>> Yuval Peres: All right. Good afternoon. Welcome everyone. So I heard
Itai give an overview of this topic in Paris a week ago, with some really
dazzling new ideas. So I asked him to show us the gory details here. Itai.
Itai Benjamini: Hi. We will comment on invariant random metrics partitions
and percolation on Cayley graphs. It's like a different slide than the ones
in Paris. So an ongoing project with Romain.
So let me first remind you the first passage percolation random lengths are
assigned to the edges of a fixed graph perturbing the graph metrics.
Place i.i.d bounded random lengths on the edges of the Euclidian grid. So
you take the Euclidian grid and you assign independently length one or two
perturbing the graph metric and rescale the length of all the edges by 1 over
N. As you know first passage percolation?
>>: No, didn't know about it.
Itai Benjamini:
Is it an interesting model?
Well, some dubious work by some people.
Anyway --
>>: [indiscernible].
Itai Benjamini: Chris did an amazing work about, yes, the most refined
understanding of geodesics in first passage percolation.
The scale metrics space almost surely Gromov merge to a deterministic and the
metric Rd just mean there's some known. L1 metric we know. L2 metric, if
there's in the tangent space there's some known. This is called Finsler
method. So it's converged to some binary space on the RD.
And the Cox and Durrett did in 1981, I don't know why because they wrote it
in two dimension and Kesten in his flow notes just wrote it in full
generality.
Now, the know, this is something that Chris is an expert, depends on the
random lengths assigned and this is only very partially understood. So the
most natural lengths to put actually are independent exponentials. And over
there, the only result known sufficiently either dimension it's not an
Euclidian ball. But simulation suggests it's not an Euclidian ball in any
dimension or any dimension and this is very much open. So the bounds on the
rate of convergence are far from the true. Though beautiful convincing
conjecture will formulate it.
So let's take the agreed, this is something -- everyone is welcome to think
about the following problem. So we take ZD. This will be ZD. Say here.
And we take say 0-0, and I mean 0-0-0 and we take N 000. And we look at the
length from here to here in the random metric, which will -- and so this is
known to be, it will be some CN. Say we put length exponential length one or
two but what are the fluctuations? What is the variance in that people
believe in two dimension the variance will be N to the two-thirds and in high
dimension, in very high dimension, people expect to be very concentrated.
Maybe the variance is sub algorithmic, maybe even tight. And I'm willing to
give you, to elect the problem, just consider, say, the distance from 00 to
NNN in high dimension and just consider even the shortest path which you just
restrict to monotone path.
>>: [indiscernible].
Itai Benjamini:
Sorry?
>>: [indiscernible].
Itai Benjamini: NNN -- NNN. So take dimension zillion and just restrict to
monotone path and look at the variance of the shortest path here, say, where
you put exponentials. So if you put just one and two you'll have 1111
because it will percolate. But take exponentials, for example, and look at
the length of the shortest path or take uniform or take something that at
least we do -- enough atom so you will not just have the minimum percolating,
understanding this seems to be long standing important problem.
So the rate of convergence very little is known on the shape. And the Chris
know something and very little is known about the convergence, and I think
there is a lot of -- I think it's not like it's a problem I recommend people
to do, to think.
I think there is room for progress. So here is a simulation of the colors.
This is if you put length one or two with equal probability and the-week
color is just the ball. I mean, this sphere.
So the variance related to this reference, which is of great interest and
related to deep think trace rhythm distribution and other things. And you
see that it's very far from an Euclidian ball. Conjecturally related.
>>: [indiscernible].
Itai Benjamini: Well, conjecturally the variance is a trace -- the minimum,
the minimum minus the expectation or the limit will converge to a trace -you take the length minus the expectation, divided by N to the two-thirds,
it's believed to converge to tracing rhythm distribution.
>>: Should preparing Euclidian or L1.
Itai Benjamini: No, no, here is just I'm asking you -- I'm asking you just
look at this, the length from 00 to N 0 in two dimension. And here the
conjecture is that after you subtract the mean and divide N by the two-thirds
it should -- N to the one-third, sorry, the standard deviation, it should be
tracing rhythm distribution.
