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[music]
>> Yuval Peres: All right. So, very happy to have Alexander Holroyd
tell us about Self-Organizing Cellular Automata.
>> Alexander Holroyd: Thank you for coming everyone. So I will try to
tell you some of what I have been getting up to with Janko Gravner who
is at UC Davis and who visited here last month.
So I don’t usually do this, but I am going to start with quite a lot of
background motivation showing you pictures. Some might call it waffle,
before I get, before I get down to things, but rest assured there will
be some theorems coming. And the reason I do this is because the
motivation for what we are doing is maybe not completely obvious to
begin with. So that’s what I will try to show you.
And let’s talk about cellular automata. So this is a subject with
quite a long history. So certainly Von Neumann and Ulam considered
cellular automata and perhaps the most famous example is Conway’s game
of life, which probably most people of heard of already, but in case
you haven’t you work on the square lat of Z2 and elements of Z2 are
called cells and each cell is either alive or dead. It’s two steps and
sometimes we call them one and zero. And so you have some
configuration of alive and dead cells.
And then there is an update rule which in this case is a dead cell
becomes alive if it has at least, sorry if it has exactly three living
neighbors. And neighbors here refers to the eight of the cells at L
infinity distance one. So it’s called Moore neighborhood and if a cell
is already alive if, and only if it has two or three living neighbors,
otherwise it dies. And you update everything simultaneously in
discrete time. So you update every cell according to these rules. You
get a new configuration and so on.
So there we go, okay. So yeah, so here is a very famous example of
Conway’s life. You start with these cells alive and you apply the
update rule. So each cell sees how many living neighbors it has and
the second step you have this and the third step you have this and so
on. And it turns out this is called a glider so this shape just moves
diagonally. And there is a huge amount of work that people have done
in setting up very special configurations in Conway’s life and other
cellular automata to do specific things.
So for example this is a configuration that computes twin primes.
Very, very complicated things and it does all sorts of complicated
stuff. After awhile there will be some output stream coming out here
and the locations of these gliders represent the locations of all twin
primes. [inaudible]
And thanks’ to an extraordinary algorithm called HashLife it’s possible
to run these things at an extraordinary speed. So this is already
getting to millions if iterations.
>>: [inaudible].
>> Alexander Holroyd: And yeah so that was the initial configuration.
Here is, I changed one cell. I changed the state of one cell in the
initial configuration. And what happens if you run that? Well it
starts off looking good and after awhile you realize something is
wrong. And well I have no idea what it’s doing, but it’s certainly not
computing twin primes. And probably the best description of this is
chaos. So it’s very difficult.
>>: [inaudible].
>> Alexander Holroyd: I have no idea. It’s unlikely in this particular
case. I think that, but I have no idea. So yeah, somehow the point is
that a lot is known about what you can do if you start from a very,
very specific situation and much less is known about typical initial
configurations. For example what happens if you take a uniformly
random initial configuration on Z2 and run life? I don’t think anyone
has any idea what the answer to that question is. I might come back to
that later.
Okay, there we go. I don’t want to have to go through that again.
yeah that was Conway’s life.
So,
So as it turns out a lot of the surprising and interesting things that
cellular automata can do can already be seen in one dimensional model.
So that’s what we will focus on; one dimensional cellular automata
rules with a quiescent state 0. So that just means there’s some state
so that if all cells are in state 0 then the next time they are all in
state 0 as well.
And I am particularly interested in what happens if you start from a
seed, which is an initial configuration with only finitely many nonzeros. And especially what I am really aiming for is what happens with
‘typical’, i.e. random seeds.
So let’s look at some examples. So some cellular automata rules, it
turns out, are completely predictable in a strong sense which I will
explain. So one example is a so called 1 or 3 rule. So remember it’s
in one dimension so lambda TX will denote the state of cell X at time
T. And if you want to know the state of XT+1 it depends on three
things. It depends on itself and its left and right neighbors at the
previous time step. And it’s simply the states of 0 and 1. And its 1
if the number of, if the sum of these is 1 or 3, otherwise it’s 0. And
of course another way of saying that is it’s just the sum of these
things, these three things modular of 2.
Okay, so we normally draw these things with time going down the page.
So here is a summary of the rule: if you have 1 or 3 of these three
guys are black than the one below it is black. So okay, here is a
picture of what that does starting from a single one at the top. So
does everyone understand the pictures? So time is going downwards and
the universe is one dimensional. So we started off at this time with a
single black dot and then it’s times 2, times 3, etc.
