>> Russ: So I'm going to introduce the final speaker of the conference, of the seminar, Geoffrey
Grimmett who is one of the most distinguished probabilists from Great Britain. He's
particularly well known for his research and books in percolation theory and other related
models and he is going to tell us why delta, or why nova rather or why not. [laughter]
>> Geoffrey Grimmett: Thank you very much Russ. I don't know whether or not I can live up to
the introduction. So this lecture title is an entry for the competition for the shortest lecture
title [laughter], of course, anybody can make the lecture title shorter but of course you have to
balance content with content, if you see what I mean, and so this seems to me to be almost
minimal. This lecture title also was partly derived from conversation with Anda Holroyd
[phonetic] yesterday. You can perhaps work out what the change might have been given that I
tell you there was one change and it was not the deletion of the hyphen. Actually there were
two changes. When was the deletion of the hyphen in the interest of brevity and as Russ says
why Nablus or why grad? The question is partly answered by this gentleman. This is a
photograph of Arthur Kennelly. Arthur Kennelly was born in 1861 and died in 1939. Arthur
Kennelly was fortunately a member of the National Academy and so when he died an obituary
was published. The obituary contained his photograph and the first paragraph of the obituary
said--we were talking about obituaries the other day at lunch--the first paragraph of his
obituary I think, well I don't know whether we would want such a paragraph or we would not
want such a paragraph, but it's a very powerful paragraph, maybe it's typical of obituaries at
the time. To few and only to the few has been granted the inestimable privilege of active and
close participation in almost the entire range of development of such an enormous and such a
useful field as that of electrical engineering, from the pioneer days of the telegraph and
submarine cable to the earliest stages of practical application of the storage battery, the
electric light and the telephone and on to the flowering of the art as represented by the present
status of electric power generation, transmission and utilization and of electrical
communication. And to few only has there been granted the friendship, esteem and
professional respect to so many outstanding contemporaries in any art as are represented by
such names as Rowland, Elihu
Thomson, Edison, Steinmetz, Sprague, Houston, and Brush in the United States; Heaviside,
Clark, Lord Kelvin, Fleeming, Jenkin, and Preece in England; Mascart, Blondel, and Ferrie in
France; von Helmholtz in Germany, Marconi and Giorgi in Italy; Nagaoka in Japan; and many
others throughout the world. But to Arthur Edwin Kennelly was granted both the privilege of
such a participation in an art and such a friendship and esteem from the principal joint authors
of its development. With his death on June 18, 1939, Harvard University and the Massachusetts
Institute of Technology lost a deeply respected professor emeritus of electrical engineering, the
profession lost one of its early pioneers and most striking figures, and the National Academy of
Sciences lost a distinguished and valued member. So what can I tell you about Arthur Kennelly
is from the point of view this talk is that he discovered the star delta, the star triangle
transformation for electrical networks or as physicists mostly put it, the Y Delta Transformation
for electrical networks or as we have just re-coined it in the Y nabeler [phonetic] transformation
for electrical networks. The obituary actually continues in my regard to be a slightly
disappointing obituary. It is purely scientific and it's purely laudatory. It has no criticism. I
hope when I die someone will right an obituary and it will have criticisms [laughter] and
furthermore, so far as I can tell, Kennelly was entirely a man of his art. That's to say no women
are mentioned. No children are mentioned. No activities are mentioned, so his life as a
scientist was obviously very esteemed and all we know about him through this obituary is his
life as a scientist and I think obituaries as you see as dressed, had he given this lecture a
hundred years ago he would have been very much better dressed than me, things have
changed. Anyway, here is the Y nabeler transformation for electrical networks, very well
known, take a triangle somewhere in a network this big network and you've got these three
edges, these three resistances that form a triangle and they have resistances and if you replace
the triangle by the star and you choose the resistances of the three red edges appropriately by
that formula than what happens nobody notices the difference. So by nobody notices the
difference at least we can say the following, in the mother network, remember there are a
whole load of other connections and potential differences in applied PDs and so on, there are
the same resistances between pairs, the same currents will flow and the same energy will be
dissipated, so nobody notices when you make this change. Of course there are two other
changes which are very well known in the series and the parallel transformations, but they are
not part of this lecture, but of course, they are very useful techniques. And this result is in the
paper referred to at the bottom. It's very difficult to get a hold of this paper; undoubtedly you
can get a hold of it in the older universities, the archives of the universities that have been
around a long time in the United States. You can't get it in Cambridge University. Google has
very recently--when I first looked on the web for it there was no track of it. There was the
occasional volume from around 1900 which had gone on to the web, finally gone on to the
web, but not this one. Google has now scanned all of the years between 1899 but
unfortunately the volumes for 1899 are only available in snippet form and so I think that means
you have to pay to get the full version. Whereas, 1898 and 1900 are available in full form, so
maybe they know that, somebody at Google knows this result and thinks that they can make
money on it. It's probably made a great deal of money already, so to start out the
transformation is a little thing, but this little thing turns out to have tremendously far-reaching
consequences for a whole variety of problems in our art, that's to say probability and statistical
physics. So starting from something which is very small what I want to show you is how it
develops and much of that is familiar to us anyway, the star triangle transformations through
other systems and then show you a recent application to percolation and it's very convenient
that Jeff introduced percolation or you so I didn't have to do that. I wasn't intending on doing it
anyway but you might find it nice that Jeff has already talked about it, and in this lecture there
are almost no equations beyond equations at that level. This is a lecture by picture and there is
another lecture which is a lecture by equations which I have never actually given which would
be longer and by definition more complex. So spin systems, EC models, Potts models, that sort
of stuff, Heisenberg models, there are also Y Delta star triangle transformations for these guys,
so for these guys the vertices are allocated spin variables and EC models are plus or minus and
the Potts model is some integer between one and Q and so on and these spins interact along
edges and the strength of interaction which is a function of the edge, so in general different
edges have different strengths. Different interactions as with the Js and the three have some
models so the model amounts to in statistical mechanics as writing a partition function, so you
have these spins. You have these interactions. You write down the energy of any configuration
and you call that E. You write down E to the minus beta times E. That's the Gibbs formula that
sets the weight of the probability distribution. You sum that over all spins configurations.