And there are heuristics. So there are some -- specific models that it is
known. So now here is another interesting thing. It's related a little bit
to critical percolation. So recall if we do bond percolation, just Bernoulli
percolation in the plan, the critical probability is one-half.
So if you look at the picture that you see that near the boundary, the
shortest path was mostly using edges with one on them. And if you look
closer, it look a little bit like critical percolation or something of that
sort. So this suggests the following conjecture that if 10 over there goes
to infinity, so with probability one-half you take one and probability
one-half you take K, which is growing, then the shape will tend to be more
and like, it will converge to Euclidian ball.
>>: Conjecture was already claimed by [indiscernible] 15 years ago.
Itai Benjamini:
Claimed by many people.
I didn't say my conjecture.
But --
>>: [indiscernible].
Itai Benjamini: Okay. But I don't know. It's a fault conjecture. And
there are good heuristics for that because critical percolation is expected
to be rotationally invariant. But no one knows. So people know that
connectivity probability is a rotational invariant. But no one also knows
that lengths like in the scaling limit I don't know no invertible connection
links this to this. If I rotate, it will be the probability of connection
will be the same.
But whether the shortest path if you go diagonally will roughly of the same
length as going here, it is believed to be correct. But in simulation
supported but it's still very much open.
So I'm just giving a little comment here. So first passage percolation is
the perturbation of the graph method. There are plenty random triangulation
for which the scaling limit is nondeterministic.
So EG, the uniform infinite planar trangulation a limit of uniform measure on
all known isomorphic trangulation with this fear within triangles. This is
just a side comment that sometimes one can get I think in some sense more
exotic and interesting object than the random geometry first passage
percolation produce. So I think this is more inspiring object.
But we want to do this talk is just reviewing invariant random structure on
Cayley graph. So we stick to first passage percolation. So the key to
establishing the deterministic scaling limit is the sub additive of Ergodic
theorem that roughly tell you that the distance from here to here, divide by
N, converges.
So this allows to extend first passage percolation to invariant random
metrics on ZD and starting with Boivine and Derrienic instead of putting IID
random variable you can just put any auto morphism invariant on lengths and
you'll get the scaling limit.
And my first proposal will be as I said I will just -- these are three
specific examples of measure group theory. So the first is first passage
percolation. So what we are suggesting is to do first passage percolation on
Cayley graph. So assign random bounded lengths to the edges of Cayley graph
in a group invariant way, in particular IID fashion. Initial steps are FPP
was studied on trees. And more recently on three times Z and some other
graphs including hyperbolic lattices. Very, very little work was studied
beyond ZT.
So as the first step, as the first step to understand first passage
percolation, on other Cayley graph, the most natural thing to do is to
consider the important groups finitely generated new important group. Over
there it's also a geometry if you take a Cayley graph of the new important
group and you scale it, you will get a sub [indiscernible] metric and you'll
get the finite dimension on manifold.
So this is a place where the geometry is scaling invariant property so you
expect that it will be analog of ZD. You will get the same picture, and
indeed with Romain Tessera we verify that fact which is there are some tweaks
involved, but it certainly philosophically it's the same [indiscernible]
property of the same mechanism. So first PP on vertex transgraphic of
polynomial graph, almost Gromov converge after the scaling to a new important
connected link group..
So if you take -- this was observed by Gromov many years ago. If you take a
polynomial vertex transitive graph of polynomial growth and rescale the
length it will converge to a new important connected group. And you need to
work a little bit hard to show after putting the random lengths the
perturbation are not changing the picture.
I'll skip that. And then I will go to your homework. So but for groups of
super polynomial growth, what paths of the FPP metrics scaled to the
deterministic metric space. So on the trees, you, as I say Yuval and others
studied first passage percolation on go, if you do it consider first passage
percolation on the lamplighter group, should I -- who doesn't know what is
the lamplighter? Okay. So well it's a vertex transitive graph of
exponential growth, but I can tell you very quickly a description of it.
So the vertices are just maps for, map from Z to two states, which is each
vertices consists of a configuration of length on Z with finitely many lamps
and in a number in Z. So it's a per.
