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>> Yuval Peres: There are many fans of sandpiles here. A couple of them are away this week in
conferences. David Wilson and Evander Holroid [phonetic]. I'm glad the talk will be recorded.
So we're happy that Klaus Schmidt is here to tell us about sandpiles in the harmonic model.
>> Klaus Schmidt: Thanks, Yuval, for the invitation. It's a pleasure to be here again after quite a
few years when I was here last. I just noticed that the running footer sandpiles and other models,
that sounds a little more ambitious than what I'm actually going to talk about, because I will not
say -- I'll say practically nothing about anything but these two models.
So what I want to talk about is -- well, first of all, let me offend you by telling you a little bit about
sandpiles. Although, everybody here probably knows much more about them than I do.
So take a nonempty finite set in Z square. For this talk, it's actually not at all important to restrict
attention to Z squared, the exact same thing works for ZD. And then a configuration is an
assignment of positive integer to every element of this finite set lambda.
And this configuration is called stable if the height of the, each coordinate or the size of each
coordinate is less than or equal to four. So as you know while all this has some sort heuristic
interpretation, if the height is too large, then the pile of sand collapses, it's unstable and it falls
over.
So the height gets reduced by four and the neighboring sites each get increased by one. Of
course, the site may still be unstable after a toppling, but you just repeat it until it has become
stable.
Then, of course, the neighboring sites may become unstable in the process. So they topple, and
you go through this process in some order. Turns out it doesn't matter in which order you go
through.
And you end up with a new configuration which is stable, which is called the stabilization, and
which does, as I just said, not depend on the order in which you do this, perform the topplings.
Then so if you have a configuration, you can convert it into a stable configuration. And in
particular you can take a stable configuration, if you like, add a grain of sand somewhere, then let
it stabilize again and see what you get.
So this is a Markov process. I'm not going to tell you exactly what the Markov process is. But the
sites are -- the edges of the Markov process are -- so the vertices are the stable configurations
and the edges are the sites and lambda to which you can add a grain of sand. That would be one
way of interpreting this Markov process.
Then this is something I've already said that the order in which you do things doesn't matter. It
also doesn't matter in which order you add or where you add grains of sand. So here's a little
example. You have a configuration. You add a grain of sand here, it becomes unstable. It
topples. So it loses two grains of sand to these two coordinates which become unstable. Then
they topple and so on, and in the end you have a stable configuration in this region, and
something I forgot to say is that if a site like this one at the edge of the configuration is unstable,
then you lose grains of sand because they fall into empty space.
So this is the reason why this process actually works. So the stable configurations under addition
and subsequent stabilization form a semi group and the recurrent configurations can be
characterized either or the set of recurrent configurations can be characterized either as the
unique maximal subgroup of the stable configurations or as the recurrent states of the Markov
process and you can describe the set of -- that's the really interesting thing, the set of recurrent
configurations.
So here is one description of the recurrent configurations. You have your finite set. Lambda.
You choose any subset E of lambda and any site in the subset. You look at the number of
neighbors of this site in the subset E and you look at those configurations for which -- for at least
one of the Ns, the height is greater than the number of neighbors of that site. So that is if -- so
you look at for lambda, you look at the configurations which for lambda satisfy this condition for
every subset of lambda and this is the space of lambda.
So, for example, in our lambda, you can never have two ones next to each other because each of
these -- so these two ones, they give you a subset of two adjacent coordinates. Each of these
two coordinates has one neighbor in this subset, but the height does not, if the height in these two
coordinates in these two locations is one, then this condition is clearly violated.
If you take two regions, one contained in the other, then if you look at the recurrent configurations
on the larger region, and project it on to the smaller region, then you get recurrent configurations
in the smaller region.
So there is consistency which allows you to -- well, let me postpone that for just a second. So
you have a group operation on our lambda and you have a group operation on our lambda prime.
And you can ask whether these two group operations are compatible in view of this inclusion and
the answer turns out to be no.
So as the size of the region increases, the group structure, which we are discussing here, is not
compatible.
So here's a picture which I think I've stolen from your survey article. So this is -- these are
squares of various sizes, and this is a picture of the identity element in the group we've just been
discussing. And you see that the picture is roughly -- at least kind of similar, irrespective of the
size of the region. So this says that the identities have nothing to do with each other, because if
you look at this big region here, then this smaller region would actually comfortably fit inside this
green square here, but it's clear that the identity here looks quite different from the restriction of
the identity to this larger square.