That's the partition function. The partition function is an object, a function which is defined in
terms of sums of a spin configurations and it's the sum and is something which depends upon
the Js. The Js determine the individual energies of interactions. The general form of the star
triangle transformation which is sometimes valued and sometimes not; it depends upon the
system. Depending on the system there may or may not be some formula of this sort here that
makes it work, is if you take a star and you replace it, a triangle and replace it by a star you have
to stand on your head to Y enabler in this one and so I suppose I should redraw my pictures
again. Then there is sometimes some function which tells you how the Ks are given in terms of
the Js. Actually this F shouldn't depend on i here. Well, in that case [inaudible]. Yes, let's leave
it as it is, so some function which tells you the Ks in terms of the Js such that if you make this
substitution then what happens? In statistical mechanics what happens is that the partition
function is unchanged. When you sum overall configurations you are summing over a slightly
different family of configurations so that the sum ands are different, but when you do the sum
you get the same thing or at least it's not usually the same thing. It's usually changed by some
multiplicative constant and then you put that into the normalization so you read normalized by
that multiplicative constant so the partition function is unchanged. This lack of change of the
partition function is an indicator to a statistical physicist that something is going on and then
you can do the statistical physics that follows from this to understand something about the
process. This was known for the Essam model to Onsager apparently, but the Onsager didn't
lecture on the subject until into the ‘70s and it used by Wannier in an early review of the Essam
model probably I would guess in the ‘40s. It's well known such functions F exist for Potts
models and for something called the chiral Potts model and there are various other classical
and quantum models for which such functions can be constructed. It's not universally the case
that whatever model you write down you can find these things, but you sit down and you play
with things in a given system to try to find a star triangle transformation and then you try to use
it. And the name that is most associated with using this construction in physics is Rodney
Baxter with his co-authors in the late ‘70s and in the’ 80s made tremendous progress
understanding the structure of statistical physical models in the presence of a star triangle
transformation. Such was that success and also the observation, the early observation actually
of Yang that the star triangle transformation is useful. The physicist, commonly called this the
Yang Baxter equation. At least for me there is some difficulty talking about the Yang Baxter
equation. It's the usual problem. We have to understand what it means and of course it
means, it doesn't just mean any equation; it probably could be distilled to an equation. It
means a construct. It means a vision and so you try to find out what it means, so you might
think that if you read the encyclopedia of mathematical physics you're going to learn what it
means. Actually the essay there by Yang is not bad but Wikipedia really in my opinion takes the
biscuit. I don't know who wrote this. Is the author of this Wikipedia page in the room
[laughter]? I doubt it. It starts off pretty well. This is the introduction, this is the blurb at the
top that is supposed to, that even the bus conductors can understand and it says the Yang
Baxter equation is the equation first introduced la de da--Yang Baxter refers to a principal in
integral systems taking the form of local equivalence transformations as they appear in a
variety of contexts. So you think well, electric networks, that sounds pretty good [inaudible]
theory. I know about that. Spin systems. I'll be able to understand this article and then it
moves on to the next line and accelerates into hyperspace [laughter] and somehow, you know,
there is a, I'm somewhere here I guess and so the one learns a little something about
Wikipedia. By Wikipedia standards this is probably a pretty good article for experts, but if you
want as a result of this lecture or not you might want to think whether Wikipedia article has the
right way of phrasing this because for us the Yang Baxter equation is this massive, and it's a
construct that involves algebraic formulations. It's a very interesting object, a very interesting
vision into pure mathematics as well as its application in statistical physics, but for us it's really
just a star triangle transformation. These generalizations for percolation even for the easy
model it's useful to see the associated algebra and geometry in certain ways I will show you,
not the algebra but the geometry, but beyond that we don't really need to know about unit
dissociative algebras for the purpose of this talk. Star triangle for percolation, so Jeff has talked
about site percolation where each site is either black or white, bond percolations for each site is
sort of either there or not. There are two states for each site independent between sites. Bond
percolation you do the same for the bonds not sites, so each bond is either present or absent.
