have a
fluctuating lattice. We wanted to work in the Hilbert space that contains both sort
of quantum gravity and labeling. So it relates to the previous talk.
And what the Hamiltonian will look like, just whimsically call it quantum gravity
Hamiltonian, is we'll have some terms which are the topological enforcers that
will enforce the fusion rules that I explained, A and B fused to C, and will enforce
the 6J symbols
We'll have a term which allows defects, so I'll call that delta capital D. Well, it
actually penalizes defects by an amount delta. And perhaps some other terms
and perturbation which I'll come to later.
So the point is if we had a fixed lattice structure like a honeycomb, I know
mathematicians don't call this a lattice, I mean it -- it's -- I mean it's not a -- it's
two lattices. Physicists do. Okay. So.
>>: [inaudible] question.
>> Michael Freedman: Yes.
>>: Are you interested in having a lot of quasi particles in this or -- why are you
talking about lattice? [inaudible] very small.
>> Michael Freedman: No, I'm thinking of ->>: Two quasi particles isn't [inaudible] right?
>> Michael Freedman: No, the -- the underlying topological fluid will be made of
this sort of string network of particle lines. But they're not the quasi particles of
the system. So they're two levels, and you're confusing the two, and I
understand why.
So the mathematical input to create a topological phase is the fusion algebra for
the representations of some quantum group. Now, on the one hand you could
think of those letters, A, B, and C, as the excitations of some system. And you're
asking me about those. But I'm not using them as excitations, I'm trying to build
the ground state of what's called the quantum double by inputting the fusion
relations from the original algebra.
So you could look at a picture like this and think that it represents a one plus one
dimensional physics of some particles moving from left to right. But that's a way
of thinking, and it's a good way of thinking, but it's not literally what I'm
describing. I'm describing two plus one dimensional physics, and the two
dimensional ground state will be encoded by some large lattice or fluctuating
lattice of bonds encoded by label decorated by particle types by some
representation theory. I don't know if that helped. But I am thinking of a large
lattice.
And I'm thinking in particular the lattice might just be the honeycomb, but it will
have to, for this type of model to function, it will have to be a provision that allows
defects to be formed and removed. It will have to be possible to flip bonds to like
heat up the kind of honeycomb and let it fluctuate in order for the 6J symbols to
function.
How am I doing for time?
So now where does Oded enter? I realize this lattice would be fluctuating, and I
thought that might be a problem, so I e-mailed Oded and told him that I had
heard that there might be some slow or moderate relaxation times associated to
the space of triangulation. So specifically what I asked him about was suppose
you take the two sphere and you take triangulations on it and just to make it
concrete you fix the number of triangles, this N, and you think of this vertices as
the graph and the edges of the graph you think of as these flip moves that I
highlighted before that packet flip. And now this is a graph and you look at its
incidence matrix, maybe minus the incidence matrix, and what I wanted to know
is the spectral gap of this matrix.
And so I'd call this lambda, this gap. It's related to the lambda of yesterday by
very simple formula. Well, it turns out that this is algebraic in N. It's probably
something like N to the minus one. With my normalization where I just take this
as the incidence matrix. I'm not taking the probabilist normalization yet.
But Oded said we could learn a lot by what was going on if we just concentrated
on the outer planar case, because everything there was given by Catalan
numbers and we could work everything out and we'd understand, and then we'd
at least know sort of roughly what we were talking about.
So over e-mail Oded gave me this lovely lecture in combinatorics. So we
replaced this with the outer case where we look at a fixed polygon, and we
consider only triangulations where we add no new vertices inside. And as you all
know, these are counted by one, two, five, 14 and so on. These are the Catalan
numbers. Cat N which is I'm sure everyone in the audience can correct me if I
make a mistake here. It's roughly equal to an exponential which again is not the
important part, and then a polynomial piece which is N to the minus three halves.
So since I was asking a spectral question and Oded thinks very geometrically, he
said we should think about this in terms of the Cheeger point of view, we should
look like where's the narrow case to cut the space of all triangulations, outer
planar triangulations? You know, we want to think of this space as roughly
looking like we want to sort of know if it's long and thin and see if we can find a
way to cut it into two pieces to see how it would vibrate and figure out what that
lambda is.
