Thank you for

the invitation so I'll be talking
about the turn to work with three D.

bus who are new about Paris and
then some other works with you while.

So the general theme is going to be
results that relate mixing times and

the cut of phenomena to he think time
self set which are worse in some sense

so this results are kind of
abstract by themselves but

there one can use them to prove you
know Father theoretically results are so

to construct counter-examples in which.

You know one control is to be ever
of mixing sorry if you think times

which is they are pretty early easy
Your And then the deuce corresponding B.

F.
here is for the mixing times.

In But actually in some cases one
can actually analyze mixing in

cutoff using keeping times and
we mentioned the K.

So if my question on trees another
class of example is the simple London

walk on Roman Nugent graphs but I want to
get into that twenty and prove cutoff for

them just by considering he
think that it's an OK So P.

would be the transition me three X.

a piece of T would be the transition
probabilities of the continuous

time Chain time he saw be working
mostly in continuous times but

everything works in discrete time
is a stationary distribution

about the state space which is
going to be finite throughout and

we consider only reversible Markov chains.

Biden's condition all.

OK And then for a function from the state
space to are we defined in the usual

fashion the word the corresponding measure
is the stationary distribution and

then for Assigned measure of
Sigma when the find the P.

norm of the same measure just
as the norm of the density

function with respect to
the stationary distribution and

then we can define the Cylon LP
mixing time as the first time T.

at which distance from
the stationary distribution

of the distribution of the chain X.

time T.

is the most silane when we
maximize over the mission state X.

and when upset on equals one half
we just emitted from the imitation

So another important notion of distance is
the thought the variation in the stance

which you know we can write the it
is Moxy more mula a new way but X.

really just one half of one distance and

then the sudden thought that evaluation
mixing time we denote the by side.

So generally the mixing times
are non-decreasing in B.

but under reversibility we have that for
every P.

a strictly larger than one that L.

P.

and L.
to mixing times are the same up to some

constant depending on the P.

So I'll be switching between two and L.

infinity all the time but they're actually
just the same up to some constant.

OK but this is actually not true for
everyone else too so the thought of

addition mixing time might be a small
of all of them and to accept that I'm.

OK So one of the things I want to talk
about is the cut the phenomenon and

this is a property of
the sequence of Markov chains and

it's the condition that the rush
of Cylon in one man is the sun and

makes in time ten to one as M.

goes to infinity for any fixed Cylon.

So when we cutoff the graph
of the worst case distance

from stationary as a function
of time looks a sympathetically

like a step function that drops
from one to zero the mixing time.

So.

You know it's because a cut off is not the
only topic I want to talk about them not

going to go through all this example is a
really the point of this very partial list

is kind of to demonstrate that cutoff is
very common it's actually more the whole

than the exception so maybe it's more
appropriate to give one example for

which there isn't cutoff and that is
simpler than the walk on the end cycle or

more generally a fixed
a mention of side length and.

OK So although it's very common usually
very finely rigorously is hard and

requires some very detailed information.

Sometimes on eigenvalues or

really understanding of what
is the bottleneck for mixing.

So I understand in Seminole
paper from eighty six

the problem of finding the abstract
condition which imply cutoff.

And though the cut of phenomena received
much attention most of the progress

was done through understanding examples
rather than developing general theory

and so one Progress was done by terrorists
who proposed the product condition

which is generally a necessary condition
for cut off and that's a condition

that the product of the spectra get and
the mixing time they verges.

But unfortunately this is not
a sufficient condition and

Miss is a relevant example as
were constructed by us and by.

And so with Buss who in
Paris we showed that you can

characterize the cutoff in terms
of concentration of times and

you know the precise statement
would be a state that later on.

So is the state not if it
turns out that the condition

kept your is a cut off when you
look at the L two distance so

you can define the notion of cutoff with
respect to every notion of these things so

when we do it with respect to that
distance it turns out that cutoff is

equivalent to having cutoff in LP For
larger than one and

this is equivalent to deceive aren't of
the condition when we replaced by the L.

two.