So as you tend to the limit, as you let this finite region tend to infinity, tend to that squared, you
get a well-defined sub shift of this alphabet one, two, three, four to the Z squared. But the
question is what happens to the group structure in this limiting process.
This is called the two dimensional critical sandpile model. Critical means in this case just the
model I've described. There are other models where you reduce the height by more than the
number of neighbors, and you also have the same effect as here where you use the edge to get
rid of surplus grains of sand and the behavior there is much simpler and certainly quite different.
So what happens to additional R lambda as lambda tends to infinity. It's clear that something has
to fail, because if you have a configuration consisting of all fours, and you drop a grain of sand on
it, then it can never stabilize. And you'll see computer animations, you can download them and
you'll see waves spreading out and coming in and the process of stabilization never stops.
So one of the questions which are very natural, which are studied to some extent, to what extent
is our infinity, how much does it inherit from the group structure given for finite regions?
If you can make sense of this group structure or at least semi group structure, what is the
interaction between addition and the shift action on this space? Which is also well defined
dynamical system. There's an interesting formula which was proved by Deepak Dhar in 2006
that the topological entropy of the shift action on our infinity is given by this expression which
deserves a star because it occurs for several other models as well.
So it's some nonobvious number. It's in fact if you're number theoretically inclined it's the value of
an L function somewhere. But it's -- if I remember correctly, it's a number between one and two
but I'm not sure.
So here's another model. Let's go back. You have to memorize this number by heart, because in
the future it will only be referred to by star. So here's another model. Again, you take Z squared
and you take T to the Z squared. So it's the Cartesian product of our mod Z indexed by Z
squared, and you look at all configurations in this space which have the property that four times
each coordinate is equal to the sum of the four neighboring coordinates but modulo one because
you're in T, you're in the circle, you're not in R.
So this is a linear recurrence relation defined by this polynomial. So every configuration in this
space satisfies that if you convolve it by this polynomial, then you have 0. So if you like, if you
plug in for U 1, the horizontal shift and U 2 the vertical shift, if you apply this expression you just
get 0 to every configuration in X.
Then in 1990 Doug Lind and Tom Ward and I proved that the entropy is star and Dan Wolf and I
proved it's Bernoulli with that entropy.
And the hard measure is the unique shift invariant measure of maximal entropy on this space. So
the dynamics of these systems is pretty well understood. Of course, there are many questions
which you can ask which have not been answered but at least these rough descriptions of the
system are available.
And it's called the two dimensional harmonic model, because this is a harmonic function for the
value at each coordinate is the average of the values at the four neighbors, but on T you can't
take average so you have to do it in this form.
Here's the third model. You take the dimers on Z squared. You look at the even shift action on
this space so the dimers means all configurations existing of exact pairings of pairs of adjacent
sites. You look at the shift action, the even shift action. So you shift by even amounts in each
coordinate. And it has a unique measure of maximal entropy. The entropy is again given by star
and -- yes, I think the proof that it's Bernoulli is confirmed.
So you have these three models. Oh, there's a fourth one which I'm not going to explain, but
which Rask can explain and many others in the audience probably. So there are four models. 42
dimensional models like this. They all have the same topological entropy. At least three of them
have unique shift invariant measures of maximal entropy which are Bernoulli. And the question is
whether there are any connections between these models.
So there are some very partial connections. The spanning tree model and the dimer model are
nicely isomorphic. For every finite set there is a natural bijection between the restrictions of the
spanning tree model and the sandpile model. But, again, you get changes in this
correspondence as the finite region changes. And there's one nice connection. There exists a
continuous group homomorphism from the space of all bounded integer sequences or
two-parameter sequences on Z squared, to the harmonic model such that it's equivariant with
respect to the shift action. So this is the shift action on the sandpile model.
This is the shift action on the harmonic model. And the map is equivariant in this sense. If you
restrict the map from L infinity to the sandpile model, it's surjective. It sends the sandpile on to X.
It sends every shift invariant measure of maximal entropy on R infinity on the sandpile model to
the normalized hard measure on the harmonic model.
And the conjecture is that the connection can be made a little closer -- so this says here that
when you restrict the map to the sandpile model, it doesn't drop entropy.
So whatever kernel means, of course it doesn't mean that the map is one-to-one or almost
one-to-one or finite to one or anything like that, but there is no loss in entropy under this map.
So in some sense the preimages of individual points in X cannot be very large. And the
conjecture is that you can improve this but unfortunately this is a conjecture not proved.