Now this picture turns out is completely understandable in a certain
sense. So it’s, it has a fractal structure, a self similar structure
and it’s possible to write down a recursive description of it where
this section here is composed of smaller sections that you have seen
earlier on and so on. No mystery about this really. What happens if
you start from a different seed?
So here is a random seed of length 12 or something like that. Well,
essentially the picture is very similar. So you still have this
fractal shape, but it’s thickened a little bit. And also the
configurations in the thickened part are predictable as it turns out.
So all these little configurations here are the same and similarly
everywhere you see it. So again this is completely predictable. It’s
something we call replication and I will come back to define things
like this more carefully later.
But essentially every seed gives a similar picture for this particular
rule. And there is a very simple reason for it in this case, because
this rule is additive. So remember it was the state of the cell is the
sum of the three above it modular 2. And a consequence of this it’s
very easy to prove. If you want to know what happens for any initial
seed.
For example the seed consisting of these three sites here being one.
All you have to do is draw the triangles, the fractal triangles that
you get starting from a single one from those three places. You super
impose them and then add the modular 2 and that gives you the answer.
So everything here is provable.
So on the other hand here is a rule that behaves completely
differently. This is rule 30. This is Wolfram’s numbering system for
elementary fundamental rules. The details don’t really matter. I mean
if you see exactly one 1 then you become a 1 and also if you see this
then you become a 1. So here’s what happens from a single seed, a
single 1.
So you might say the best description for what’s going on here is
chaos. Although kind of it’s not quite as simple as that because you
can see things are periodic along the left hand side, but certainly
there is no simple formula that anyone knows to predict what happens a
long way down. And not surprisingly if you start from some other seed
like this one then you still get apparent chaos.
And even in one dimension some of these rules are sufficiently complex
that people have proved that they are two incomplete, capable of
universal computations. So in particular rule 110 which is another one
of these 1s. Cook, Matthew Cook proved that it is too incomplete,
previously conjectured by Wolfram.
But once again this is, you know, the theorem says you can produce a
Turing machine if you set things up exactly right to begin with, that’s
very important. So it’s nothing to do with what happens for typical
initial condition. Now in some ways the most interesting types of
rules are 1s which can exhibit different types of behavior depending on
how you start them.
And this is such an example. So called the “Exactly 1" rule. It’s
simply, you know, you look at yourself and your two neighbors left and
right and if you see exactly one 1 then you become a 1. And here is
what it does from a single one. It looks kind of similar to the
previous pictures, a nice fractal. Here is what it does from another
small seed. Also we call replication so a nice fractal, although it’s
a little more complicated because the configuration here is not the
same as the one here. But still, you know, this is completely
analyzable.
Now on the other hand if you start from certain other small seeds, like
this one, you get apparent chaos in the sense that there doesn’t seem
to be any way to predict what happens in the middle. And also from,
again the same rule, from some specifically chosen seeds you can get
something else, a periodic pattern in space and time. And you can even
get mixtures of several of several of these things. Here it’s periodic
in a non-trivial wedge at the side, but apparently chaotic in the
middle.
And rather amazingly some of these sorts of things can be proved.
Gravner and Griffeath recently proved that many of these behaviors for
this particular rule I just showed you; exactly one replication
periodicity. Some of these mixtures occur for infinitely many initial
seeds, even in exponentially growing family of seeds. [inaudible]
But the conjecture here and this is sort of the point I want to get to.
The conjecture here would be that chaos is the typical things. In the
sense that if you take --. If you start from a uniformly random seed
of length L, you know there are all 2 to the L binary strings that are
equally likely, and then with high probability with probability 10 to 1
you see chaos; at least within some cone or wedge. And what does chaos
mean? Well, yeah, there is no universally accepted definition in the
subject for what chaos ought to mean.
And here is one thing you could say here for example that if you look
at local patterns, strings of length 10 and look at the densities of
them then they converge to something. And if you look at all possible
lengths they converge to a non-trivial probability measure which should
have some sort of mixing properties. So what happens over here should
be [inaudible] independent of what happens over here.
So you know you can write down more precise versions of this. The
conjecture would be that something like this typically happens. And by
the way nothing like this, nothing like this has proved for any
cellular automata that I know of, even though this sort of behavior
seems very, very common. And you might argue this isn’t quite strong
enough because somehow chaos implies it’s not predictable as well.
There is no simple formula, but in any case, even nothing like this is
proved.
Okay. So you might at least imagine that something like this situation
is a universal law, something akin to the second law of thermodynamics.