And so this is under versatility and
this is due to change in.

The.

OK So this is really why we look at
the cutoff with respect to the total

variation distance because it
is in a sense not interesting

when we look at the distance for
the larger than one

OK So as I'm going to describe
further it's Classico that took that

evaluation distance sort of mixing time
can be understood up to constant factors

using and Nitro to probabilistic
concepts such as heating fans and

such as a stopping times whose
distribution the distribution of X.

at the stopping time is
the stationary distribution and

the total valuation mixing time
is also very related to copy

which is another neutral
probabilistic concept.

Converse Lee.

We don't really have any natural
probabilistic interpretation for the L.

to mix in time right and
moreover we don't have any

bound on the ultimately found which is
always sharp up to a constant factor.

So similarly we don't have really
a probabilistic interpretation for

I per contract if it is weather and OK so

we various We showed that both
the Earth to mix in time and

the local constant can be characterized
using heating times up to constant

factors.

OK so we say that a parameter is
robust if changing the edge weights

of the chain by a constant factor can
change the value of these parameters or

only by a constant factor
then similarly for

random walk some graphs we can talk about
a bus in this Under off a zombie trees.

And so the point is that the methods we
have to control the Infiniti mixing times

are using a whole bust your metric
quantities like a the Cheadle constant or

the eyes of parametric profile
the specter of this profile or

the locus of constant but these are all
analytic quantities which are robust

So it's very natural to ask whether or

not that mixing time itself you
know infinities robust OK so

this was because my but
also by various other authors who like

they are consistent set of course and.

Once you understand that
infinity mixing time

is the same can be understood in terms
of fitting times then you're in a very

good point to construct
a counter example because

you only have to show that mixing
times are non robust which is easier.

So this shows that it's impossible
to obtain a shar bound on

this quantity which is non robust using
these analytic quantities which are robust

OK So
they said Classic because that heating

times are related to anyone
mixing a under reversibility so.

Early work of others says that

variation mixing time is up to
universal constant the same as the.

Time they set times the size of
the set the stationary sides.

And this was recently refined by Paris and
so C.

and independently by only fair R.

who showed that actually it's
enough to consider sets of sizes

one half right so if the sizes of
one half we don't need this here.

Yeah.

Yeah and we must surmise over initial
states an over the oil target sets.

No no I'm thinking Mark Seymour.

Yeah.

So this means we're starting
that day wrong story.

You know is the term the stick
will take worst day.

Yeah thank you.

OK so.

So the following example
is is very useful So

first of all it shows where
the threshold one half comes from and

I mean the threshold right here on
the size of the set so if we consider sets

that are larger than one half then we
would heat them in constant number of

steps and this would not capture
though that of the mixing time and.

This is more important point is that
actually here the relation between heating

and mixing is mostly fine then just
determining the older of the two objects

and here for every time T.

is up to smaller all their terms

their worst case since from
stationary the time T.

is just half the probability of its size

you know if it's of size a half
plus Epsilon and vertices.

Then it would have a two step Cylon and
the vertices on each click

here so
in expectational would be one of a Cylon.

And.

Yes So the point is that the Cylon is
not allowed to tend to see what we will

fix it and.

So so so that was just this side comment
that in this example we see that we cannot

improve this we can replace one half
by a quarter but not by three three.

Quarters.

Where you are.

A.

Little late yeah.

So.

So.

You.

Are you.

Close.

And.

Yes.

I want.

OK.

So.

The point is that I'm making here and

you know here this is a really simple
calculation you can do by hand

is that suggest that maybe what
we should look at is instead of

these expectations look at
the tastes of times and

maybe get a more refined
connection with the Nixon times so

this multivariate defining a mixing
parameter using tells of hitting time so

hit the Cylon is a ten day
maximizing over initial states and

then for a nation state X..

Hit epsilon parameter is the first time T.

at which every large set is
hit with probability a case to

one minus the Cylon and when I say large
I'm saying the size is at least one half

OK So it turns out that the son took
the evaluation mixin time is the same as

hit the Cylon this if we are willing to
change the steps Cylon by a little bit.