So you can -- there's a closed shift invariant subset of X such that the entropy of the action
induced on this quotient group is the same as before. If you look at the map -- now, there is a
map from this -- yeah. If you restrict this map to the sandpile model, then it's equivariant with
respect to the shift action on this quotient group. It's still surjective. It sends every shift invariant
measure of maximal entropy to hard mention of this quotient group. But this is just a repeat of
what I've just said before.
But the map is one-to-one almost everywhere with respect to at least one of these measures of
maximal entropy and that would imply uniqueness of the measure of maximal entropy on the
sandpile model, and that factors -- well, that is still an open problem.
So this is something we've been thinking about for some time. At the moment the problem is
resting, but we haven't made sufficient progress to give an answer either way.
Now, what I want to explain to you is how one gets a map like this from the space of bounded
integer configurations to the harmonic model which, then, when you reduce it to the sandpile
model gives you something at least close to what you want.
And the method of proof goes back to an idea of Vershik. I'll explain it in a much, much simpler
setting; namely, for a single plural automorphism, hyperbolic automorphism of the two torahs, a
an old friend, the entropy is the golden mean.
And you can look at it expanding and contracting, subspaces. So here is the expanding sub
space. Here's the contractive subspace. The matrix is symmetric, so these two are actually at
right angles to each other.
And you have intersections, because this is the two torahs. So this point here is the same as this
point and the contracting subspace through this point will intersect expanding subspace through
this copy of the origin and here is one of these intersections.
There are many -- this is a dense set of intersections because these lines wrap around. And this
point is called a homoclinic point because on the backward iteration it will remember that it lies on
the expanding subspace. So on the backward iteration it will contract to 0 exponentially fast and
on the forward iteration it will contract to 0, again infinitely fast. So this is a point whose forward
and backward orbit converges to 0 quite rapidly.
This line here, well, it extends and wraps around and in fact it comes back here. So here's
another intersection of this expanding and this contracting subspace. So this is another
homoclinic point.
Now, if you have one of these homoclinic points, then you can define a map from the bounded
integer sequences to T squared by doing this. So you take this point. You move it through the
auto morphism alpha, and you multiply it by the nth coordinate. This is an integer. So this is
something you can do in this group.
And you get, by doing this, a map from L infinity to the group, which has this form. Which is
clearly equivariant. That's why you have a minus N here.
And Vershik showed in a couple of papers in '92 to '94 that the restriction of this map to the
two-sided beater shift coming from the large eigenvalue beta is -- you don't have to know what
the beta shift is, it's the golden mean shift. It's the sequences of 0s and 1s with no two 1s next to
each other. And if you restrict this map to these sequences, it's surjective.
And if you choose a good X, the map is almost one-to-one. So you have a Markov shift, a
topological Markov chain, and an equivariant continuous surjective map to this hyperbolic auto
morphism such that the two systems are practically identified under this map.
Now, a good homoclinic point is homoclinic point whose on or about generates the group of all
homoclinic points. So I'll show you here a good one. This is a good one.
There are other good ones. But I haven't actually worked out which other good ones are visible
in this diagram.
So such a homoclinic point is called fundamental. And this is essentially how the map from the
sandpile model to the harmonic model is constructed.
So in order to do this, you have to find a homo clinic point in the harmonic model and then you
use this homo clinic point to define a map from L infinity z squared Z to the sandpile model.
>>: Anything from the harmonic model.
>> Yuval Peres: To the harmonic model. I can't remember what I said but this is what I meant.
Now how do you get homo clinic points of the harmonic model?
So you take the polynomial which we looked at before. This is a polynomial with integer
coefficients -- well, this is the integer group ring of Z squared and you define an element F in this
integer group ring which has these coordinates at the origin. It has the coordinate four. At the
four neighbors of the origin it has coordinate minus one. And it's 0 otherwise. Then this element
in the group ring, in the integer group ring, is not invertible in little L1 of Z squared. But it's
invertible after a fashion -- it's just that the inverse doesn't lie in little L1. The inverse is in fact a
scalar multiple of the potential function of the elementary random walk in Z squared.
If you take 1 over F, so this is 1 over -- take one quarter of that. Well -- so let's take one quarter F
if we can't invert that, then we're still in good shape. So this is one at the origin and one quarter
at the neighbor's minus -- yes. Yes. So if you take one over one quarter F, this is equal to one
over one minus U. When U is the equi distributed measure at the four neighbors of the origin. So
this is equal to some new to the K. K greater than or equal to knot, and this is how you get the
greens function of this probability measure which is distributed at the four neighbors of the origin.