The vertices is just you have some N in Z and some configuration of a
function from Z to 01 with only finitely many elements here getting one. So
you have the configuration of lamps consists of finite lamps that are on and
all the rest are off. And a position of the lamp lighter. So you have to -this is Z. You have all the lamps are off. You have some finite
configuration of lamps that are on. And then only 0s and somewhere you have
the position of the lamp lighter. So these are the vertices, and the edges
is the lamp lighter can either move one side or change the position at the
lamp over here. So with this graph has exponential growth as one can see.
It's amenable. I don't know if you know what is amenable. Never mind. But
nice, okay, but a nice fact about it is that it contains geodesic sub tree
which is almost like the binary tree. Each time think of either the
lamplighter either moving one step to the right or turning the light on.
So it starts from zero can either move to the right or turn and move to the
right. So all these possibilities give you the sub Bernattchi [phonetic]
tree which is the growth rate of the Bernattchi number and embed geodesic.
So you might think if you do first passage percolation on this tree it will
be like first passage percolation on the binary tree. But unlike the binary
tree, you have also loops that you can go around to connect to that place.
So understanding first passage percolation on the lamplighter, or other
lattices seems interesting because we don't know maybe still the metric is
deterministic.
So I mean if you take two realizations, on the regular tree if you take two
realizations you would have raised that the ratio will not go to 1 by large
deviation. But it's not clear what is going on the lamplighter. So I just
want to indicate that this is a very natural process, first passage
percolation on polynomial setup the field we should be partially analog to
ZD. But on trees we have some very nice understanding. But it will be nice
like was done for random walk and all that to try to consider first passage
percolation on the other graph.
Here inside you have copies of the Eisenberg group sitting inside.
>>: Second question, when you have positive inference [indiscernible].
Itai Benjamini:
vertices.
But this is not only random walk.
Some sequence of
>>: The second question for random walk.
Itai Benjamini:
Right.
>>: And then so when the walk has positive speed, it would follow that it
follows from several [indiscernible] in the perturbation.
Itai Benjamini:
This should be one, uh-huh.
And the other thing, too, we have this conjecture that in high dimension the
variance of ZD in high dimension the variance of first passage percolation
should be very, very small. So try to do the variance of first passage
percolation point to point. Point to point in the lamp lighter. Maybe the
fact that it's infinite dimension or something will allow to get very good
bound point to point.
>>: [indiscernible] the tree the variance is linear.
Itai Benjamini: Yes, but I suspect that -- I suspect here that it will be
very tight. On 3 times Z we know in the Z direction the variance is at most
log N. This is recent result with Pascalme.
But anyway, okay so now so we discussed -- I have one side question. So to
wrap up invariant random metrics on Cayley graph, I have a side question
which it's purely geometrical question which I'm passionate about. So in
metrics space is C roughly transitive. So metrics a graph transitive or a
metrics basis is transitive if at any two points there's a isometric of the
space mapping X to Y for every two points. Then the graph is called vertex
transitive or geometric space homogenous or symmetric. So a metrics based is
C roughly transitive if for every pair of points XY there is a C quasi
isometry sending X to Y. Quasi isometry is isometry up to some fixed
multiplicative constant C and other constant.
So what we say is the space is roughly transitive if there is some numbers
seven. For every two point in space there is seven quasi isometry mapping X
to Y. And the first passage percolation metrics on Cayley graph rough
transitive when the sign lengths are bounded, of course. If the sign banks
are bounded, then it is quasi iso metric to the original Cayley graph and
therefore in the original Cayley graph are transitive. But the question is
there an infinite C roughly transitive graph with C finite which is not quasi
iso metric to homogenous space. So the question is whether this is really a
genuinely larger family or the only way to get roughly transitive spaces is
to perturb spaces which are all of the transitive.
Answers? Okay. So we are done. We are wrapping up invariant random metrics
now moving to invariant random partitions. Partition and infinite graph to
infinitely many connected infinite subgraphs such that each path may only be
finite many others paths.
So by partition, a graph, what I mean specifically here is if I have a graph
think about Z 2. I want to partition Z 2 to infinitely many subgraphs. Each
subgraph is connected and infinite, and each one of them neighbors only
finitely many others.