If some cellular automata rule is capable of producing chaos, what ever
that means, then chaos ought to be the norm in the sense that from long
random seeds you should get chaos with high probability. And it’s, you
know, it’s easy to postulate a mechanism for this where somehow chaos
can start from some local configuration. And once it starts it should
win. It should take over everything, chaos should take over.
So, yeah, you might guess this is what always happens, but it turns out
that’s wrong. So some cellular automata are self-organizing in the
sense that the opposite thing holds. And the first real evidence that
we know about for this is another paper by Gravner and Griffeath that
says they showed that for certain one dimensional rules some seeds give
you chaos, but apparently all long seeds give you predictable behavior,
but predictable in a non-trivial way.
And what we, our goal here is to give a rigorous version of this
phenomenon. And now I want to be careful because a lot of the words on
this slide, you know, you hear all the time with elaborate claims, and
sometimes with not much substance behind, not necessarily. And I don’t
want to fall into that same trap. So what we are proving is something
very specific about very specific models and this is just motivation.
But I just want to give you the idea that we are motivated by some kind
of deep questions, even if they are not necessarily mathematical
questions.
So this is where we are coming from. And I should also mention in a
slightly similar vein to this there is this famous or infamous positive
rate problem. So there is a result by Gacs, very, very difficult to
prove. It’s so difficult that very few people other than the author
claim to understand it. But --.
>>: Who claims to understand it?
>> Alexander Holroyd: Well, sorry Larry Gray has written this thing and
–-.
>>: [inaudible].
[laughter]
>>: [inaudible].
>> Alexander Holroyd: I don’t understand. So the claim is that there
exists a one dimensional cellular automata model so that if you run it,
but with random noise, so you take an epsilon and at each step, every
transition, you have probability. Absent of making an error and
changing to a random state then there exists a rule which nonetheless
has multiple stationary distributions. Think of it as like a plus
phase and a minus phase. So in --.
>>: It’s supposed to be running on the infinite.
>> Alexander Holroyd: Yeah, that’s right. So for two dimensional
models it’s well known that something like that can happen. For
instance for the [indiscernible] model, but in one dimension it’s
apparently true, but a very, very difficult thing.
>>: [inaudible].
>> Alexander Holroyd: Yeah, so that’s aside from it, but it’s the same
sort of thing that we are trying to get at. Can you --. Can cellular
automata or other similar systems do complex things even in the
presence of randomness? Okay, so what we prove is, and I will have to
come back and define terms here, but what we prove is that for one
dimensional cellular automata in a certain class, started from a
uniformly random seed with probability going to one you have
replication; which I will have to define. But you should think of it
as fractal like pictures, while some seeds do not give you replication.
They provably do something more complex and apparently chaos in some
cases, although we are not able to prove chaos.
Okay. So obviously for this to be a theorem I have to tell you what
each of these three things mean: certain class replication in a more
complex behavior. Okay, so I will give you one simple example of a
rule, which our theorem applies to first of all. It will have three
states 0, 1 or 2. And they will, 0 will be white, 1 will be black or
grey and 2 will be red. And your state at time T+1 is given by a
function of the three things above you; yourself and your two neighbors
at the previous time.
And the function f (a,b,c,) is if you see 1 or 3 ones in your
neighborhood than you are 1. And otherwise you look at A and C, so you
just look at your left and right neighbor at the previous time. If
exactly one of them is a 2 than you become a 2, otherwise you become a
0. So the way you can think of it is that the 1s are performing this
1or 3 rule, which was the nice predictable one or the additive one.
And the 2s try to perform Xor, just on the points not occupied by ones.
So Xor meaning they look at their left and right neighbors and take the
exclusive all of them. Okay, so that’s the rule.
So here’s an example of what it does starting from a very simple seed.
This is an apparently chaotic picture, although we can never prove
that. So the grey is the ones and they are just doing the additive 1
or 3 rule. So you can think of them as being decided on to begin with.
And then the 2s the red things are somehow seeping through, but here’s
what happens for a typical long seed. So this is a random seed of
length 30 or something like that, totally different. This is
replication, so basically you have this fractal picture that we saw at
the beginning, except thickened and a few reds around the edges. And
you can already see how we might go about proving something like this.
And here are some pictures where you don’t have replication. Your
provably have something else. Here is an example of something we call
quasireplication. I will come back to what that means. There are some
non-trivial fractal configurations of 2s. And here is another picture
of quasireplication. It turns out that this picture as well can be
completely analyzed, this specific seed. One can completely see what’s
going on here. I will come back to that, okay.