So they'll tell you smarter than
a Cylon and one man to subside on and

this is up to an additive order is
the inverse of the spectral gap.

The the local

time agrees with the expectation
Yeah yeah it's different.

OK So so I'll get through that again soon

but the result I want to talk about today
actually an extension to the infinity

case so
you know this you know maybe he finds

earlier results but
in that sense is maybe less surprising.

But so this character is ation of

infinity mixing times using heating
times and the claim is that for

reversible Markov chain stolen from the
mixing time is up to universal constants

just the first time by which
every small set is escaped from

with probability at least a one minus
three half the size of the set.

OK and here it is again using math.

Instead of words.

So the point is that one
their action is really easy

namely the fact that the stymie is
the lower bound on the mixing time.

So if you assume that this
condition fatalism for

some initial point and some set.

Then clearly the probability
of being in a time T.

is a key least three halfs of A but

then using the beach in order principle
there has to be a point inside such that

the probability of being in this point
would be at least three halfs a PI of a.

And from that just using the definition of
the infinity distance we see that we're

not mixed So this is in the sense a very
Eve way of trying to bound the mixing

time but it turns out that this actually
gives you also the reverse direction.

So another result we have is
that you can also characterize

the mixing time in the relative
entropy distance but

then you need to change these three
have a constant over the log one of our.

And so as far as we know these
are the first sharp bounds and.

Mixing times.

And OK so
here is a very simple a corollary

which I'm not sure how you
would prove otherwise and

it's that the mixing times so
we suspect to both total value

of entropy and
infinity of discrete time lazy chain.

Up to constant the same as the one that
a state acts as are the probability

P X where the peer are uniformly
bounded away from zero and one

OK so if you believe that all that matters
is hitting times rather than mixing times

then of course that changing the odds in
probabilities shouldn't change the heating

times distributions by more
than a constant right so

this sounds like a very simple fact so
you know maybe you don't need all these to

mix machinery to prove this but
it turns out that this may fail without

reversibility So a counter example is
due to books South Paris and city.

It.

So I actually wrote it in a way that
can allow laziness but yeah I mean

you can assume piece not lazy so this is
why I wrote it like that that even if P.

is lazy then doesn't
probability wouldn't be the X.

it would be P.

X.
plus the times the ordering of it.

Yeah.

Yeah.

Yeah.

So here we are only allowed to
add loops or weighted loops and

in cosmic conjecture we are allowed
to change the edge weights or

a do rough is only three so this allows us
to replace the edge state by two edges so

this can actually really help us they
wreck the walk to do walk in direction

it wouldn't work either wise and then
we can kind of set up traps to eat but

just adding loops just makes it
spend more time at each vertex but

it kind of follows the same trajectory so
it doesn't effect heating times.

OK So
again this is a more precise version of

the total valuation inequality Iad So
it's the same as heat a plus

minus don't up plus this term was over
the is the inverse of the Specter you get.

So one comment I wanted to make is that

this term is generally smaller
than the two other terms.

For reversible Markov chains.

But you know when you consider
the case that the product condition or

its so
called This is the condition the T.T.L.

is lifted off the mixing time
which is a necessary condition for

a cutoff and then you can just kind
of forget about these two terms and

then it really tells you that the meek so
upset I don't like heat up side on.

Under the condition this is
a much stronger inequality.

So a conjecture I have is that you can
actually go beyond reversibility and

then carked arise the order of
the mixing time up to consent is

the maximum of a heat at
silence heat quater and

this parliament to hear
a call to star which is just

an extension of the notion of the
relaxation time so it's the first time T.

such that device.

P.T.S. is at most one over East Square
a device for any function F.

So if P.

was reversible than this would be
precisely the read exception time OK.

So yeah I think that this would
be pretty interesting one.

The mini most time you need to heat
every set of size at least one half with

probability at least three quarters and

if you can take the previous one which is
the notion of doing it in expectation.