This is what I called partition. Now, what I'm requiring, which Cayley graph
admits invariant random partitions? So what is invariant random partition?
It's a distribution on partition, on partitions and the distribution is
invariant with respect to auto morphism of the graph.
So give me an example of infinite Cayley graph that do not admit invariant
random partition just that we feel that we are on the same grounds? So I
want infinite Cayley graph that cannot be partitioned to infinitely many
connected infinite graphs. Even ignore now the invariant. Z. Z. Z you
cannot partition to infinitely many connected subgraphs. But the question -so, for example, the tree can be partitioned to infinitely many if I take
array, move the array and then just each component goes out from that ray, I
just declare it as a component, then for tree I can do this. But can I do
this, put position on distribution that are with respect to auto morphisms of
the tree. So variants of this question can further require that the paths
are indistinguishable or removing the finite number.
So the natural edition, this is like geodecity requirement which is natural
but I just don't want to go there. So we have one example of a graph
which -- one example of infinite Cayley graph that admits invariant random
partition is Z 2. Just lines. Lines or if you also want either these lines
or these lines, this gives you a distribution which is invariant with respect
to auto morphism of Z 2. But it turns out that, okay, there is this theorem.
So Cayley graph of the group with positive first -- so a surface group or a
Cayley graph that, a Cayley graph that emits harmonic function with finite
additional energy do not permit such a partition. As I was talking to Lewis
in Paris, also higher length lattices do not also emit.
So it seems a lot of restriction to admit partition. Lattices in H3 do admit
partition, and this is highly nontrivial. But let's try to do together you
Yuval thought I should prove something. So let's do together the exercises.
Show the regular tree -- I showed you regular tree admit partition to
infinitely many connected infinite graph that neighbors only finitely many
others.
But I claim that you cannot do it -- you cannot put a measure on such
partition which is invariant.
The reason it follows is if you have in the tree, if you have connected
infinite paths that neighbors -- it neighbors only finitely many others,
other paths, then the intersection with the other parts should be finite set.
If in the tree the paths are connected and they meet, they can meet only at
finite place. And if the paths meet finitely many others, then yet finitely
many connection to other paths, and then you use the miss transport principle
you define a function that if there was such an invariant thing, you will
move the miss from every point is distributing the miss between all the
vertices that are neighbors of other paths.
Since you will have infinite miss coming to finite set, this is impossible.
So this gives you at least [indiscernible] between you and a quick argument
why it does not exist. So apparently this invariant random partition is
tricky.
So try to think with bare hands of an argument why you cannot put the measure
on partition of the tree, which is invariant with respect to auto morphism of
the tree.
But --
>>: The tree but what's the argument with the -Itai Benjamini: For surface group, you just move, so you will have some
chunk, if it's a surface and you will just push the mess to the boundary. So
for surface group it's topological thing. You push the mess to the boundary
and it's similar mess transport argument that it's easier to do for the tree.
>>: But if you have this positive -Itai Benjamini: So essentially it's very much, you just abstract this thing.
I don't want to go there. I'm not an expert. But essentially surface group
is not far away from this.
So in mine I conjecture that lamplighter over Z, this graph, do not admit
invariant random partition.
But we couldn't do it. And what about -- so Lewis Born thinks he might show
using measurable normal subgroup theorem of Margulis, a thing beyond my ->>: Beyond your pay grade?
Pay grade.
Beyond your pay grade.
Itai Benjamini: Yes. Ah. Okay. Yes. [laughter] yes. So yes exactly.
some are good. So maybe both cannot, will not admit invariant random
partitions.
So
So here is the question. Given an invariant random partition, when is it
possible to further partition each path infinitely many to infinitely many
connected infinite subgraphs? So, for example, if we take Z 3, if we take Z
3, we can partition it to plans and each plan then further partitions to
lines. So you can get a chain of refinement of partitions.
Now, we don't know in Z 2 there is an old theorem of Burton and Keen that
tells you that you can just do it once. With these lines, you cannot further
partition. But in Z 3 it should be obvious that you can do it at most twice.