Each bond is either present or absent with a certain probability independently of all the other
bonds. Some bonds are present and some bonds are absent and you asked the question what
is the probability that there is a very large connective cluster of bonds. The bonds form a
subgraph; what is the chance of a very large cluster in that subgraph? If the graph is a lattice
then you might ask what is the probability of an infinite subgraph formed by the connected
bonds, the bonds which remain present. So a bond is present with a certain probability and
these are the probabilities associated with these three bonds, and if we replace this triangle by
the star and the right probabilities to use over here are these probabilities so P1 here becomes
one minus P1 on the corresponding edge here, and then you ask is it enough. And the answer
is no. You also need this tree vector to satisfy the equation. The kappa is zero where the kappa
is this function, and that's essentially if and only if. I guess it is if and only if, so essentially is
one of those weasel words, isn't it in mathematics. It's like generally that means always if not,
always but not always. And I like, what's the best, it's generally I like, which means it's true
always except when it's false. So whenever the P satisfies this formula and, so whenever the Ps
are replaced satisfy this formula then when you replace the Ps by one minus the Ps in a natural
way then all of the usual things happen. The local connectivity has got the same law.
Remember these? The triangle is sitting in some other graph and you are interested in what is
the probability that two vertices connect it or three vertices are connected and the fourth isn't.
When you make this change and those vertices are off this picture, in fact they can even include
of course the vertices of the triangle, and you have to be a little bit careful because you can
choose a new vertex in this transformation and you have to keep track of that in the later part
of this lecture. Then those connection probabilities are unchanged by this transformation. I
think it was probably observed first by Sykes and Essam when they wrote down the critical
probability for the bond percolation on triangular and hexagonal lattice, but it might go earlier
than that. What you notice is it only works under this assumption. So far there is nothing here
about criticality. This is just a certain--you can work this out in 10 minutes. It's a 10 minute
computation just check that this is correct. But there is the implication from talking about Yang
Baxter from statistical mechanics that star triangle in some sense tells you information about
critical points, and so it is the case actually that if you take the star triangle transformation on
one of these graphs, for example, what am I saying? Take the triangular lattice here, that's the
triangular blue lattice and it's got a--and think about P as a constant for the moment, each Pi is
the same, then there is a critical intensity for bond percolation and that critical intensity is given
exactly by kappa of PPP when all of the Ps are the same equals zero. The validity of the star
triangle transformation in some sense characterizes the critical point of the triangular lattice
which is formed in many of these triangles. So there is an interplay between critical points,
critical surfaces and the validity of the star triangle transformation.
>>: [inaudible].
>> Geoffrey Grimmett: You're not going to do to me what you did to [inaudible], are you
[laughter]? Have you still got a question? [laughter].
>>: The spin systems you didn't need any equivalent of this condition [inaudible].
>> Geoffrey Grimmett: Well, I didn't write that down, if you care, I didn't say. I didn't answer
that question. Is that a question or a statement? [laughter]
>>: Yeah.
>> Geoffrey Grimmett: I didn't hear the question mark, sorry.
>>: Does it work, the spins system work for any…
>> Geoffrey Grimmett: Well, it's slightly complex. It's a very good question and let me just
come back to it later because I'll probably talk about that in a minute in the context of random
cluster models, but it is absolutely a very good question, and it's something that I think has
been, well, maybe some people have explored it sufficiently, but I haven't explored it
sufficiently, and I think there is something to be said about that. I mean, we understand the
answer to the question, but the answer is sometimes yes and sometimes no, and I haven't
really understood why yes and why no. You can see technically why yes and why no, but I don't
know the implications of that. So one thing now is that of course, this is just replacing one
probability by another and the laws of the same, but of course what we do in probabilities is
the laws in a sense are much more than just in the sense of the partition function. The partition
function being the same, you have to do a lot more work to get, extract information from this.
But these laws are the same, much higher dimensional object, and furthermore a useful thing
to do is not to do this so analytically but to do it by coupling. What you really want to do is
have a configuration on this guy and replace that by configuration on this guy such that the pair
jointly has this property of invariance, so there is a coupling of the star triangle transformation.