So that's the art. You have to have some inspiration to figure out how to cut
these infinite dimensional graphs, you know. And he told me to cut it into the
thick part and the thin part. And I asked him what that was, and he said, well, it's
-- he said every outer planar triangulation, so I won't draw the vertices around
here, has what's called a central triangle, and it's unique. Well with, there's the
detail, you say there's an odd number of vertices on the boundary and it's unique.
So I don't know if this is a standard notion. But what the central triangle is, it's
the one triangle in the picture where the fraction of vertices on the three sides is
all less than 50 percent. So you have N1, N2, N3 vertices here, and each of
these NI is less than or equal to N over 2. Okay?
And it's quite clear that there can't be more than one of these because if you
have one and you go to the side, then one of these edges from something on the
side will involve the sum of two of the previous terms and they'll be much too big.
So uniqueness is easy.
>>: [inaudible] classic triangulations are you looking at?
>> Michael Freedman: Outer planar.
>>: Outer planar.
>> Michael Freedman: That means there's only the vertices on the edge of the
polygon, you don't ->>: [inaudible] tree structure.
>> Michael Freedman: It's the equivalent to the dual -- there's a dual tree
structure. And those are counted by Catalan numbers.
So there's -- but then Oded said a lovely thing, which is very, very much what he
would say. He said, well, now of course you know why there is a central triangle.
It's the fixed point theorem in the disk. So the point is you imagine each of these
triangles to be endowed with a sort of a vector field where it flows in on the long
side and flows out on the short sides.
And if there was no central triangle, you'd retract the disk to the circle. So that's
how you show there's a central triangle. Now, the definition of thick and thin is
you just take the smallest ends, call it N3, the -- you know, say N1 is greater than
or equal to N2 is greater to or equal to N3, and you just say N3 is less than or
equal -- is greater than or equal to N over 10 other N3 is less than N over 10. 10
is more or less flexible here.
And this is called thick, this is called thin. And it basically cuts the graph, the
hyper graph or it will be high dimensional graph of outer planar triangulations into
two classes. And now you can just kind of count away to your heart's content.
You only need Stirling's formula and this approximation to the Catalan number.
And you can figure out that roughly you have order one thick ones, order one thin
ones, you can figure out how many are near the boundary and you can figure out
exactly what the Cheeger constant is, and the conclusion is that you get an upper
bound, you get N to the minus one-half is greater than or equal to this Cheeger
isoperimetric constant K, which of course then gives a bound on this lambda I
was asking.
And this translates immediately to the mixing time being greater than or equal to
N to the three halves. It's sort of the reciprocal of this exponent, but you get an
extra one there because talking mixing time you should go to the probabilist
normalization of the incidence matrix.
By the way, there's and important difference there if you do some physics as well
as probability you should be aware of. In probability it's very natural that if you
have D guys you're going to, then the probability is like one over D you'll move to
any one of them. But if you're thinking physics and you have D interactions with
other people, just because you're interacting with a lot of people, doesn't that
weaken those interactions. There's no reason to divide them by anything, you
know. You have a lot of [inaudible] in stabbing you with knives, so I'll hit you with
the same strength. [laughter].
>>: [inaudible] think of other interactions. [laughter].
>> Michael Freedman: No, actually from computer simulations I think this is -[laughter].
>>: Typical physicist generality.
>> Michael Freedman: Yeah, not the math world, right?
>>: So probability of looking at the shoes of one of your neighbors [inaudible]
smaller, smaller.
>> Michael Freedman: So by the way, so, all these numericists who I don't think
I mentioned them at the beginning, I wrote them on the board. So this is -- what
I'm discussing is all joint work with 4 numerical physicists who have been trying
to see if anything I'm saying makes sense. Matthias Troyer, Simon Trepps
[[phonetic]], Charlotte Gills and Sergei Ifsikof [[phonetic]]. And they've done
numerics on all these topological issues that I'm now going to.
>>: [inaudible].
>> Michael Freedman: I think they're still out. I won't speak for them. So by the
way, I think even in the outer planar case this beautiful idea of Oded's maybe
didn't hit -- I'm not sure if it actually hit the best cut. Found some cut. Because
the simulation seems to give a slightly slower mixing.