Now, in Paris, Lewis Born constructed for me a partition which is not just
plans and lines but something which is different. So but I don't believe you
can in Z 3 do more than twice and we don't know how to show that you cannot
do infinitely many.
And in T times Z -- so in T times Z you think either just take each path will
be a copy of T or each path would be a copy of Z. This is the only two ways
I know how to partition. But we don't know to show that maybe there are
others -- because what we really think, the number of possible iteration
might be some invariant or some dimension. But it seems hard. Well, I mean
it's not -- it's not too many people thought about it but it's not hard -it's not telling you a problem exposed to the world just last week.
>>: There were 100 people.
Itai Benjamini: Right. So now I did invariant metric. Invariant partition,
and now we go to the classic invariant percolation. So in Paris we had the
invariant majority processes to entertain you why I'll change this section, I
change to invariant percolation which is a classical, by now classical, it's
from the previous millennium.
So invariant percolation, including percolation is a nontopic. Therefore we
will only remark about one random walk question in this context.
So equaliative properties of random walk of the property of some interest, in
particular stability of such properties under perturbation and mono tonicity
with respect to inclusion of graphs. So, for example, zero speed of random
walk. Say we know random walk in our group is zero speed and now we consider
random walk on clusters of invariant variation of the group. Can the speed
suddenly turn to be positive? So almost two intersection of two independent
random walk path. Could it be that on Z 4, for example, you have invariant
percolation for which on it random walk path will not intersect almost truly.
No. So now this is considered collision of two independent random walks. So
I think okay I have this slide to explain this one. Okay. We will get to so
collision of two independent random walks is I let two random walks walk, and
unlike the classical question of whether the paths intersect I'm asking
whether the two paths actually reaches infinitely often. This is called
collision. And so I will comment a little bit about these three. So assume
simple random walk on Cayley graph G has zero speed.
That's simple random walk on any invariant subgraph of G has zero speed.
This is a special case of an open conjecture from many years ago.
And so we suspect that it will also have zero speed.
>>: [indiscernible] for this question?
Itai Benjamini:
No.
>>: [indiscernible].
Itai Benjamini: It's a general question. I can't come up -- now,
specifically with an example. But I will -- okay. So one can further
consider monotonicity of the speed exponent. So I think this was not
considered. But you have a notion of speed exponent. So on lamplighters
with lamps with dimmers, if instead of the lamps being zero and one, they can
have an integer value, then you can take an exercise to make sure that speed
is equal to the three-quarters.
>>: [indiscernible].
Itai Benjamini: Integer. Because after N steps, the lamplighter roughly
covers N to the one-half. And in each place it did N to the one-half steps.
So the dimmer go to high to the one quarter. So the distance to just, to go
one coordinate and to set the lights to zero, it's N to the three-fourths.
So here you have speed exponent like that. Now consider some Bernoulli
function on the lamplighter shows always the speed cannot exceed N to the
three-quarters. So is there monotonicity of speed with respect to invariant
percolation.
Now, mind you, Yuval proved the two independent random walks on the comb.
This is the comb. You take Z and you put -- so just Z and just these lines.
These are not allowed. So this graph is not a Cayley graph. So these guys
prove the two independent random walks on the comb will collide only finite
many times almost truly. So what's nice thing about this is it's recurrent.
It's a subgraph of Z 2. Therefore it's a recurrent graph. So every vertex
will be visited, and if it was a group, then recurrence is equivalent to
collision.
But here it's I find it surprising that two random walks will almost truly
meet only finitely many times. So for symmetric random walk on Cayley graph
collision and recurrence of equivalent, we conjecture that it is equivalent
also for invariant subgraphs.
So more generally there's the notion of unimodular random graphs which are
stationary with respect to delay the rooted graph and if you move the rooted
graph using a delayed random walk you see a stationary process on this space
of graph.
But ->>: Invariant subgraph [indiscernible] if there currently would be two
dimensional?
Itai Benjamini:
Sorry?
No you can have invariant.
>>: The model can be.
Itai Benjamini:
The unimodular can be anything.
Invariant.