It's very natural. It's very obvious what you do and that coupling is very useful for what's going
to happen next. That's what makes this work, the work I'm going to tell you about intrinsically
probabilistic rather than statistical mechanics. This is just a reminder that graphs in two
dimensions have duals, the square lattice itself dual. The triangular lattice is dual to the
hexagonal lattice and of course the triangular lattice has the property that if I change every
other triangle into a star, that also changes the triangular lattice into the hexagonal lattice, so
there are two ways of getting from the triangular to hexagonal, and so you get an equation and
those two ways give you an equation and those equations tell you exactly that the critical
surface of the triangular lattice is kappa of the, should be non-bolded. Should be when all of
the Ps are the same; it's not a vector P. All of the Ps are the same. Is exactly given by kappa P
equals zero. Random cluster models, this is partly in response to [inaudible]'s question. Just to
mention this, I think it's probably two transparencies, no more and we are not going to come
back to it until the very end. This is percolation, so you have a graph. You have an edge E, and
edge E is either over closed, black, white, zero, one, whatever you like, so configuration is a
configuration [inaudible] where each edge gets a zero sitting on it or a one sitting on it
reminiscent of Jeff's lecture, so if Q weren't there, is if P to the number of open edges one
minus P to the number of closed edges, so that's percolation. Now we muck it up by putting in
this rather [inaudible] Q. Q is not one minus P necessarily. Q to the K of [inaudible], where K of
[inaudible] here is the number of open clusters, so you look at the graph, you take a
configuration, throw away the other edges. That graph has a certain number of components
including the singletons. Count the number of components. Call it K, put that up there and
then you end up with another measure. It's a very interesting measure. It's an interesting
measure because as was noticed by Fuldtime and Kestalane [phonetic] that it corresponds to
the Q state Potts model when Q is a positive integer. So Q equals one is percolation. Q equals
two corresponds in a certain interesting slightly geometric way to the EC model to the 2 state
Potts model. Q equals 3, 4, 5 corresponds to the Q state Potts model, so this is a kind of
geometrical representation of Potts models in which connections correspond to correlations, so
it's a good place to explore the geometry of the Potts model. Now this has a Y Delta star
triangle transformation and actually it's written down explicitly here so each edge in this
picture, in this formula has got a parameter P but the P could depend upon the edge, so expand
the generality of each P, so this is the product over all open edges PE of PE where P is the
parameter of the edge and the right way of parameterizing this model is you take the edge
parameter P and you take P over all minus P and call that Y. You see Y because you've got a one
minus P to the minus and you get that and that's why that happens. Then the capital Ys have to
be given in terms of the little Ys by this formula and you require a certain function of this vector
of Ys to be zero where this is the function, so this looks a little bit like the triangular lattice
critical point. It's not quite the same because my Ps have become Ps over one minus Ps and my
Q equals one has become a general Q. So there is a Y Delta transformation here under this
assumption. And this question is a very interesting one. Here we notice that this is also the
critical surface in the situation where all of the Ys are constant, this is the critical surface of the- so we take all Ys constant over the triangular lattice so tile the plane in the natural way we
start with these triangles, copies of these triangles, quadrilateral triangles. Then there is a
constant Y so there is a critical point in the random cluster model and the [inaudible] theorem
says for triangular lattice that is the critical point. This gives the corresponding cubic in Y gives
you exactly the critical point. It strongly believes that if you tile the plane with these triangles
but you retain Y1, Y2, Y3, this generalized vector of parameters, then this gives you the critical
surface of that two-dimensional random cluster model but we don't know how to do that, so
the [inaudible] theorem really seems to need all of the Ys to be equal. We don't know how to
get away from all Ys being equal. In fact, one of the reasons we started this work was to try to
answer that question, but we haven't yet succeeded. Now in particular, this works for the Q
state Potts models. Q just had to be a real number, a positive real, well, bigger or equal to one
if you want the FKG inequality so you want all of the paraphernalia that predicts critical points,
but Q could be an integer. If Q is an integer this is in some sense related to the Q state Potts
model and the Q state Potts model I've already told you has a star triangle transformation on it
by implication here, but the funny thing is that the Q state Potts model has a star triangle
transformation whenever Q is not 2, but when Q is 2, sorry, when Q is not 2 it works at the
critical point. When Q is not 2 it works for all temperatures, so there's some magic about
working in Z2 as it were, Z sub 2 which gives you a star triangle transformation for all parameter
values, but when you work away from, when you get Q bigger you need to work at the critical
surface; you have some extra equation. So this is even stranger in the context of the random
cluster model. The random cluster model with Q equals 2 has a star triangle transformation
only when this equation holds, when this equation is equal to zero. On the other hand, it's a
geometric representation of the EC model which has a star triangle transformation for all
temperatures and that is to say that there is no such equation. This I think is not, well it may be
fully understood by some, but its implications are not grasped by me. It's some kind of
algebraic property of the sets that you are working over. It's not difficult to see why this is true.
What I don't fully understand somehow is why it's true. Now for something completely
different, this is a rhombic tiling of the plane. In this picture there are many different types of
rhombi. All of my rhombi are going to have side length one. Of course when you rescale and
make the side what I want, so I am going to refer to one as the length of the side of a rhombus,
so this is a rhombic tiling of the plane and what I'm asking you to accept and actually I very
much doubt if it can be extended to the whole of R2, so just accept for a moment that this is an
image of a bit of a rhombic tiling of the plane, and rhombic tilings have very beautiful structure.
One very famous theorem about rhombic tilings is the theorem of Penrose. One gets so
confused about these Penroses. This was Roger Penrose who established the existence of a
rhombic tiling of the plane which is aperiodic . A rhombic tiling is any two types of rhombi such
that there is no rotational translation of the plane which conserves the tilings. This is very
famous tiling. It's in fact an example of a whole family, a whole continuum of tilings. His paper
which was what you might call a popular paper was followed by three very serious papers of de
Bruijn in the early ‘80s, published in the proceedings of the Royal Dutch Academy. They are
very, very beautiful papers that show the connections on how to generate sort of the
generalized Penrose tilings and how to show connections between two-dimensional tilings and
slices through five dimensional cell systems and a variety of other rather beautiful things, one
of which I am going to show you in a moment. Think about this for the moment as something
completely different and unless you know the connection, well even if you know the
connection, you will see the connections in between this and the star triangle transformation.