OK so.

We can define the notion of hit the cutoff
but just imitating the definition of

the evaluation cutoff where I said
yes requiring that this rush one

in the limit and then the result we
have is that the evaluation cutoff

under reversibility is
the same as heat cutoff and

this is almost the media from
the two sided inequality

once you realize that a cutoff
implies the product condition OK So

we showed that but for thought
the relation cutoff this is known so

you can assume the poor the condition or
its cause otherwise need their cutoff of

course OK and then under the condition.

You can forget about these two terms and
then it becomes definition chasing.

So it's this condition that So
this was kind.

Of a distance from stationary were
the only test we're looking at is

is there a set that there is large
probability we still didn't hit it

OK.

So we have OK So there is a mole.

In a sense need to a version of this when
you fix your starting distribution and

you only want to understand whether or
not you have cutoff for

distribution or
you know abrupt convergence so for each.

Distribution you can look at the set that.

Of size at least one half that expected
heating time of the set these maximized

OK and then.

The condition becomes whether or not the
heating time of the set is concentrated.

But this doesn't work when you don't
fix your initial distribution so

this is a bit soft that.

OK but then it's really about
saying that war sets have

their heating time is
concentrated is cut off.

OK So

the next result I want to mention
is about characterizing the logs so

will have constant So first I need
to define the spectra to get a so

it's just the smallest
eigenvalue of the N.

But then given by this variation
of formula I involve being

the form and
take of all functions with variance one.

And by the point in the quality
when we scale time according

to the spectral gap then to
distances from stationary K.

exponentially and also variances the K.

explained in Chile when we
applied them P T two functions.

So there are some one of consent these
defined using a similar external

characterization but we know me I
saw the entropy if square is one

this is the film enough for the entropy So
this is important for two reasons

the first one is that the carked arises
I per country give it the so it's the.

Largest beat us such that piece of P.

is a contraction from the Toonami
to the one for beta T.

norm for every T.

and the other reason is
that it captures the N.

infinity distance up to
the studio global fucked or

a where I guess the lower
bound here uses reversibility

So one of the licks ation you can do
is instead of having i per contract to

fifty full of functions
you can escalate for

indicators and
the following these even the farther

Alexei Sion it's a hitting time version
of five per contract with the and

all I want is that the probability office
keeping a set starting from station earthy

would be some constant times
a the size of a race to some

exponent strictly larger than one so the
reason I'm saying this is I per country.

Because I'm raising the exponent
to be larger than one and

then a pretty easy which I
don't want to do tells you that

this is actually weaker than usual i per
contract to fifty even for indicators.

So.

This idea Fahy think I per contract with
the Month events looking at this picture

and get perfects this
because calling it a is just

the smallest I didn't value
of the restriction of to a so

it's kind of like the generate
her of its restriction to a of

the generator of the chain killed that one
need to scape say and the point is that.

They kept her is the specter will get
one we take a minimum of a smaller sets.

So if we do not by conditioned
on a by eight then

the law of the reverse ability that of
this keep time from a starting from

condition on a a cause it's
a completely monotone law

which is a fancy way of saying mixture
of exponential and the smallest

parameter in the mixture would be
precisely the spectra to get off a OK So

this is the general effect and
this gives us

a tail esteem by just plugging in
the head of the exponential is to be.

OK And
then when we were here we had it for

conditioned on a beast immediately
translate into the meat condition

was starting from pi right because we
have to not to scape A We first of all F.

to start in the set a time zero which
happens with probability by eight and

then we can use the previous A teammate.

OK so if and now we consider
a weighted version of the spectrum

get we wait things according
to the size of a so

we punish smaller sets more so we divide
by the log one over the size of a.

OK and then take in from all
our smaller sets so this

is exactly what we need for the hitting
time version of five per contract evictee.

Know it very easy so
as the probability of not escaping a by M.

of a starting from station earthy so
we plug in this estimate but

then a can sense with the Lambda
a here and we're left with this log

and we get some exponents larger
than one which is what we want that

is OK so you know this is
just a part of the proof but.