So what’s replication? So first of all lambda is the 1 or 3, the
support of the 1 or 3 rule started from a single 1. So it’s this nice
fractal triangle thing we saw before. And lambda R is the set of all
space/time points within distance R of that set. Okay, so just flatten
the fractal by R. And we say the configuration is a replicator of
thickness R and ether eter, where eter is a doubly periodic
configuration of 0s and 2s on Z2, so space and time periodic. If every
bounded component of the compliment of lambda R is filled with a
translate of the ether eter.
So here is the picture you should have in mind. You have the black
which is the nice fractal triangle and you shrink away from it by
distance R. And you get these triangular spaces. And each of those is
filled with a doubly periodic pattern eter, which we call the ether.
So in particular for example this implies, if you have such a
configuration that the density of the whole thing is just the density
of eter, because these white triangles occupy proportion one of the
area. They have density 1 in the forward light cone of the origin,
because the black is just a fractal. So it has non-trivial fractal
dimension. And also if you have such a situation it turns out it’s,
you know, if you have any replicator it turns out that it’s possible to
describe the configuration everywhere, even close to lambda.
And Even within lambda R and that’s because in these regions in between
the white triangles those have bounded width. And over width of about
R so there are only finitely configurations that you can see in those
places. And it turns out on the basis of that you can describe the
whole thing. And this is by Gravner, Griffeath and Pelfrey.
So anyway, the take away message is that if you have a replicator
everything is predictable. So, here are some pictures. This one we
already saw. This is a replicator with 0 ether. You have the gray
fractal and you have the white triangles. And in the white triangles
you just have 0s. And this is a picture of a replicator of different
ether. You have some thickened version of the fractal and each of
these voids is filled with this doubly periodic picture. This is for
another rule and that our theorem applies to.
And this is now for the same rule as the previous picture. That’s
different ether, which you can get, so a different doubly periodic
configuration. And that would be a replicator with 0 ether as well
because you are allowed to have complicated things happening on the
fractal as long as these triangles contain 0s in this case. And
according to our definition this is something which is not a replicator
because we claim --. And again this is another rule that our theorem
applies to, but this is an exceptional seed. Our definition said you
are only allowed to have a single ether. You have to see it everywhere
for the particular seed.
Okay, back to my theorem. Here’s what the claim now says. It says,
“For certain cellular automata if you take a uniformly random seed of
length L then there exists a random number RL which could be infinite
and a random ether, eter L so that you see a replicator with thickness
RL + L and ether eter L. And furthermore, so okay, if RL was infinity
than this statement is vacuous because it says the distance that you
pull away from the fractal is infinity.
But the probability that RL is infinity goes to 0 and even RL is tight.
So the distance that you move away from the fractal is basically just L
+ something very large. And the same holds even if we change some of
the 0s and the 2s, some of the 0s to 2s in the initial configuration
with the same ether and the same R. Okay, so that’s the conclusion of
them. And now I have to tell you which things it applies to.
So think about a configuration of the 1 or 3 rule. Again this is the
simple additive rule. And we will define an empty path as just a path
of 0s in the configuration that takes steps down left or down right.
And an empty diagonal path is a path of 0s where you only take diagonal
steps diagonally to the left diagonally to the right. And don’t worry
about this too much, but we also define a wide path as a path which is
empty, but you can’t move diagonally between two 1s like this.
Okay, so the definition is we have three states which we call 0, 1 or
2, 0, 1 and 2. It looks like there is a bracket missing here. We have
a range 2 rule, which means you can look at yourself or your two
neighbors or your two next newest neighbors at the previous time. And
the assumptions are if you just look at ones then they obey the one or
three rule. If you are blind to the difference between 0s and 2s then
the cellular automata obeys this simple 1 or 3 rule.
And the key thing is information about the distinction between 0s and
2s can only pass along diagonal paths. So the 1s you can think of them
as being there already, because they just come from this very simple
rule that we know about. And then anything that’s white here could be
either a 0 or 2 and how do you decide? So if you want to know whether
this is a 0 or a 2 then it’s going to depend on stuff above it, but
it’s not allowed to depend on whether that’s a 0 or a 2, because it’s
only allowed to look diagonally.
And so here whether this is a 0 or a 2 can depend on whether that’s a 0
or 2 because that’s a diagonal step and it can also depend on the fact
that there is a 1 above it. And there is no reason to prevent that,
but information can only flow along diagonal paths. Okay, so --.