So you add the powers, the convolution powers of this probability measure, and that gives you
the potential function of greens function of the elementary random walk. And because of
recurrence, mass piles up at the origin. So you add up the first N convolution powers and the
weight that the origin increases because of recurrence of this process. You have typically
infinitely many visits to the origin in this random walk. So you have to rescale the whole thing all
the time, and you end up with a function which, when you go to the limit, has the value 0 at the
origin and drops off ->>: Scale, you subtract.
>> Klaus Schmidt: Yes, you're right. You just subtract -- you subtract constants and you end up
with a function which is 0 at the origin and slowly drops off at logarithmic speed to minus infinity
as you move out to infinity.
Now, you take this function. So this is this W here. And you look at multipliers of this function.
This is in the convolution algebra of functions, real valued functions on Z squared. And you're
trying to find polynomials which have the property that when you convolve this potential function
with this polynomial, it becomes summable.
And in fact such things exist. You can explicitly describe this ideal. For example, if you take one
minus one times one of the neighbors of this, of the origin -- the notation isn't very convenient
about talking about it without writing anything down.
So you look at the binomial term. One minus one of the neighbors. And you raise that to the
third power. And that's an element of this ideal. And the ideal is generated by precisely the third
powers of these binomial terms. So you can calculate this explicitly. Then for every age, for
everything which lies in this ideal, but which is not a multiple of F, we obtain a homoclinic point by
taking HW, this is now an element in little L1. So the coordinates of this point DK and you get
something summable. This point is an element of little L1. You reduce the coordinates modulo
one and then this is an element of the group X.
So this is now a homo clinic point of the group -- in the group X in this harmonic model. And one
result is that every homo clinic point which has sufficient decay to be summable actually arises
like this.
Now, this homoclinic point defines, this is now exactly like Vershik, once you have a homoclinic
point, you can define the same map as before. So you take again the bounded integer arrays
and you define the map. So we have indexed the homo clinic point by this multiplier polynomial
H. So that's the index which you see here. You take a bounded integer array. You take your
homo clinic point XH. Move it around and multiply it by the corresponding coordinate of the
bounded integer sequence.
And the covering map is -- well, this is already a more ambitious version of this. You take this
map. Then it turns out that when you restrict it to the sandpile model, it's surjective, and it's
entropy-preserving.
Okay? Of course, it may not be the most economical map for doing that, because if you take an
edge which is unnecessarily large. So suppose you have an edge which actually gets you into
little L1 and you multiply this edge by something else, then that new thing will still get you into L1.
But it is unnecessarily large.
In other words, this map psi edge, a bigger kernel than before. And since we're interested in
getting the map to be as close to one-to-one as possible, you are forced to try and be
economical.
So what you can do is you take the generators of this ideal. You can combine the
homomorphisms coming from these various edges, combine them into a single map, and you get
something which is entropy-preserving and which is a candidate for being almost one-to-one. But
we have not been able to prove that it is almost one-to-one.
And this construction works for any dimension. In fact, things get a little easier with dimension
greater than or equal to three because there the greens function is bounded. But that's no major
advantage in this game.
For the -- how am I doing for time? Yes, I think I will not abuse your patience unduly. So for the
participative sandpile model, where the topic operators remove more than four grains of sand
from each site, where you therefore have a reduction in the number of grains in the configuration,
you get a nice result. The one we were hoping for for the participative system.
>>: The one you were hoping for the nonparticipative system.
>> Klaus Schmidt: For the nonparticipative system, that the corresponding sandpile model is
almost a one-to-one cover of the corresponding, superharmonic model algebraic model where D
times each coordinate is equal to the sum of the four neighbors.
So there everything works fine. Also, the various other aspects of the dynamics of both the pa
participative sandpile model and this algebraic model are much simpler, and the theory really
works very well there except it's the less interesting case.
But let me still dwell for a moment on this and just discuss very briefly what more you can say
about the connection between the dynamics on these harmonic or maybe on this somewhat
simpler version of the harmonic model and the sandpile model.
So, for example, what corresponds to dropping a grain of sand on to a sandpile configuration.
For this group it means that you just pick your favorite homo clinic point which you used to get the
covering map, and you add that homoclinic point to the group X. That's the exact analog.
If you add something to the sandpile model, you get -- the image of this under the map psi will be
precisely the sum, the addition of this homoclinic point.