I wanted to explain this more for
intuition sake so

together with Paris we show that this
parameter is the same is the locus So

one of constant up to a constant factor.

So.

Then we had this in this

general lower bound under reversibility
a whose proof is very analytic so

you'd want to have some probabilistic
explanation of this inequality and

the proof of the previous proof was using
Stein's in TURP relation theorem for

a family of finality cooperators.

So now we can understand
it using hitting times so

for every set a there is a distribution
called the quasi distribution

on a such that starting from the
distribution of this time is exponential.

With barometer of the spectra get of A and

then what we had before is
an inequality becomes equality

and so I mean we already saw that.

In order to be mixing L.

infinity we need to have that tail
off not escaping a set would be

proportional to its size so
this shows that right so

we already know that the time that we
need to get that is the Laura bound and

because of that we see this is it or
just from hitting time

considerations so I'm referring to.

To this so we saw that this this was
trivial this was a one line calculation

to show that this is a band on infinity
mixing time using that we see that.

One over.

Is a lower bound on the fittings.

OK.

So remember we had this product condition
which is necessary but not sufficient for

a cutoff and then generally a problem is
to find large families of Markov chains

for which we do have the simply cation
that for the condition implies cutoff.

So I want to talk about Markov chains and
trees so this I just

mark of change such that the graph
supporting the transitions is a tree.

And then with Busta in
Paris we showed that if for

such a Markov chain as long as the Cylon
doesn't tend to zero two up at Lee.

The difference between the Cylon and
one minus the sun and

mixing times is that
most some constant times.

The Joe metric mean of T.

Meeks until there are.

No good side on.

OK so in particular a tells us that

inside the cutoff window if we
scale things by Joe metric mean of

the mix we have subclassed in
convergence to stationary thirty.

So generally you can't do better even for
birth and death chains.

In another comment I have is that
actually the condition that the chain

is the tree can be.

So we kind of can allow it to take
bounded Distin jumps on a tree.

OK And then another result we have for
trees is that the end

infinity mixing time is up to universal
constant just the maximum of the L.

one this total of the thought of
a time in the inverse of the locus of

its right to require that this
is always the lower bound and

an infinitely mixing time but
here we have the converse inequality.

So earlier it was shown that first trees
the thought that evaluation mixing

family is robust and there are bonded
perturbation of the edge weights and

because this is always true for

the locus of constant we see that
infinity mixing time for trees is robust

so hoping that then to explain why a trees
are very good candidates to apply these.

Relations between heating times and
mixing times because for

trees we can actually say
what are the worst sets

Yeah OK so

the main tool we used in the proof
of the theoretical results

is the Stars maximally inequality.

So to state the for any function

from the state space to our we can
define the max amount of function F.

star a as taking supremum of the order
times of absolute value of a piece of T.

X.

which we can also write in
terms of expectation and

then the result is us that.

Under reversibility dippy norm of
the maximum function is the same

as the norm of the original function
times the conjugate the exponent of P.

and then there is a version
that bounds the one norm

of the maximal function
using a one of the F.

times slow gaffe and I see

another gust inequalities to what you
would have been dupes maximal inequality.

OK so I one of the one proof and that's.

To the proof of this result.

Yeah you need.

You need I mean this works.

You need reversibility you can be funny

even in this kind invert space set up and
then it has to be self a giant

So if you consider
a walk on the cycle with

the fixed biassed you see that it
can't be true when you take F.

to be an indicator of
a point just not true.

OK So recall that we had a parameter that
was the first time that every small set is

escaped from with probability at least one
minus the half the size of the set then

we showed that this inequality
actually even we showed these for L.

infinity instead of fell to and
I want to prove the other direction

OK so so fixed T.

to be nine times the.

Time parameter and then.

Right for every set B.

by definition fall so
using the Markov property a.