>>: [inaudible].
>> Alexander Holroyd: Yes, [inaudible] in 0s in the 1 or 3 [inaudible].
Okay so I won’t write down the formal definition, but you can look at
the paper if you want to see it. So, okay our conditions as for one of
these things that is either compliant with diagonal paths or wide paths
we can do as well. This is the conclusion and that includes all the
pictures I showed you at the beginning.
So what about this provably more complex behavior? So I will be fairly
brief about this. So we call a configuration a quasireplicator if
there is some exceptional set of space time points. So that each
bounded component of its complement is filled with an ether eter. And
furthermore this exceptional set if you scale it by A to the N by some
fixed number A, which is normally 2, but not always, then it converges
to something if non-trivial [inaudible] dimensions. So it’s really a
fractal. And, yeah, so it turns out you can prove or rather Gravner,
Griffeath and Pelfrey can prove that certain specific seeds are
quasireplicators by inductive schemes.
So here is a very simple example of a quasireplicator. So the
exceptional set is the gray and the red together and you know you can
completely understand this picture because you can see what happens
here, and what happens here and there are some inductive recursive
scheme. You can prove it and it’s not a replicator on the other hand.
And rather more surprisingly examples like this you can sometimes prove
are quasireplicators as well. And it’s much less obvious what’s
happening here by the picture, but it is completely predictable.
And that’s a bigger picture of the previous one. And here is another
example of a replicator where the ether --. I am sorry,
quasireplicator where the ether is this non-trivial periodic patter,
but it has these disturbances which nevertheless only live on a thing
of non-trivial fractal dimension. So they occupy 0 densities in the
bulk, these disturbances. And, yeah so for specific cases it’s often
very, very hard to decide whether you have a quasireplicator or chaos.
So for instance here we probably think it’s chaotic, but we have no way
to tell for sure.
Right, so, okay, so that’s the last part of the theorem that for many
of these rules that the first part applies to some seeds are
quasireplicators and also some are apparently chaotic, but of course we
can’t prove that part. Okay. So, and we can say some additional
things as well. So you saw in this pink picture of the rule that I
called piggyback, which I didn’t define for you and I am not going to,
that you can have different ethers, different periodic patterns in the
triangles.
And basically if you see some ether at all, if for some ether eter it’s
possible for some seed that you get that ether and this R [inaudible]
finite. Then you see it with positive liminf as the length of the seed
goes to infinity. And in fact we can compute rigorous lower bounds on
probability of particular ethers. So for instance, for this rule we
know that at least 100 ethers have non-trivial liminfs for long seeds
and we believe there are infinitely many.
Also we can prove this is a slightly kind of different result if you
change --. If you take a seed and you change 0s to 2s, some 0s to 2s
in the initial seed than I already said it doesn’t affect the ether or
the R, but in fact it has no effect at all within some region about log
L from the forward light cone.
Okay, so what’s, oh wait I should briefly mention this. So although
perhaps the particular cellular automata rules that I am looking at may
seem a bit contrive, it turns out they are relevant to other things
people are interested in, so they have implications for certain two
dimensional rules that people are definitely interested in,
solidification rules via something called “extremal boundary dynamics”.
So I probably won’t have time to discuss that, but that was part of our
motivation.
So the key tool is one of my other favorite topics, percolation. So we
do percolation on the space time configuration of a cellular automata
and it’s this 1 or 3 cellular automata, this particularly nice one. So
here is a theorem. So if you take again the 1 or 3 rule, this additive
cellular automata rule and you start it from a uniformly random
configuration on the whole of Z.
So in other words if you take fair coin flips 0s and 1s on Z and you
run it then, well diagonal paths do not percolate. In the sense that
the probability that there is a diagonal path from time 0, so anywhere
along the top line to a particular point at time T, for instance T
down, goes to Z and decays exponentially to T.
And the same applies to wide paths as well, which I didn’t talk about.
So here is the picture. Here is the set of points reachable from
diagonal paths, reachable by diagonal paths from this interval as for
wide paths. And on the other hand if you look at empty paths, which
were the ones that can go down and diagonally in the two directions,
then they do percolate. So the probability I should have said, yeah
there is an infinity missing here. It should have said the probability
that exists is an infinite empty path from the origin is positive. And
that’s the picture.