I'm now -- why was I worried about this probabilistic sounding thing or mixing
time? If you want to construct it, a topological phase, you really need a gap -- a
gap above the ground state to protect the information. And I was worried that
any model that had quantum gravity aspect to it would be gapless. And the
gaplessness would damage the robustness of the topological information.
And in fact, these models are slightly gapless. That's -- and this isn't a very.
>>: [inaudible] plane is gapless?
>> Michael Freedman: Well, I mean you could have a much worse situation.
You could have the gap decaying exponentially in some parameter. I mean, it's a
low polynomial. I mean, it doesn't meet ->>: It's in [inaudible].
>> Michael Freedman: Yeah, I mean -- right. I mean, it's -- it's not good. It's still
not good.
>>: No, it's not good.
>> Michael Freedman: It's not good.
>>: [inaudible].
>> Michael Freedman: Yeah, it is gapless. It's just different communities don't
call those powers gapless. But I think of them as gapless. Yes?
>>: You were originally thinking about general triangulations, right? So is there
any reason why you would expect the outer case to be the same as ->> Michael Freedman: Oh, Oded was just sort of warming us up with this. You
know, that was his guess that they had the same exponent. And the numericists
actually we did on the -- not the -- we did them on both, and they were the same.
Okay. So the last thing, the last part of this talk is to try to explain ->>: Mike?
>> Michael Freedman: Yeah?
>>: So one thing you did, the Cheeger bound didn't give you a sharp bound.
>> Michael Freedman: Yeah, I know there's a square that could come in.
>>: [inaudible] square [inaudible].
>> Michael Freedman: Yeah, I know. I know.
>>: [inaudible]. Yeah, I think -- I guess, I might have distracted Oded with the
outer planar because that was the case I knew about and in fact David
[inaudible].
>>: Yeah, [inaudible] this has been known for 15 years. It's actually the
[inaudible] first thought of this.
[brief talking over].
>> Michael Freedman: Okay. Okay. Yeah, I -- actually I did know of -- I did find
out about your work, and I should have mentioned it.
>>: Oh, [inaudible] no. [laughter].
>> Michael Freedman: But you know my usual practice was just to ask Oded
first [laughter], you know, wasting time typing anything to Google or Wikipedia
[laughter] or Bing.
Okay. So the question is what could be done about this gaplessness. Can you
make a topologically protected model off lattice? And basically there are two
answers, two ways to proceed that both work. And one is called I think of as the
timid approach, and one is the bold approach.
So the timid approach in the Hamiltonian that I wrote down is just to make
defects very rare by setting delta toward infinity, make delta very large constant,
really penalized detects, then basically you're only working on the honeycomb
with tiny quantum fluctuations around it. So that it would cost a lot of energy to
change the honeycomb, and you only get virtual processes that drag you out of
the honey come into other lattice.
Now, when you do this, you actually can prove that you get all the topological
phases you expect, assuming you trust the perturbation theory, which I think can
be made rigorous in this case. But you get a spectral gap for the topological
phase on the order of one quarter raised to the 55th power times the bare
energies of the terms.
Now, this is a very small gap and probably not useful for building a computer.
Now, why is the gap so small? It's because in this picture the quantum
fluctuations around the hexagon tiling that are important are the ones that would
eventually do something topological, rotate this placket B all the way around A.
And you can ask how many F moves, how many 6J moves were required to do
that.
Now, the first one would be just to rotate this spoke to there. This spoke to there
would be a second one. This spoke to there. Now we've done three. You do
three more and you've moved -- you do a total of nine, and you've just moved B
to B prime. And then you have to do that six times to get it back to where it is.
So you get nine times six equals 54. So you get 54 F moves. And you have to -then there's a creation and annihilation step so it turns into 56. And then you get
a 55 in the denominator.
So this is only intended for those who are familiar with perturbation theory and
know about energy denominators and so on. So this is -- so timidity gets its
usual reward. [laughter].
