I just want to write some version of it.

Right.

Well thank you so much for
the great pleasure to be here.

The topic of this talk is basically
torture and I want to start with too much.

More abstract model but will follow so the
first one is called it said to my Greek.

Are my letters begin Are they OK OK So

this is basically random to talk so
we have and current.

And each card has a weight card I.

Has weight.

W.I. so that means W.

eyes are positive in this case and
very hard to one.

And we pick a card
according to its weight.

So pick a card I with probability.

W.
I and move it to the top.

So this is one of the by
the most stark famous and

basic models in car chuffing And
of course this is not

a good way to shuffle a deck of cards
it takes forever but as most of you

know that much better than me this comes
from sorting algorithm and because

of that we know everything there is about
this model like we know the eigenvalues of

the transition matrix we know all sorts of
stuff I will only mention the start that

are related to what is about follow so
when we have the uniform weights.

Are So for all I then we know
the mixing time very well of this and

that account is.

Proof that thanks and Logan steps to mix

the black our I mean it is one of the
first examples that we can cut both and.

I'm not going to write a more precisely
what that cut the means right now it will

come back later or what mixing it and

then the other thing that I want to
emphasize right now is a result.

Which says that we have an optimal
strong stationary time for

this model I'm not going to define
strong position times right now it will

come later hopefully
strong stationary times.

So these are two nice features that
will carve out later on in our model.

Just what retirees make and stock are on
Thursday this model was really he didn't

between becoming random to random which
is what Meghan that will be talking about

on Thursday now I want to talk about
another court shuffle as well which is

people actually shuffle a deck of
cards which is riffle shuffle zz

or in this case inverse riffle struggles.

So what is that you know
we have a deck of cards

how do you say shuffle a deck of cards you
cut it into two piles and you refill and

you do it again and again and again and so
this is what this model is going to be so

we have let's say again and cards.

And we flip a fair coin for
each one of them.

And it has.

Marked the card with zero.

Tails market with.

One then we move the zero
cards from the tag and

move them to the top keeping
the relative order fixed so.

Move the busy row.

Of cards to the top keeping
the relative order.

And this is some version of you know if
you think about it subversion of random to

top but now you somehow move
more cards to the top and

this is the famous off of far corners.

Coonass that it takes three hundred US.

Lock to one of.

Four for the mix of time and
it has cut off this is from ninety to.

About then I also want to emphasize.

That this is that for the mixing time for
down for the total variation of X.

and time if we look at separation time of
the definition will follow soon so for

separation.

It will take too long too and.

Steps and this this result is a
combination of something that obvious and

diakonos did in this paper and.

A lower bound due to
a soft diet communities.

And some direction.

Where you more recently from
two thousand and eight OK And

I want to talk a little bit about this
this gap between the constants a little

bit so let's keep in mind this to
mortals because that will motivate.

The hydroplane arrangement walk so
what is a hyperplane arrangement so.

Hyperplane arrangement work
is a minor collection.

Of hyper planes in Oregon.

At least for
our case it is a finite collection and

it's going to torics they do
whatever they want with it so.

OK I can only draw such a picture but if
we're going there three you must imagine

planes coming out from from the board and
so

this is our collection and looking in the
picture we can see all sorts of geometric

features so
we see these regions for example.

Which we are going to
be calling chambers and

I will define things are more
carefully in a second.

Are we also see you know points for
frind So

all sorts of geometric spots so
all of the lower dimensional features.

Will be called Faces.

And the set of chambers.

Which I will be told by C.

will be.

Configuration space of the market chain
and I'll be using the faces to define

the marketing OK So let's start to find
those things a wee bit more carefully now.

Two are defined the chambers so.

Let's look at the picture again and if
we just focus on the specific hyperplane

let's call it our first hyperplane we see
that it cuts the space into two different

half spaces Let's call
this one of the class.

And this one of the minus.

OK then we go to the next one this is
the plus area this is the minus area

then we go to this guy I'm not saying
that every above is the plaster and

the area below is minus and
what we get is vectors on plus and

minuses so the chambers.

So here you can tell.

The size of the plane arrangement so
we thought of factors we've coordinates

as many as the number of
five per planes then.

We have signed class and minuses like
this and for the faces all we do.

Is that if we live in a specific
type of plane like this one

then the first chord that will be zero
Actually this is the last chord and

will be just a zero and
this just indicates that this

geometric which are lies in
the third hyperplane So faces.

The inside this doesn't make
sense to set up our so for

I know we have a product because
our Summit Group structure here so.

The faces act in the chambers
on the following way.

So if we have a face of.