High This should be nine instead of
six in the exponent a right to can

try to escape from its nine separate times
and condition where we wore so we get this

and the point is that because it be
smaller than one half I want to get rid of

taking three half so and
you can verify that this is true so

the point is that even we even
though we define draw a so

here it was eight to
the power one we can assume

that we have this with an arbitrary power
so three would be good enough for us.

OK And then we generally can write second
moment often non-negative on them via.

Integrating PI where the set on
the viability is more than S..

OK So we want to control

as to distance so I mean first of
all we can write it up to this

constant term as just the two
norm square of the distribution.

And then right by definition this is just
the second moment of the density function.

And then using this
a dentity we can write it as

the integral from zero
to infinity of twice S.

a PI of the points in which
the density function is at least this.

So the point is that I don't have to show
that this integral is very small it's

enough to show that it's bounded by some
constant because by the point of quality

sheltie after it's at most some
constant it becomes very small Ok so

in particular it would be enough to
show that into a grand is the most

something we can integrate So let's say So
one of rest of the power three have.

OK so.

So let's consider the set vs which is

all the points why such
that supermoon of all time.

The probability of being a S.

at time K.

is larger than S.

over to the size of a.

Right so this this is

a set that these defining terms of some
level a set of a maximum function.

OK So this is the thing we want to
get the control over its size it S.

by S.

So by the definition of a S.

you know if we sum this of
the a while you will see that

I mean this in the sense
Markov inequality we see that.

Is that most.

The probability of being at the S.

and then we can write this
probability as terms so

the first one is the contribution coming
from the case that we didn't escape S.

and the other one is the contribution
coming from the case we did the scape S..

So here we have this general.

That's precisely the way we define T.

so we can say that this is the most
a PI B S to the power three

Well you know provided the B.S. is
smaller but let's leave that for now and

and the other term we define the fit B.S.
such that if

we escape from eat then we know that
the probability of ever being at S.

again at the later time would
be at most this quantity here

from the definition so that's
precisely the reason we defined B.S.

like that right so
then we can as substract the S.

and hold we have to show is
that PI of B.S. is small.

We might hope to be able to show
this because this is defined as

it's basically the set of a Y.

such that some maximal function
is large at those points.

OK But unfortunately this doesn't work so

you have to do things to
actually different T.

And so
here it was very convenient to define B.S.

like that because A it gave us immediately
control over this term in a way

that was very convenient for us but if we
define B.S. slightly differently then.

We can actually control its size OK so
if we define it to be the pointed to which

a P K Y S of A is larger than the square
of log one of or size of a yes.

To which.

And so after multiplying by some
calling this term delta S..

So exactly as before we
get this inequality.

And we want to show that it's
small the right inside this model.

OK.

But then this maximal function.

It's one arm is most kind
of the one that first

moment of this times to log of this and
then relation tells you

that the maximal in quality tells you
this is at most eat times to log by a a.

OK.

So then mark of the inequality
tells you that pi of B.S.

is at most one over a square with S.

because.

This is such that one or
devalue of the maximal

function it why is larger than square with
their one norm of the maximum function

OK So this is good for us right if we
substitute the tear than it's what

we want provided the delta is small.

Delta S.

is smaller than we're good.

And then when you work out the details
you see that if we don't have this then

we're already in a very good situation so
if those tests is not much smaller then S.

by S.

It actually means that this quantity we
want to bound is this stretch explain and

child to begin with so it's some bra.

OK so so
either you are very good to begin with or

you can forget about the S.

OK so that there is more algebra but

the idea is like the cheating
proof I did before.

So it's the first time T.

such that every smaller set.

Is this from with probability
at least one minus three half

the size of A So
the probity of not escaping it is that and

then instead of taking a ball I
took like nine times out of all and

they said that I'm getting this
estimate to some exponents so

I can forget about this and
I can write three instead of one.

Here

yes.

Or.

No it was so for entropy.

So it's the same expression we had
before but before we had here three have

a then here it turns out that this is the
relation you need to in the probability of

not escaping in the size of a or you know
the right punishment to take for the size.

And then it's the same result.

So.