Okay. The bolder approach is to set delta all the way to zero and sort of face the
quantum gravity head on. So now you're not in the crystalline phase where
you're working around -- and now what happens is the -- in this phase, you made
the cost of a defect so expensive that the kinetic energy benefits associated with
proliferated defects were balanced and the defects were confined or didn't even
happen. Here kinetic energy takes over and the whole thing becomes just a sea
of defects, and you're in a quantum liquid. And in fact the measure should be
basically some -- one of the measures that was discussed in the last talk. You're
sort of randomly sampling at random triangulation. And you will have to then
think about these gapless gravity waves which we call them.
So the result is a very novel phase which succeeds in protecting some
topological information. But on the other hand, has gapless gravity waves. So
normally you'd expect this to be very risky. But the point is there's a gap to
topological excitations even though that there are another class of excitations
which have low energy. So they seem to sort of pass in the night. I mean, it's
not physically impossible. I mean, for instance, you could have the disjoint union
of two systems, one is topological and, you know, one is just a vibrating string.
And one's gapless and the other isn't. So it's something akin to that that's
happening here. Thank you.
[applause].
>>: Any questions?
>>: Rather ill informed question. Why decompose the structure of that Hilbert
space that way? Why defects? Why not embrace more of the topology of the
first term I guess is another way to say it?
>> Michael Freedman: Well, the point is that the topology of the first term
involves, you know, these two ingredients ->>: Right, right.
>> Michael Freedman: -- of the 6J symbol and the fusion rules.
>>: Right.
>> Michael Freedman: And the 6J symbol can't operate unless there's a defect.
That's the point. You know, if you start with -- imagine this gets completed. I
won't draw it all out. But imagine you start with this picture, and you draw the
honeycomb. And then you insist that this happens.
>>: Yeah.
>> Michael Freedman: What that does is it turns this into a seven-gon and this
into a seven-gon and turns this into a five-gon and this into a five-gon. The rest
are all sixes out here. So this is what I'm calling a defect.
>>: I understand.
>> Michael Freedman: Well, so the point is the way of inputting the topology was
by the form of the 6J symbol, and the 6J symbol has no scope for action unless
there's some defect, at least virtually.
>>: Yeah, I guess I'm questioning whether the neighborhood is all six-gons.
>> Michael Freedman: Well, it will be.
>>: Well, but is there some topological framework in which the 6J rule could
apply but it's not -- it's not a defect in a sea of hexagons, it's rather just a sea of
that stuff where the things flip and migrate but it's ->> Michael Freedman: Yeah, yeah, no, that's --
>>: But it's more regular, it's not noisy or chaotic, it's actually more topological.
>> Michael Freedman: Well, you see, this is actually goes right to the heart of
quantum gravity studies and what I was trying to learn over the open mike just
before the talk is, you know, people have -- people have hoped, you know, to
describe ultimately the physical universe by probability measures on
triangulations and what they were hoping for is to get something that wasn't
boring, wasn't just a fixed crystal with maybe exponentially small fluctuations
around it that would allow there to be galaxies and whatever, you know, some
variation on the background. So it should be somewhat flexible. But it isn't have
Hausdorff dimension, you know, twice what's expected, it should be proximately
flat space, in this case two space or four space for real physics. But they sort of
searched in vain for 20 years trying to find a reasonable homogenous
Hamiltonian that would give a face that was halfway in between a crystal and a
wild sea of everything.
So that's sort of the big failure quantum gravity. Now, there's something called
causal dynamical triangulation which has been big on the quantum gravity scene
in the last 10 years where they break the symmetry -- they break symmetry.
They have a three directions picked out of space, one direction is time, and they
gave weight to the edges of the simplices differently, depending on whether
they're space like or time like, and with that structure they actually can succeed in
enforcing flatness in the correct dimension.
The problem with adapting that to condensed matter is that causal structure
which is so useful in allowing fluctuation but enforcing global flatness doesn't
allow braiding. You can't braid excitations around each other when you enforce
this causality. So we looked at it and enjoyed it, but we didn't try to incorporate it
since we're, you know, at work on a serious project. [laughter].
>>: [inaudible].
>>: I think we have to finish at this point.
>> Michael Freedman: Okay.
>>: Thank you again.
>> Michael Freedman: Thank you.
[applause]