Okay, so how do we prove this? So it’s on the face of it potentially
very tricky because this is a highly dependent percolation problem. We
are looking at the space time configuration of the evolution starting
from a random configuration. And that’s not our [indiscernible] in any
way. It’s highly dependent, because everything is a deterministic
function of what happens on the first line. So problems like this
highly dependent percolation problems in some cases are very, very
tough indeed.
So coordinate percolation, Winkler percolations are some of the words
and there are a number of impressive results here. I particularly
mention this recent result by Basu and Sly, which is on a model of this
kind. But it turns out in this case we are lucky. It’s not one of the
cases where it’s fiendishly difficult to analyze and there are some
other similarly well behaved cases that have been found as well,
including these.
So the key idea is this. So the space time configuration is highly
dependent, but on the other hand we can find certain sets within it.
Set’s of space time points where we see independent randomness. So
suppose, don’t worry about all the notation so much, let me just
explain. So suppose you start from an initial configuration in which
some set is uniformly random. Some set A, not necessarily the whole of
Z. And suppose you are interested in some set of space time points
down here.
Well because the rule itself is simply this additive cellular automata
rule the configuration that you see down here on S is a linear function
of the initial configuration; linear function modular 2. And what we
want to do is basically find cases where the matrix that tells you what
linear function it is, is upper triangular with 1s on the diagonal.
Okay, so if that’s the case than this set of sites you are looking at
down here will it be uniformly random. If you just look at that set of
sites you see IID coin flips, fair coin flips. And what that basically
involves is how do you find out what the matrix is? Well it basically
comes down to just looking at the configuration started from a single
1. That’s where all the information lies and this, you know, this
picture one can understand.
Okay, so that’s the idea. So we sort of now try and find the right
sort of percolation arguments that will adapt to the setting. So here
is the proof of no percolation for diagonal paths. So here is a
diagonal path of length T. The number of such paths is 2 to the T
starting from the origin. And I claim for any given path the
probability that it’s empty, in other words the probability that all
the sites in it are 0s is exactly at 2 to the minus T as if it were
everything IID coin flips.
And the reason is a dual assignment. For any given path, so I guess I
didn’t really define what these pink arrows are, but this is basically
how you find the upper triangular matrix. The idea is that each step
you take there is a new random site in the initial configuration that
you get to look at. That you haven’t seen before. If you are going to
the right you have to look that way and when you are going to the left
you have to look that way.
Okay, so this is true. And of course that doesn’t help because, you
know, 2 to the T times 2 to the minus T certainly isn’t going to give
me exponential decay, so I have to just squeeze just a little bit more
out of it, because this is critical. So, you know, there are lots of
ways you can try to do this, but eventually we find one that actually
works. So I claim that if you take a given diagonal path the
probability that it’s a leftmost, or the leftmost diagonal empty path
from the origin, well you can bind it by something a bit better.
So the idea is that if it’s leftmost then whenever it takes a right
left step like this then that site has to be a one, because otherwise
it wouldn’t be the leftmost path. So there is some more information if
I know this path is leftmost. And well, it turns out if I only
restrict my attention to these right left steps at odd times then there
is a dual assignment for them. So in other words if I look at the
whole set, the diagonal path and these black squares that occur at odd
times then that set is uniformly random. You see uniformly random fair
coin flips on it.
And the key factor that you need remember, you know, you have to go
back to the configuration starting from a single one and the key factor
is that down the second diagonal you see alternating 0s and 1s. So it
just works. So, basically this allows you to squeeze something extra
out of the argument and you get exponential decay.
And, yeah I will just show you the pictures, but I won’t say anything.
So for wide paths, wide paths don’t percolate and it’s a superficially
similar argument, but the details are very different. We just had to
find another construction that these dual assignments work for and then
proving that empty paths do percolate; again uses a dual assignment,
but in a different way.
Okay, so then what do we want? So now I want to deduce, so what I just
said was starting from randomness on all of Z. So now I want to deduce
percolation properties for what happens from a finite seed. So you
take a random seed on a finite interval and on the face of it again
this seems more difficult still because the space time picture I get is
infinite and yet I only have a finite number of bits of randomness to
work with. But it turns out this was somehow the surprising thing
here. We were just always able to prove exactly what we needed and
almost nothing more.
So here is a simple result first of all. If you actually start from a
random seed of length L then there is, with high probability, there is
no empty path from anywhere in the initial configuration in times 0
into this forward light cone by more than distance long L. And how do
we prove this? Now we really exploit the long-range dependence. So
here is the picture starting from a single 1. If you look at a strip
along the diagonal, a strip of width K along the diagonal then the
picture that you see is periodic in time.