These are rhombic tilings. Rhombic tiling of the plane doesn't have to be aperiodic. It's just a
rhombic tiling of the plane. From the rhombic tiling of the plane there are two graphs that you
can construct and it is a dual pair of graphs. The first thing is you notice that this rhombic tiling
is bipartite. Every rhombic tiling is bipartite because every cycle has length four, because every
cycle is a union modular two of the quadrilaterals that form the rhombic tiling. Every side is
length two so it's bipartite so we can color the vertices red and white in such a way that every
edge joins a white vertex to a red vertex, vertex or edge, anyway, there we are. What I'm going
to do is join up the red vertices across diagonals. When I do that I get this red graph. That red
graph has a property of isoradiality. It's a word I believe coined by Rick Kenyon. So this red
graph, concentrate on the red graph. This is an isoradial graph in the sense that if you take any
face of this graph it has a circum-circle which passes through all of the vertices of constant
radius, radius one in this case. Proof? Well, look, this vertex in the middle is not in the graph,
but it's distance one from each of the vertices, so if I draw a unit circle center of this vertex it is
the circum-circle of the triangle and since this is true for all of these faces, every face here has a
circum-circle in exactly the same radius and therefore this qualifies for the name isoradial
graph. Isoradial graphs were first introduced in 1968 by Duffin in the very simple but beautiful
paper. Duffin showed the relationship between rhombic tilings and isoradial graphs in discrete
holomorphic functions, so he asked for theory of--holomorphic is a function of a complex
variables and it has certain properties. How can we discretize this? What is the right way of
having to define the functional diversities of the graph which has some property of reminiscent
closely related to that analytic functions and this led Duffin to consider rhombic tilings and
isoradial graphs because isoradial graphs are the right places, or rhombic tiling is somehow the
right places on which to have these discretely holomorphic functions. Very nice paper and very
easy read. Of course I could have joined the white vertices instead of the red vertices and had I
done that I would have obtained a pair of isoradial graphs which are dual; one is dual to the
other. Each one is dual to the other. Duality we know in probability, two-dimensional
probability physics is a very useful concept. This is how duality arises in rhombic tilings. One
more thing that de Bruijn pointed out which turns out to be extremely beautiful and useful
there is something else you can do on a rhombic tiling. You can drive a train across it, a railway
train and what you do is you come up with a railway train and this is one track. So this is one
rail and this is the other rail and you drive your railway engine across here with your left rail and
your right rail and when you get to this edge you have to turn a corner to make that your left
edge and your right edge and so your wheels traverse these tracks and you end up driving your
train along this blue line here. Whenever you cross from one rhombus you're driving across the
middle of a rhombus and whenever you get to the opposite face you just reorient your
direction to the next rhombus and you carry on in that way and that gives you a blue track and
of course, each rhombus is the meeting of two train tracks, so you cross each rhombus through
two train tracks you can make the map of train tracks and there are three of the train tracks.
These train tracks are just the way, a very nice way of encoding rhombic tilings. Any rhombic
tiling has got a set of train tracks. It's convenient to think of train tracks in every possible way.
First of all it's just enjoining the midpoint straight line segments, joining the midpoints or being
the actual sequence of rhombi that you traverse as the train moves across it, or alternatively
just extract the blue graph and allow yourself any holomorphic image of that blue graph, so you
don't create any crossings. Just think about it as a graph and so you can ask about the graph of
train tracks. We call these tracks. De Bruijn called them ribbons. Some people call them train
tracks and there is another notation around, but these train tracks are really neat things. Let
me just summarize this very briefly. Rhombic tiling generates an isoradial graph, in fact, two
isoradial graphs which have the primal dual property and in addition, they generate families of
train tracks. Now train tracks, if rhombic tilings give you train tracks, rhombic tilings are pretty
difficult things to draw. Well actually, they are not difficult to draw, but you want to set off,
you only know your rhombic tiling, it's a big object and so it's quite worthwhile sometimes for
you to draw the right train tracks. So what are the characteristics of train tracks? Train tracks
at the bottom here, de Bruijn pointed out that no track intersects itself because each track
crosses rhombus sides and all the sides it crosses have got the same angle because that's the
definition of a rhombus. Because of that property you can't self intersect because you carry on
around and you are going, something nasty is going to happen when you cross and that can't
happen. Two train tracks can cross of course. They cross a rhombi, but they can't cross twice
because once again, you would be contradicting the conservation of angle. So de Bruijn
pointed out that those are two properties of train tracks and Kenyon and Schlenker proved that
those are actually necessary and sufficient to generate a set of train tracks which comes from
rhombic tiling. So those are two interesting properties of train tracks. Now as I say if you write
down descriptions of train tracks, I mean examples of train tracks there are two examples of a
set of train tracks. I don't want you to think of train tracks as being a rigid embedding of a
graph in a plane. I just want you to think of it as a graph so you can move it around a bit. I just
don't want to create any extra intersections. This is a set of train tracks of the square lattice. A
square lattice is always, it always generates itself. You take the dual. You take the square
lattice; a set of train tracks is still a square lattice. It's very hard to get away from the square
lattice. Actually the square lattice of which this is the set of train tracks is the usual square
lattice. It's the horizontal, vertical one. It's just the train tracks give you some angles; they give
it some funny angles. Just draw it and it's easy to see. Instead of taking two of these sets of
parallel lines, if I take five of these sets of parallel lines and I adjust them a little bit. I move
them around a little bit so that you only get pairwise intersections. You don't get any three
ways are four ways or five way intersection and this is a set of train tracks and it's a set of train
tracks of a generalized Penrose tiling. An example of that, here is a Penrose tiling ripped from
the web, bit of the Penrose tiling, the original. Here is the isoradial graph that it generates.