Chamber C.

we want to see what the product is going
to do that all we need to do is specified

each co-ordinate of the chamber so
if age is hyperplane in our collection

then we define age cornet of F C.

To be whatever abscess if
that's a nonzero coordinate so

this is equal to the eight coordinate.

Of.

If not zero otherwise we do whatever C.

sets if we get to zero we do.

H.
coordinate.

Prophecy.

What is this product let's
look at it geometrically so

assume we are at this chamber and
we pick So this is our seat and

this is a our earth and we want to
multiply them what this multiplication

does is it moves you to another chamber
that is adjacent to this phase so

either this one or this one and closest
to the chamber that you started from so

the product will result here
that's what this all products.

And I just noticed that
this is also product coming

face is I mean you could do exactly
the same thing if sees a face and OK you

might still get some zero coordinates but
that's OK because faces accordance.

That's why we have a set like of
this this product is associative and

that's why we have some sort of group or
structure.

Arm the mark of chain
now is the following.

So we assign weights.

To the faces.

And then we start say let's
say we start at sea not.

A specific chamber we pick a face F.

with probability the face that's
called the W.F. and we move to F.

time c.

This is the mark of chain are the Gen Ys
is this car truffles or it also

generalizes the lazy walk on the hyper if
you put your types of walks and so on.

OK so.

Just from a quick notation and

are you a bit about the history
of the model in the second so.

So we start at C.

not and we start multiplying by faces.

So let's call F T R The face picked.

Base picked a time T.

and this is our configuration a time T..

Just For
Today Show on then you know we will have

things like the transition matrix will
just be R.P. The probability of going.

From C.
to D.

in one stop and the question has a mix
of type of question so all I know that

usually people work with total variation
distance the so let's define that so.

If we start from zero C.

not then this is just the L.

one distance divided by health.

I haven't even told you what
the stationary measure is right now and

I'm trying to avoid that
because it's a nightmare.

And so I want to tease the marks.

Over see not.

And but I will be playing with
separation a lot today and

basically I'm kind of giving lower
bones for this model so our.

Separation distance is
the maximum would be.

One minus so

just think case people are not familiar
with the separation distance but watch.

What here this is again like
the probability of starting at C.

not and being a D.

of to after T.

steps you are missing a team.

And that makes the time pressure bomb so
is it can give me an absolute on.

What is the minimum T..

Well.

What is the minimum piece
of that this thing the D.

T.
is less than absolute one or you could

have seen her case which is the maximum
for start of starting to give a shit OK.

So.

OK I was not the one to introduce
this model this came on from bigger.

Rockmore.

This one thousand nine.

That transition matrix
are noticed by the way that

mostly this is a family of
nonreversible month mark of chains.

I mean apart from some hyper tube type
of walks mainly we get not averse

stuff therefore we cannot use eigenvalues
to say anything about mixing.

So far at least.

Then ten Brown and corn is.

Found the spectrum on a different way.

And they characterize
the stationary measure and

they are condition of the weight so that
this walk is a periodic and irreducible.

And then the first time that there
is some mixing discussion here is

in an generalization of Brown who
lifted all this to some family of

family groups and basically he says
there is a monotone coupling here

what does the momentum coupling we're
going to be using it at the monitor

coupling is so t's the first time.

That this product over
here is a chain but.

What does this mean if this is a chamber
then we don't really care what

the starting configuration is if we keep
on doing the same moves if we have to

configurations and we keep on applying
the same faces on our configurations

then we're going to end up going to couple
at the end no matter where we start from

OK so that there is a lot of money to the
city in this type of model just like there

is money to me city on the tube if you
prefer to think about it this way.

They sell This is the first time that
we have some mixing answers OK And

what is the problem with a cup
when someone should say we're done

with upper bounds the problem is this.

You know three halves and this two or

four the hypercube we know that
the monotone coupling on it's own gives

a little gain while the sharp mixing time
is one half and will dance so this is not.

The sharp Unser.

For people who care
about cut off at least.

Yes.

It's a stopping time.

No is what you run grandma
got Markov chain so

you pick one face them another and
so on when you

multiply the faces that you have picked
all this time if that's the chamber.

And in particular like for this thing
this corresponds to having picked up and

minus one cards because if you pick up
one card and you move it to the top

you have no idea what the top card is
a second time that you bring a new card to

the top the first two cards or ring any
possible order to look at that they're

equally likely to be small big or big
small and so on so once you have touch and

minus one cards you're shuffled
that's basically the analogy here.

So the.

The thing is that OK So but the recently
some upper bound discussion here so

this is by the way from two
thousand then OK Just some

more literature that a fantasy of
the synthetic corn is once more.

Characterize a spectrum
with different techniques.

And then finally for
the first time John Pike.