So you see that the addition of a grain of sand corresponds to something extremely simple to this
compact abelian group, it's just translation on this group.
So if you choose a clever element, then you can say the dynamics is very simple. You get unique
invariant measures for this. You get all sorts of nice things you can say about this.
So let's go back for a moment to the critical sandpile model and see how much of that survives
there. So you've got the critical sandpile model. And on the critical sandpile model addition is not
well-defined. But it's clear that for some configurations addition is well defined. So, for example,
if you take something very simple like a configuration which nowhere exceeds three and you add
grains of sand in various places, then the resulting element will still be an allowed configuration.
Now, if you look at the image of this under the map psi H which you saw earlier, it's precisely
group operation. Once again, you add various copies of your homo clinic point or various
translates of your homo clinic point to the group.
Similarly, if you take two configurations and you add them, well, in general you can't do it in the
sandpile model. But if you can do it, then it corresponds to addition downstairs.
Now, downstairs you always have addition. So if you go upstairs and you take configurations and
add them, then the preimage of the sum downstairs will be in a sense modulo the kernel of this
map the sum of the configurations upstairs.
Well, you can play with that. But if for this to really make sense and to really give you information,
you should know more about the injectiveness or lack thereof of this map from L -- from the
bounded integer configurations to the harmonic model.
So this is another nice feature of the participative sandpile model. It actually has homoclinic
points. There's a single homoclinic point that actually generates all of them. You don't need any
multiplier ideals. And you can do exactly what Vershik did for the two torahs.
Now, let me waste a few more minutes by going a little further into the background of this. So
criticality for the sandpile model corresponds to nonexpansiveness of the harmonic model. So
the harmonic model, if you think about the definition of the harmonic model, assume these equal
to four, if you assume four times this coordinate is equal to the sum of the four neighbors, then
you will see that this model has unaccountably many fixed points.
Each constant configuration will satisfy this condition. Therefore, this harmonic model cannot be
expansive. If you have an expansive action, it has to have a finite number of fixed points
otherwise you run into trouble. Expansiveness means that you can have not have two points in
the space, in this case in the group X, whose orbits remain close together at all times. Now, if
you have fixed points, well nothing changes in the orbit. So if they're close, they city close. Or if
they ever get close, then they must have been close at the beginning.
So the group X for the harmonic model is nonexpansive, whereas for this model here it turns out
to be expansive. This requires proof but it's not all that hard.
And one of the really interesting problems is when you have homoclinic points for nonexpansive
models like this. I'll give you one example and then I'll stop. So there may not be an immediate
natural candidate for a homoclinic point. You may have to do something to it like what we saw
earlier. You may have to multiply something. So you write down something which you think it
would be nice to be a homoclinic point it turns out not to be one. Then you multiply by something
and you improve it.
So, for example, four minus X minus Y minus X inverse minus Y inverse, that's our old friend
we've been talking about. Look at this polynomial in two variables. And you look at the variety of
this polynomial. So you look at the set of 0s. See in the complex numbers such that F of C is
equal to 0. So this is the complex variety of the polynomial F.
Now, here we are using inverses. So we better be a little careful and avoid 0. So we shouldn't
take 0 here. But something like this. Now, in there you've got the unitary part. So this is VC of F
intersected with S squared. Don't worry, I'm not going to go much further in doing this kind of
stuff.
So these are the complex numbers of absolute value.
>>: Anywhere below that would become invisible.
>> Klaus Schmidt: Okay. I'll stop now. In fact, I think this is the last line I'm going to write. So
you look at the points in this variety for which all the coordinates are of absolute value 1. Then
the expansiveness is equivalent to the hypothesis that this intersection is empty.
This is analogous to the statement that the torah automorphism is expansive if it has no
eigenvalues of absolute value one. In this case there's a point which lies in this intersection. It's
where X and Y is equal to 1.
Now, the question whether there exists homoclinic points or whether there exists, whether you
can produce homoclinic points by a method like this depends on the size of this set. In fact, it
depends on the dimension of this set in S squared. If the dimension of this set -- now, this is for
two possibly replaced by D. If the Cody mention is greater than or equal to two, then you can do
exactly the same thing and you can happily define homoclinic points and covering maps of
discrete models and do all this stuff for any kind of exaltic model you might be interested in. But if
the code dimension is one, then nothing works.
So I'll stop with something on this line, which you can still see. So we had this polynomial, this
was the harmonic model which corresponds to the critical sandpile model.