There is the isoradial graph. There is the set of train tracks and there you see according to the
colors you see exactly the five sets of colored, not parallel, but non-meeting, doubly infinite
non-intersecting lines of constant color that constitute the train tracks of a Penrose tiling. Now
what's this got to do with percolation. So a variety of people have worked on systems of this
sort, certainly Baxter and his successors and his collaborators but not in the context of Penrose,
well in a related kind of the more combinatorial context, but without explicitly realizing or
understanding or at least expressing their understanding of what they were doing was
connected to isoradial graphs for rhombic tilings, so probably I guess, Rick Kenyon and coauthors, certainly Rick where the people to write down explicitly the correspondence between
these rhombic tilings, isoradial graphs and the models of statistical physics. This is percolation.
So we are going to do percolation on isoradial graphs. The gray circles are the circles that
demonstrate that this is an isoradial graph underlying this is a rhombic tiling. So we've got this
graph here, this red graph and I want to tell you what the right Ps are. Here is an edge. That
edge has two faces abutting it. Those two faces have circum-circles. Those circum-circles have
centers. Because these are isoradial, all of these black lines have got radius one and so this is
another rhombus and so these angles are equal, so call it angle theta. So P for that edge, the
parameter of one percolation is given by this formula. That's the right formula. If you wanted
to do the EC model, then there would be a different model for J, E in terms of the angle theta.
If you wanted to do the random cluster model there is another formula. In fact there is a
general formula which has got a Q in it. For percolation that is the right formula. Let me show
you why it's the right formula. Now I'm going to go back to, well, one thing you'll notice about
this is that if you go to this dual--let's just establish the notation. An isoradial graph G each
edge has got a certain probability of being open called P and I'm going to call that measure P
sub G, so there's a canonical measure P sub G in an isoradial graph. If you go to the dual graph,
the dual isoradial graph corresponds to flipping this. P of E of one minus P becomes one minus
P over P. That's just the geometry. Theta goes to whatever it is, minus pi minus theta and so
the dual graph G star has got the isoradial measure which also comes from the isoradial
formula. So this formula preserves the natural duality relation of percolation measures, so
that's one reason why it seems okay. It passes the first test. The second thing is that it satisfies
and gives a very harmonious explanation of the star triangle transformation. So this has been
understood for some years. This is not because of our work. Our work is to exploit this
statement, so first of all let's think what the star triangle transformation does. Here's a triangle.
A triangle, because this is an isoradial graph each of these edges corresponds to a rhombus, so
this triangle comes from three rhombi stuck together in that way. Now if I fixed the outside--I
guess I've heard David talk about things like this. You're looking at me in a baffled way, but I'm
sure I have Dave. When I fix this external hexagon, that's the word you were missing I think
Asef [laughter]. But you made me smile. Then the inside, there are exactly 2 rhombic tilings of
the inside. There's this guy and this guy. One is switched and one is switched from the other;
it's an interesting move. If you make this move then the isoradial graph instead of having this
triangle in it has got this star. So the switch of the, so this also switches the measures to the Ps
here and are now given by this new isoradial graph and those Ps are exactly those given by this
formula. This is the formula under which this mapping is exactly the star triangle
transformation percolation. So what that means is if you have a sequence of graphs that are
isoradial graphs you can make these star triangle transformations either arithmetically or
geometrically, and so the geometrical embedding of the star triangle transformation is a very
useful thing to do because you can think geometrically in doing it. So you can make graph
theoretic changes to your graph and you know that the connection probability is going to
remain the same so long as this is the nature of the graph theory that changes. They are
basically flipping the rhombus configuration within a hexagon. Was that chit chat, or was that a
question? By the way, I wanted to tell you is that you're really good audience for two reasons.
You are almost all here since and only one person has his laptop open [laughter] and it's not me
for once. Now so here's the theorem. This is the main theorem of the lecture. This theorem is
called criticality in universality, so we have this isoradial graphs. We want two conditions on a
set of isoradial graphs. Let me just see what it says, 5:15, almost time to go, so did I stop on
time?
>>: Five minutes late.