>>: Invariant subgraph is maybe -Itai Benjamini: So, yeah, okay. So the conjecture was I didn't want in the
talk to introduce the general conjecture, but there's a notion okay I have a
comment. Natural setting to look at invariant subgraph in the questions
above regarding partitions and random walks, is set of unimodular graph and
graph limits.
So I conjecture that if you have this more exotic situation, then collision
should be equivalent to recurrence. So way beyond the situation of vertex
transitive graphs.
>>: So the invariant [indiscernible] seems too restrictive because I think
they have to be two dimensional.
Itai Benjamini: Ah, well to be recurrent -- they can be kind of trees. You
can have uniform spanning tree in some general graph. So you will get -- you
can get all kinds. But they might have to be recurrent.
>>: [indiscernible].
Itai Benjamini: Yes. Now, since the slide to a confidence random walks on
group, I added to please the people wanted to see some random walk. I added
a little section not on invariant random stuff but just about random walks on
groups.
So we end with a suggestion to consider harmonic measure on Cayley graph.
Given a set S in a graph start random walk from some fixed V until it hits S.
So the probability the random walk hits S first at U is a probability measure
on the boundary of S and is called the harmonic measure. Okay. So if I
have -- this is my graph. I have some set. I have some starting point. And
I do random walk till I hit the boundary of the set. And this gives me a
distribution where I hit the set first, and this is called, this is called
harmonic measure. Harmonic measure is well studied topic in Euclidian space.
We have expert here. And with remarkable result and still fundamental open
problems. So it's a comment let G and B unbounded sequence of finite
connected vertex transitive graph with bounded degree such as they have large
diameter. So the size of the graph is little of the diameter square or
diameter to the 7.
Then for any set N of in GN harmonic measure is supported in a set of size
little O of G. So just these assertions tell you again liking the philosophy
that first passage percolation you expect the polynomial well to be like ZD.
This assertion I have to tell you that also for harmonic measure, the
phenomenon of the Euclidian space that harmonic measure will be supported on
a little O part of space lets ignore hopefully sharp conjectures, about how
big this can be. You have the same phenomenon in. And here are two
observations with real on the way to understand harmonic measure beyond
polynomial growth. So for sets of at most half the size on expander, when
averaging over all starting vertices harmonic measure is supported on a set
proportional to the size of the set. So unlike Euclidian space, for every
set that you take, you cannot find a little O part of the set that will
support say half of harmonic measure.
So you have strong converse. But there is a interesting phenomenon if you
are not in the polynomial case, not in the expander case but you are in the
intermediate world, then you have some hybrid behavior. On the lamplighter
over the cycle, so instead of having here the line, you have just a cycle,
size N, there is a set of size G over 4 for which harmonic measure is set on
the size proportional to the size of set because it was impossible here in
the polynomial where you have the graph is polynomially small by the
diameter.
And so on the one hand you have behavior like in expander ->>: [indiscernible].
Itai Benjamini: For every set inside of GL the -- GN. Little of GN. Right.
Yeah. Yeah. Little of GN. Yeah. Yeah. So on the one hand you have a
hybrid behavior. You have a set that behaves like an expander, but on the
other hand it's easy to take for the set, to find set for which the harmonic
measure is supported on small path.
We still do not understand qualitatively harmonic measure in the regime
between polynomial and expanders to that. Thank you.
[applause]
Yes?
>>: So you said that three times Z you can prove the Z direction.
Itai Benjamini: The variance of first passage percolation we believe it's
bounded. We have good heuristic that it's bounded on the 3 regular 3 times Z
and we can actually show the variance. We can show if you divide it by log N
it's tight.
>>: I see.
So that does seem like variance, but it's close.
Itai Benjamini:
It's close.
>>: How do you show that?
Itai Benjamini: Well, you use the trick of decking and the hosts somehow you
break space and you do some recursive argument, and you also know that the
path that go up will not go more than log away. So it's a one-page proof in
the archive from a couple of months ago, which is sort of the best motivation
we get so far or with whatever in high dimension it should be very
concentrated.
But of course it doesn't give any of that because it's a tree.
>> Yuval Peres: Any other questions or comments? So Thai is here for
another -Itai Benjamini:
>> Yuval Peres:
[applause]
Yes, the end of this week and next week.
Let's thank him again.