If you do this, that's the participative sandpile model, which is well behaved. And there this
intersection is empty because you can't get to five with pairs of numbers of absolute value 1.
But if you take this here, then nothing's known. Because this intersection here is a curve. It's
something like a slightly irregular -- well, not irregular, but dented circle.
And there are no summable homoclinic points. So nothing I've been talking about works even
remotely in this case. I should also say that once you have homoclinic points, you can do a lot
more than this. You can, for example, prove very strong specification properties. So you can
prove that if you look at configurations in some part of the space, so this is true for sandpiles, for
example, if you look at configurations on regions of the space and then you look at another
arbitrarily large configuration in some other region, if these two regions are separated by some
specified amount, you can, within a small error, approximate anything you can see here and
anything you can see here by a single point. And this kind of specification works if this dimension
has co-dimension greater than or equal to.
But it doesn't work for this example now. And we don't know whether this system still has these
properties.
Of course, no sane physicist would do sandpiles with this property where you keep getting -- what
happens here? Yes, you keep getting more stuff all the time, yes. Which is not very realistic but
you can still play with it. I think that's it. Thank you very much.
[applause].
>> Yuval Peres: Questions or comments?
>>: So in your covering of the sandpile over the harmonic model, what's the inverse of 0?
>> Klaus Schmidt: Yes, very good question. So basically what happens is so if you take two
configurations in the -- yeah, so let's -- we have to go back a little. Take any configuration in L
infinity Z squared, some bounded integer array. Suppose you have two configurations which
define only a finite region. Then you can interpret this difference as a lure on polynomial.
Finitely many coordinates. Each of these coordinates can be identified with a monomial, where
the first variable -- if the coordinates are KL, then it's X to the K, why. To the L.
So if you take two configurations which define a finite region, then the differences are polynomial.
There may be some negative powers. And you can ask whether this Laurent polynomial is
divisible by polynomial F which we keep talking about, which is the one with four here.
And the map which we are constructing identifies points which defer by a multiple of this
polynomial. The difficulty is that points also get identified if they're infinite multiples of these
polynomial. So if this polynomial times -- if the difference is this polynomial times an infinite
configuration, so, for example, all threes is such a thing. I think.
Anyway, I withdraw that remark. But you can easily write down -- sorry. Here is one way of
getting something like this. So here is ->>: [inaudible].
>> Klaus Schmidt: Am I too low? Is this too low? So take this configuration and add to it a
shifted version of this configuration. So now we have minus three. Minus three, one, one, one
and so on. So you can build a strip consisting of minus threes, ones, and it continues.
So this would be -- and the rest is zeros. So this would be an infinite configuration which is a
multiple of F. So anything like this will get killed under the map psi X. Now, if you have -- you
can have two legal sandpile configurations which differ by something like this. And they will get
identified.
So the ->>: I guess I forgot a key part of my question which is looking at the recurrent configurations map
to zero.
>> Klaus Schmidt: Yes, but you can have two recurrent configurations which differ by this,
because you can have -- so this would allow you to -- oh, I see. Now, okay, so we have to do one
more level here.
So this is also a multiple of F. But you can see the bottom now. So two horizontal rows of 2s and
1s. So you can have two legal sandpile configurations which differ by this expression therefore
they don't get identified under the map.
You look doubtful.
>>: I just don't see the answer to my question here.
>>: You want to know ->>: First, it's one thing -- what's the image and really asking about what the entire image -- just
give me one thing in the image.
>> Klaus Schmidt: I give you one thing in the preimage. But I'm saying that all possible
multiplies of infinite multiples of F will get killed by the map psi.
Now some of these -- so this is, of course, a huge set. And then you have to decide whether
there are points or sufficiently many points in the sandpile model to which you can add none of
these preimages without getting something disallowed. That's ->>: To be more specific, a critical integration that maps itself to 0, not two different maps. So
critical ->>: Add all 3s to this.
>>: You can't then you have two 1s next to each other.
>> Klaus Schmidt: No, if you add all 3s to this, then you -- oh, I see, all 3s. Okay. Okay. If you
take the point minus 2 everywhere, rather boring point, I must admit, you could add all 4s to it.
And there you get 2s everywhere and that's an allowed configuration. And you can do more
complicated things. It's just that --
So it's -- I don't think it's obvious how to get an explicit description of these things. But you can -it's not hard to write down specific examples. Although I stumbled when I tried to do it.
>> Yuval Peres: Any other points or questions? I think maybe we'll do that in private. So let's
thank Claus again.
[applause]