>> Geoffrey Grimmett: Five minutes late, thank you. So the two conditions are the square grid
property and the bounded angles property. The bounded angles property says you've got these
rhombi. You don't want it to be too flat. At least you don't want it to be infinitely flat. So the
angles of the rhombus have to be bounded uniformly away from zero and pi, otherwise you can
have a lot of really flat rhombi that could accumulate somewhere and that would be very
unnatural and it would cause trouble with the estimates. In fact, it changes the theorems too.
The theorems cease to be completely true in such a situation. So this condition is necessary,
the bounded angles property. The square grid property is that the track system of the graph
contains a square grid. Remember, the track system of Z2 was a square grid. The track system
of the Penrose tiling was five sets of parallel lines, so it contains five choose two square grids,
so I think you see what I mean by a square grid. This we think is not necessary, but at the
moment our proofs need this. We don't regard it as a big assumption because in all of the
graphs that you thought about proving this theorem for have the square grid property and a lot
more.
>>: [inaudible]?
>> Geoffrey Grimmett: Yes. If you look at the track graph, then the track graph contains tracks,
then there is a set of, there is a subset of that set of tracks which form a square grid. That is the
assumption. So now the theorem is a theorem from Manolescu that says that for any G in this
family the following three things hold. Well, actually the first two things hold and the third one
shouldn't really be in the begin list. First things says that these canonical measures have the
box crossing property. Remember Jeff talked about crossings of boxes. Crossings of boxes are
really useful things to have because you can combine box crossings on bigger and bigger scales
to form infinite clusters. And so the first says that the statement of P of G has the box crossing
property basically says that the measure PG is either critical or supercritical. The statement
that it is dual, because this follows if every isoradial graph has this property then certainly that
means that dual has that property because its dual is preserved by, it’s the last these two
conditions up there then the fact that dual has the box crossing property says that the primal is
either critical or it's subcritical and the union of the intersection of these two statements is
critical. What critical means but in some sense it's a generic formula for the P is the generic Ps
that comes from the isoradial embedding are the critical Ps. If you take one of these generic Ps
and you kind of, if you increase every P a little bit by epsilon or by a ratio one plus epsilon you
will be supercritical. If you decrease it by epsilon you will be subcritical. Second statement is,
so the second statement is that, these are the critical measures, so this statement includes of
course the square lattice. It includes a lot more than the homogeneous square lattice. It
includes, of course, triangular and hexagonal. It includes, well it includes just about everything
you know except those constructions of [inaudible] about self dual graphs which tend to rise in
this category. There are self duels and hyper graph structures that based on the conjectures of
Ziff and Scolard which Baylor and Oliver have made rigorous in recent years. Those are of a
different nature. But apart from that family this includes everything you thought of so far and a
few others and furthermore what it shows is that what you are using usually improving these
things, you use graph auto morphism using variance of the translations or rotations or
reflections or something like that, but there is absolutely none of that in this. There is no graph
auto morphism. These isoradial graphs have generally no graph auto morphisms. This works
on Penrose tilings, so there's no periodicity, no rotation invariance. It's just a local invariance
with respect to the star triangle transformation that makes this work and furthermore, this set
G is a universality class to which you will say what is universality class? And I will say well, it's
the following in this case. So here are some critical exponents. One over rho is the exponent
that Jeff talked about. It's the one arm exponent. One over rho is the one arm exponent. Delta
and eta are two other exponents that are present at criticality, so those are three exponents
present at criticality. They are also the arm exponents which are, the alternating arm
exponents are okay, and though one arm exponent is okay. We can't do the three arm, we
can't do that four arm non-alternating exponents. They seem to be much harder to study, but
the alternating exponents are the one arm exponents they are exponents and those exponents
are constant across the class script G. so statement says we have no theorem that says they
exist anywhere in script G, so the theorem says that assuming there exists some G in this class
for which, well, some subset of this set of exponents exists; I think it's enough to assume that
eta exists, but I have to check this. I think you need eta and maybe the one arm. Eta in the four
arm exponent I think. I can't quite remember now, but there it is either one or two of the
exponents in this family such that if you know that these exponents exist for any one of these
graphs, such as homogeneous percolation on Z2, on percolation on Z2, then they exist for all of
these graphs and they are constant across the entire family of graphs. So this is a conditional
universality statement. If they ever exist then they are constant. Now these are only
exponents at criticality. All of these exponents are at criticality. We can't do the exponents
away from criticality. The reason is because the star triangle transformation only works at
criticality and we need to make essentially a very large number even in infinity of the star
triangle transformations to go from one graph to the other. Basically, we have a mechanism for
taking the one isoradial graph and mapping it to another isoradial graph, but in doing so we
have to perform a certainly a large number, and possibly depending on what we do an infinity
star triangle transformations and if you're away from the critical point, each of the star triangle
transformations introduces an error and these errors gang up on you and we can't control
them. So the moment we don't know how to get away from criticality, but we believe this will
be true. And the alternating arm exponents are right, so let me just remind you that these
things are known for site percolation on the triangular lattice, but site percolation on a
triangular lattice is not one of these graphs. Bond percolation on the triangular lattice most
definitely is, but not site percolation. We have one system for which we understand as I was, as
was pointed out to me the other day, well, he didn't quite say this. Everything, well, at least
almost everything, I mean almost everything that we asked the question to more than five
years ago we now understand for site percolation and triangular lattice, but we've got this
other enormous class of graphs which none of these theorems explore, though they probably
should because we don't have S and E for these things. Yet for this other class of graphs we
know how to identify the critical surface and we know that we have universality. I don't have
time to show you what's going on, but broadly speaking this is an example. You've got a bit of a
square lattice. That's the red graph. The red graph is a bit of the square lattice. You embed it.