And that has to be true for any cellular automata actually because
there are only 2 to the K possible configurations you can see as you go
down. But in this case it’s periodic with period or decay. And that’s
because it’s this special additive cellular automata. And all these
things, things like this, properties of this picture are easy exercises
to prove. They are easy exercises.
So what that means is that if you start from a random seed again you
have these additive properties so this picture is simply a bunch of
those superimposed on each other [inaudible]. Then again this diagonal
strip has period, the same period as before or decay, or to its width.
And furthermore if decay is less than L basically each row of it is
uniformly random. So at each time you see fair coin flips, but of
course there are a lot of correlation between different rows.
So that is basically enough, because if there was a path from up here
to somewhere in here then it would have to go through this strip, but
for every, you know, each of these is uniformly random so it’s unlikely
that you can make your way through it and down a bit. And the picture
is periodic with period about K. So if you apply a union bound then
you don’t need to look at infinitely many cases, you only need to look
at order K cases. So you have the constant K because of the period and
you have the exponential decay. So this is small.
And basically by the same sort of methods we can prove this more
elaboration application of the same ideas. Take the 1 or 3 rule, start
it from a uniformly random seed, then with high probability you have
these triangular voids of 0s. Above each one there is a strip, which
blocks diagonal paths. And by the way, everything I say for diagonal
paths is true for wide paths as well.
So, right, so there is a strip for which diagonal paths cannot get
through into the void below. And furthermore all these strips are
spatially periodic, so periodic in the horizontal direction and with
the same repeating pattern for all of them.
And this is basically where the main theorem comes
that’s true and you have one of these rules in our
each of these voids you are not influenced by what
here. All you are influenced by is this strip and
the same periodic strip in every case.
from, because if
class then within
happens outside, up
this strip is just
So all you can get is an ether, the same ether in each void. And the
way we prove that again it’s just sort of a sequence of miracles, semimiracles where you just want them somehow, because --. In the
configuration starting from a single one, above each void you have a
periodic --. If you go a distance a power of 2 up then you have
something that’s periodic with period 3 times 2 to the M. And it’s
this very specific periodic thing where you have 1, 1, 0 spaced apart
with 2 to the M0 as in between.
You know again, anything like this is an easy exercise to prove. It’s
not a mystery why it’s true, but it turns out to be the thing we need.
So that means that if you start from a random seed then you have some
thing similar. Above each void if you go up distance 2 to the M you
have something that’s periodic and the same period no matter where you
look. And every interval of length 2 to the M within this periodic
thing is uniformly random. That’s the way to say it. So then it’s
essentially as before. So this is, as I say, what’s behind the main
theorem.
So as I said, somehow just enough things work to enable us to prove
this, but many seemingly closely related questions we don’t know the
answer to. So for instance we do not understand very much about super
critical percolation started from random seeds. So remember empty
paths were in some sense super critical in the sense that if you start
with random RZ then infinite empty paths exist.
And the way we prove that is we actually prove a bit more that if you
take all the points you can reach starting from, oh let’s say the half
line origin to the left then it has a frontier. And that frontier has
drift, positive drift a quarter. But, for instance, what happens if
you start from a random seed of length L? And you look at where can
you get to by empty path starting from the left hand end, say?
So here’s an example of the picture that you get. Again there is a
frontier and the question is at time T how far over has it got? And
what we believe is that for typical seeds this gap behaves like a nontrivial power law of T. Now that makes sense because the, remember the
black thing is essentially fractal and you presumably have speed a
quarter when you are inside the fractal, because it’s kind of random.
And you have speeded one when you are in the void. That part at least
we can agree on.
But it’s a fractal most of the time you are in the void. So that’s
kind of why this power law makes sense. But we certainly can’t prove
that. And again it’s not true for every seed. So here’s a specific
seed and here in this specific case we can prove everything. If you
look at this frontier then it’s a microscopic distance from the edge.
So even though for the moment we looked at these pictures for a while
and we thought this must be essentially the same picture as this,
because when do you look at the first thousand [indiscernible] it looks
very similar, but it’s not. In this case it’s, this distance is
strictly --. This distance over T is strictly bounded between 0 and 1.
And in fact the frontier converges to a variant devil’s staircase in
the scaling limit. That’s provable for this particular case.
And we can prove that this power law holds for some very simplified
version of the model that we came up with. And probably something like
this power low is behind some cellular automata pictures that we can’t
analyze. So here’s an example for one of the rules in our class where
probably what’s happening is you have this frontier that obeys a power
law and then it causes some other things to happen just on the
frontier. So if one could prove the power law then it would prove that
this was a quasireplicator probably.