Tried to say something about lower bounds
so he found the eigen values of this

when I get those I can functions
he discussed some I get functions.

And he you fuck you for some more
intuitive way to think about the spectrum

like a you know as a probabilistic
sometimes the representation theory

arguments are too difficult to grasp or
understand but

he basically said look this walks are lump
a ball so we can project into easier

type of chains find eigenfunctions and
I going to I was there and

with them back somehow so you he really
made things a bit more clear what

is the issue with his eigenfunctions what
he tried to do some Wilson's lemma For

example he doesn't get stuck on source and
he cannot do Wilson's lemma for.

For such a big generality go for Wilson's
lemma we need to bind the violence

the various can be a massive this
space is changed too much so

it was a good approach but
in the first approach for lower bounds but

he didn't give anything so
the theorem that I want to write

now is the following and.

So the first part at least is the one but

I will emphasize the most which
is that the separation distance.

Is bigger or equal than the tails of
this this strong stationary time.

Time.

That is given that A is central
If A is central this is to

what essential meaning it means that all
hyper planes pass through the same point.

And B.

this is our this is not what I want to
emphasize today I want to emphasize

this one but under some nice symmetry
conditions that are not too restrictive

in the sense that all the examples like
under it there are symmetry conditions

so under the nice symmetry
conditions we have equality.

So under some nicer conditions this
is a strong stationary time and

basically this theorem says that it is an
optimal strong stationary time just as in

the case of still for the thetan library.

OK so I was really trying to get
some structure on surface and

in this case I can have them.

Now.

I want to actually prove Part eight
because I think the reason nice idea and

this idea if I have time at
the end I want to apply for

global dynamics on a modern or
monitoring system does like using model

which is something that
you guys care about.

So and reprove a theorem of being
in Paris so the proof here.

So C.N.D. are just itching bursts to
configurations I want to understand.

The probability of being at sea.

Starting at C..

Being at D.

after T.
times given that T.

has happened I want to understand
what this thing is OK What is T.

T.
means that the faces picked are a chamber

I told you earlier that this means that

we that we don't care
about the starting point.

Like the probability of being a D.

given that week of separated
that we have really.

That given that we're
multiplying by Chambers does not

depend on the starting configuration so
this is a function.

Of D.

So this does not.

Depend on c.

OK And what is interesting about
eighty while it's a measure No

it's not a car no it's something into one.

OK so we're going to pick a D..

So that they say of D..

Is less or equal to the station measure so
I have no idea what the stationary measure

is but given that there are both measures
there should be are a chamber D.

that satisfies that this is
last week when this OK Does it

make sense what if this does not depend on
the starting configuration because that's

basically the main idea one can do it with
monotonicity two but I think that this you

know what it means that would be
an overkill so what is this our A D.

Why do we care about it well so
we picked D.

C.

is going to be the bar
which is the chamber

the difference the first from.

D.

in all coordinates Why is this possible

it's possible because a central So
you know if this guy is D.

then this guy B.

C.

they differ on all coordinates so

that's why we have this
assumption that there are.

Lingering two in central OK And the point
is not now we're going to do some coupon

collecting So we look out
the probability of starting at C.

and being at the after T.

steps and
we say this can only happen if T.

has happened.

So if we do base the extra term is
the zero that's all I'm saying OK And

therefore we can plug this
into separation distance.

Which is bigger equal number one minus.

This probability.

It.

Is and what we have done
is that we have chosen D.

so that this guy is less than
the stationary measure so

this quantity here is less than
the stationary measure times are P.

of T.

So this is bigger or
equal than one minus P.

of D..

But of course.

It's the probability
that hasn't happened yet.

That's happened divided by the station.

So the fact that we chose this
to be less or equal than this

gives us this inequality and this is
really equal to the tales of teeth.

So we started with separation
distance we just plug it in C.

and D.
in the form of law we said that look this

guy is less are equal.

Than the stationary measure
times the probability that T.

has happened and this gives us the tails.

Even if there's looks confusing I'm
about not to do it once more for

global dynamics and using So it will be
a little bit more intuitive there and

maybe more interesting to you OK So.

It will raise this line.

Yes.

Yes and.

Yes Yes Yes And this is what I meant by
thing with three conditions by the way so

if you have some group Action Arctic
on the R.N. preserving the hyperplane

arrangement and strands it on
the chambers then the uniform

of the space in a measure is uniform and
I didn't I just didn't want to write it to

because I'm not going to talk about
the upper bound in the songs they seem to.

Get Yeah.

OK so now after this I'm gonna want to go
into how much time do I have by the way.

OK good so perfect.

So well the last maybe even five minutes.

I want to apply this idea for
global dynamics.