You add another edge to make it a triangle and that edge you can kind of propagate via a series
of star triangle transformations all the way down the row. And you do this repeatedly and this
gives you an operation of ultimate, of swapping tracks. So remember tracks have charismatic
angles of things that they are crossing so each track is labeled with an angle and so this gives
you a way of swapping angles. It gives you a way of moving between isoradial graphs as you
swap their triangles, their tracks, and it gives you a way of moving between a bit of the square
lattice which is kind of semi regular in a sense to a bit of the square lattice which is totally
irregular by swapping tracks. There's a track across here. There's a track across here. I can
swap them using this sequence of star triangle transformations and while I'm doing that
repeatedly, I can move the bottom section up and the upper section down and swap, smooth
error is the space around. By doing it the other way I can move this into the regular square
lattice. We can relate everything to the regular square lattice for which you know box crossings
exist at P equals one half. You have this mechanism of moving between graphs and that's the
oven and in the oven you have to put food to be cooked and the input, the energy is the fact
that you know quite a bit about duality for Z2, self duality. You put that in. You cook it
repeatedly and out come these theorems for isoradial graphs. These isoradial square lattices,
and then you have to deal with general isoradial graphs and so here's a track system, a
complicated track system of a general rhombic tiling and you want to find a square lattice in it
and so what you do is through star triangle transformations you turn it into something that
looks like a square lattice and sort of move the tracks, you know, you got this kind of square
grid. You take the horizontal lines in the square grid are not necessarily horizontal, you kind of
move them down by making these star triangle transformations which are operations in the
universe of the rhombic tiling. That gives you a bit of square grid. You've got some information
about square grids and then you release everything back and you see what you've got. And
what you've got, you've have to prove two probabilistic estimates which is actually most of the
pages in the paper, which I'm not showing you what they are. And out of this come the
theorems and to just conclude and also give you a summary of errors to be discussed in the
future. First of all this isoradial embedding produces critical percolation and you have
universality within this class if the exponents exist. Contrast that with site percolation, for
which we know an enormous amount but it is not one of these systems. So the further
directions are I think are obvious ones. First of all of star triangle transformations preserve
interfaces, so understand the geometry of interfaces. Interfaces are of course key to SLE, so
can you find, could you find conditional exploration process. Could you show, for example, that
if the exploration process converged for bond percolation on the square lattice, then it
converges to SLE for all of these isoradial graphs? We think that might be possible, but we
don't know how to do it. Find a connection to the good guy, the guy of which we know
everything, and then add it to that. Remove the square grid assumption, probably technical,
probably a bit messy, and not very appealing, we don't know how to do it, nor do we feel--we
feel more compelled to work on some of the other problems. Prove the universality of the near
critical exponents. Again we don't know how to do that. The only way of doing that at the
moment seems to be in the Keston papers of the ‘80s, but the Keston papers of the ‘80s used to
get away from criticality, they used properties, they used auto morphisms of the graph. They
used some rotation invariance of the graph. Isoradial graphs don't have rotation invariance. If
you assume rotational invariance as well, then somehow you are defeating the purpose of the
work which is to show that it's the isoradiality that counts rather than the auto morphisms, so
that's not something that we wanted to do. Nevertheless, we would obviously like to be able to
move away from the critical point and understand those critical exponents. And the next thing
is all of the, most of what I've said and the probabilistic estimates which I have not told you
about are valid for the random cluster model. Why isn't there a corresponding theorem for the
random cluster model? The answer is the usual, because we can't control boundary conditions.
Percolation you've got product measure. You don't have to worry about boundary conditions.
They are irrelevant. For the random cluster measures they are highly relevant because you
don't have FKG. I'm sorry, you have FKG but you don't have independence and so that's what
we can't control at the moment. This would be very beautiful to understand how to do this
with a random cluster model. Finally, I just want to remind you that isoradial graphs have
featured prominently in work of Chelkak and Smirnov on the EC model, and I guess to a lesser
degree on the random cluster model, and of course they are connected because of the
connection to potential theory, through discrete analytic functions through potential theory.
There is an early paper by Chelkak and Smirnov proving that random walk on isoradial graphs
have obvious functions [inaudible] theorems. I mean you have to prove them and so you have
to understand why they are true but moreover [inaudible] the [inaudible] theorem is a very
natural thing to expect and of course by the same token I would expect the uniform spanning
tree because of its relation to [inaudible] random walk has a very nice theory on isoradial
graphs, but so far as I'm aware, that's not been explored. And that's the end of my talk. Thank
you very much. I am sorry to of overrun. [applause].