It turns out that not only that the same
results translate into the discrete time

case you can also translate them into this
Jane that makes only one lazy step and

we could that every chain so because
a reversible change if there are per

year the period to it makes sense that
making one step is enough to overcome or

is new the city issues so
this was a conjecture by and feel.

So you know because he think times
are the same for this change in for

the continuous time chain in the sense
that it's very reasonable to expect it and

this actually gives us
a way to prove it show for

total evaluation distance so we show that
up to changing the time by square with T.

factors a this is true for

any initial state OK so the term we
have on both sides ten to zero as M.

goes to infinity and.

And then for this effort chain.

And the relative entropy mixing times
are smaller here there is in the converse

inequality because for example for simple
abandon walk on the end and clique a.

Strictly smaller or
their right to making two steps for

the every change it's
already completely Meek's but

the continuous time Chen because
we saw a small probability doesn't

make any step then it has to
spend more time until it sneaks.

OK so I don't have any time
to discuss counterexamples.

Though some of them are very nice.

OK So one of them briefly explain
why trees are very good candidates.

So because we can use the tree structure
to roughly identify the warthe sets and

first of all it's easier to think
about birth and death chains and

if I ask you on in what is the worst
set of size one half to hit

then it's like hitting the center of
the mess it's like hitting the point

such that disconnecting eat leaves us with
two intervals of size less than one half

its precisely this point so we were
reduced hitting time of four sets to

hitting time off one point
which is the center of math is

so it turns out that we can
generalize this to trees so for

any tree there will be one point such
as the if we delete it from the graph

then every connected component is
a stationary mess at most one half.

So we fixed one there can be
at most two we fixed one and

we call it the central vertex or
dilute the not that by zero.

So I don't have time to expend it but
there is a fairly simple of

the duction the total tells you that
a concentration of fifteen times of

sets of size one half is like showing that
the heating time of the center of mess is

concentrated in one direction is easy.

For the other one you
have to do something.

So instead of finding the worst initial
state and proving concentration

starting from the worst the nation state
we can actually prove concentration for

every nation state such
that the expected heating

time is large to the center of mass so

we don't need to worry about what the what
exactly is the worst initial state

and then we can use
the tree structure again.

To decompose this heating time.

So the hitting time from X.

to the center of mass we
can write it is a sum

of crossing time off edges so the time
it takes us to get from here to here

plus the time it takes us to get
from here to here except or up.

OK but the point is that this crossing
time off edges are independent London via

Bellus OK so.

So to get concentration of the heating
time it's enough to show that the variance

is the variance is smaller all of there

than the second moment
Square right then we take X.

such that this is larger than it ever.

OK So as I said before we can
decompose the heating time

into some of these
crossing times of edges.

Right to an V.I.I.I. I thought
very ticks and on this path then.

The difference of the sitting times
is the time after we got here

it took us to cross this.

So as they said their independent.

And thus the variance it's enough to get
control over the individual variances.

And then you using general considerations
de Law of these crossing time of

edges is completely monotone it's
a mixture of exponentials and.

Parameter is at least two by
general considerations OK and then.

This is enough to get the control we want
our differences because the violence would

be at most the second moment of
each of these crossing times.

And then for a mixture fix potential is
the second moment looks like that but

then we can pull out.

So each of these guys is the most.

Over to sorry twice so we pulled that
out and then we're left with the first

moment right and then when we substitute
it into the some of the variances.

Then we get some of the four

times the first moment and
putting this outside we're getting.

Just the estimate we want.

Yeah so similar consideration apply for
two but I think I'll stop here thank you.

One.

Of.

You mean the constant Yeah well
actually we don't really have even for

the station time right so the relic
station and you can approximate it using

spectra get both appear restricted
to sets and this you can

understand in terms of fitting times when
you start from Kweisi stationary but

it's actually a conjecture that
instead of that you can start from

by conditions on a journey you only
have one sided inequality so even for

the delegation time there isn't a heating
time characterization involving pi.

Yeah.

Yeah.

OK.