With.

Full reasoning or any mono tone.

System that you want so
we know this are thier own by.

Being in pairs.

Which was I think.

Two thousand and eleven although I
do like the archive version better

from two thousand and thirteen you
have a nice sort of proof there so

if you don't mind I'm going
to write that OK Thank you.

So what it says is that.

I'm glad that we call of
the double variation distance

odd definition here too.

That.

Well they apply excellent
equals to one four So.

They say that if we take one half and

Morgan steps OK let me just say yes so
we gotta grab the G.

and the vertex set is and we do cooperate
dynamics there so let's say we plus and

minuses on the vertex
just just presentation so

they say that if you take one half and
Logan stops minus

you have lost a very specific window
I want to write it down we log.

Log and then the total duration
distance is bigger than one force

if I remember it correctly but
I want to write.

So I want to improve it
you think this idea.

But also just say that also getting
a bone from separation distance so

just saying that in this paper that I.

Submitted last year.

I have that for separation distance
on any graph any temperature.

And Logan steps minus C.

N..

Separation distance bigger than
one minus eat the minus C..

OK And.

You can retrieve the total
variation distance a ball because

we are dynamics in.

Reversible mark of change and there is a
specific inequality that I will write soon

that links separation distance and.

OK so the proof goes goes as follows so.

We're going to start from
the configuration which is all classes

of thirty C's.

Classes OK And we want to see what
is the probability of ending up at.

The state.

Where all Berkeley sees.

Of my.

OK so.

Probability that say X T S
are configuration at time T.

we start at X.

What is the probability that we are at Y.

after death so let's call T..

To be the stopping time the first time

that all coordinates all vertices have
been picked first time all vertices.

Have been picked.

OK it's the same idea basically and
this says now that

there is no way that I will start with all
classes and end up with all minuses unless

I have touched all thirty cities right
either Plus I have updated all of them so.

Why give them that teaches happened
times appropriately that T.

has happened the extra term
the probability of this given that T.

hasn't happened is so the extra time for
base is just zero OK and

all we get from person Winkler for free.

This is actually less than
the stationary measure of the all minus

distribution so
this is from two thousand and thirteen.

This is due to the monetary City this is
due to the censoring that there is in

the easing model and I mean it's not
the whole essence of the paper but it is

an operation of one of the lemma thought
that one of the numbers that they have.

OK And then again would plug it
into separation distance now.

That the monotone part that's
why it won't work for parts.

Unfortunately although I think it's
a nice idea you never know if someone

can manage to prove Well it won't
be old processor no minuses but

if someone manages to say
if I start with all red and

I can find a bad configuration that
will require touching many Bertie's and

they prove this inequality then yes
we have pots but it's not easy.

I mean starting from All right I'm
going to all blue still hold but

does this inequality hold no most likely
no but I mean if you had to I was thinking

that if you had many call Earth and you
started from all reds if you wanted to and

up with each vertex having a different
color then something like that maybe

works put Still it's for many many colors
and I don't know how interesting that is.

Anyway so
just like getting into separation distance

that now says that we are bigger than
the probability of starting at X.

and Y..

Look like this inequality and this really
stands that the separation distance

is bigger than the tails and
now this is coupon collecting if T.

is analog then minus the end then and
then this bone told form to scoop and

collecting and just to finish in terms
of separation discussions there is this

inequality due to I think of the same
type of Conan's that if you have recent

reversibility then the separation
distance is I'm doing it the other way.

The total variation distance
is bigger than one half.

Price to teach I usually
butcher this inequality so

forgive me I hope I wrote it correctly.

And this is how one can retrieve
the bar argument of taking and

purrs from people collecting they saw I
think I'll stop here thank you very much.

I.

Yes you can see.

So so so the thing is that here you cause
whatever way so it can be something

crazy you know you might not be
going to have concentration if for

example for the second library.

Random stop assuming that you put
almost all the weight in one card you

have to wait until you touch that card so
you cannot write an explicit ban I cannot

going someone else might be able to but
in some nice cases I have

written down when you can hope for
cut off in this case.

You can write some nice stuff

I think it's a bit too complicated
to write it right now now.

Of course it does.

Are.

Over and on the number of the depends
on the faces that are used

it definitely depends on the number of
hyper planes and things like Lots Of

course all the geometric pictures
pictures coming in the answer now but

I think it would be too complicated to
write down so the condition I mean for

this case would care about the probability
of a specific coordinate being

picked during your draw and then
the probability of picking two specific

coordinates during a draw and you have
some condition of these two probabilities.

What.

About.

What.

You were what have one.

Yes yes yes.

You know.

Which mention.

Or.

Miss.