And probably something somewhat similar is happening in the chaotic
picture I showed you at the beginning. Again, this is one of the rules
that our theorem applies to. So long random seeds are nice, and
predictable and well-behaved. This one is apparently one, but
presumably there is a power law frontier here and then all sorts of
complicated things happening behind it.
That’s just a bigger picture of the same thing. And you can see there
is just a tantalizing mix of order and chaos because you sort of have
this power law repeated in lots of places as well and what’s going on
here? Well it doesn’t look exactly random. So nothing to say
rigorously here and it also completely opens our cases where the
percolation is apparently critical. And this is a particular case like
that.
So lots of problems and we don’t know how to prove that any of these
rules have infinitely many possible ethers, although we believe it to
be the case. Also very interesting question is trying to do space/time
percolation for other one dimensional cellular automata rules, not
additive ones. And there are plenty of others. For example rule 30
for which uniform measure fair coin flips is an invariant measure. So
that would be a natural place to start with some of these questions.
Yeah, so we don’t even know what happens to empty paths if you start
from random configuration on the half line and 0s on the other half.
And of course proving existence of chaotic behavior in any of these
cases is very, very difficult apparently. And, yeah, we are coming
back to were we started. Also presumably impossibly difficult, what
does Conway’s Life do from a random configuration on Z2?
And so the reason I think that’s difficult is on the one hand, you
know, you can come up with these very elaborate computers that do
interesting things, but they are not at all robust. If a little bit of
chaos runs into one then it’s the end. But on the other hand, if you
are on Z2 perhaps there are local configurations which do interesting
computations and also are very good at defending themselves against
attacks from chaos on the outside.
You know you have the whole of Z2 to play with, so. And who knows
whether those win over the chaoses. So that’s, yeah that was sort of
our motivation, but I will stop.
[clapping]
>> Yuval Peres: Questions or comments?
>>: Do you actually prove that anything is not [inaudible]?
>> Alexander Holroyd: No, well I guess strictly yes, but in a very, in
a very trivial example. The one that was a different ether on the two
sides. So it’s a mixed replicator. I mean, and this is more just a
symptom of how we define things, but, yeah.
>>: Okay, so [inaudible].
>> Alexander Holroyd: Right, right, that’s right. There is a bit more
evidence besides eyeballing the pictures than that, which is the,
although its round about evidence to say the least. The program that I
showed you right at the beginning for simulating Conway’s Life it uses
a remarkable program itself which is called Golly by the way, a highly
recommended free software.
And it uses a remarkable algorithm called HashLife, which somehow
stores configurations that it has seen before. And it’s very
interesting to see that for some configurations it does very well. If
you give it a replicator it goes very, very fast. It gets into
trillions of generations in a few seconds. For an apparently chaotic
case it’s not much faster than just computing everything as you go,
although it’s a bit faster.
And for quasireplicators it seems to be
into the millions. So there is kind of
apparently chaotic ones really are more
quasireplicators, but yeah. And I mean
could be investigated further.
kind of intermediate. It gets
some evidence maybe that the
complicated then the
I think that’s something that
>>: Do you expect any classes of rules for [inaudible] same behavior
going outside? Now you have [inaudible].
>> Alexander Holroyd: Yeah, I mean [inaudible].
>>: I mean there are just too many rules.
>> Alexander Holroyd: Yes, but --.
>>: [inaudible].
>> Alexander Holroyd: I mean yeah, I don’t think anything particularly
would change if you started looking at bigger neighborhoods, but you
would just have to be very careful how you define things. And, you
know, all we are doing is --.
>>: Is there any theorem which says that if your rule depends on a
finite neighborhood that would satisfy certain convictions then you can
encounter the following [inaudible]?
>> Alexander Holroyd: [inaudible].
>>: [inaudible].
>> Alexander Holroyd: Yes, yes that’s absolutely essential.
>>: [inaudible].
>> Alexander Holroyd: Yes, Yes.
We --.
>>: [inaudible].
>> Alexander Holroyd: Yeah, that’s right. The rules we prove things
about are very, very special ones and so one could extend the class of
rules for which the theorems work, but would still be very, very
special.
>>: [inaudible].
>> Alexander Holroyd: Yes, sometimes it does, but typically it doesn’t.
That’s what’s new really.
>> Yuval Peres: Let’s thank Alexander again.
[clapping]