OK thank you for the introduction.
As a great motivation and
thank you everyone for
having me here and
my pleasure to be here and.
So as kind to suggest that I'm going
to be talking about the dynamical
properties of the pots model at their
critical point mostly in the two
dimensional lattice But in order to
motivate that I'm going to start off with.
The mean field Potts model.
Which So let's start off with and
verdict Cs and each of them are in
one of Q Different states and so
frequently I'm going to jump back and
forth between referring to these of states
or colors so you can think of these as Q.
different colors and we have a market
chain on the state space one up to Q.
to the N.
which you has signed uniform ID clocks to
season whenever the clock of a site rings
then it switches its state to the J.
with probability proportional to each
of the beta times the fraction of
sites total that are in state so
if most of the sites are in the J.
or C.
read then with high probability
the vertex the updates stays and.
Flips to a color are red and
here is a parameter which indicates
the inverse temperature so
as you can see the larger beta is
that corresponds to lower temperature
which means the more you favor aligning
with the majority opinion of like and
varies and if you run this
market chain for long enough for
forever it will converge to some
stationary distribution which we're going
to do no meu and that's the pots
measure on the state space one up to Q.
to M.
and it has probability
given by exponential in
this inverse temperature parameter beta.
I did buy an and
you sum up over all edges so
since we're in the mean field set up
sum up over all pairs of verdicts sees
the indicator that those
two vertices agree OK so
this probability measure favors alignment
and is symmetric with respect to the Q.
different states and when Q.
equals to two This corresponds to
the easing Curie Weiss model which is
a classical model of statistical physics.
And that has a phase transition at
Beta critical equals to two and
it's a continuous phase transition so you
transition from a high temperature regime
where with high probability if you sum up
the spins of all the different sites and
you look at the magnetization.
OK.
I guess I'll have to move
the mouse every once in a while.
So if you look at the magnetization
of the system which is just the.
Average state among all over
to seize then in the Q.
equals to two case the average state
is no more likely to be plus or
minus when you're in a high temperature
regime whereas when you're in a low
temperature regime it's either
with probability you at say
one half the average magnetization is
positive and with probability one half
it's a negative and what the continuously
of the phase transition indicates is that
at that critical temperature be
the sequels to two you have no magic is so
when you sum up over all spins that
converges to do rock masses hero.
And so the reason I mention
emphasize this continuity of
the transition is because
when you move to Q.
bigger than two then immediately you
lose this continuity property so.
What we're going to be mostly Can
concerned with is the Q State pots model
when there are more than two states and
so in the mean field model which I just
described with this probability measure
there is an equilibrium phase transit.
And again similar to that using one.
And when there's Q.
bigger than three colors are states the
phase transition is not discontinuous so
what that means is that when I look
exactly at the critical temperature and
I send and to infinity then
the configuration is equally likely to be
in a disordered phase where on average
all the states are equally likely to be
red green or blue this is the three
state pots model and when I'm in a high
temperature phase so this is beta strictly
later less than the beta critical
on the exponential scale I'm much more
likely to have a disordered phase where
sites look at sensually as if I just flip
coins and in the low temperature regime
where I have data strictly bigger
than data critical I start to see.
Either in a regime where I'm mostly red or
a regime where I'm mostly green or
in a regime where I'm mostly blue and
that's what like these three
humps correspond to OK and.
So what actually ends up happening
is because at the critical temperature the
four different phases co-exist the four
different phases being three ordered
ones corresponding to red green and
blue and one disordered one corresponding
to essentially flipping coins at random.
Slightly above the critical
point the disordered
one which is the one where they're
equally likely becomes mess stable but
doesn't get any mass under the Gibbs
distribution so at the exponential
level it has exponentially less mass than
the three different ordered States but
it still has exponentially more
mass than the states near it OK And
so that's what supplied it here is you
have the net a stable say in the middle
which corresponded to this one at the
critical temperature and then passed some
threshold that matters to belittle
completely disappears and in the high
temperature regime you get similar
medicine ability for the three ordered
states which then eventually disappears
once the temperature is high enough.
And so that's the equilibrium picture and
because these safes
are only met a stable and
they have exponentially less mass than
the dominant one they don't show up in
the equilibrium picture you're all with
high probability going to be over here but
if you look at the dynamics on such
models then you'll get stuck in
Madison able regions and it'll be hard
to get to where that gives mass lies.
And so.
That's the question we're going to ask
about the mean field Potts model is how
does this equilibrium picture correspond
to a dynamical picture OK the market
chain I defined on the first slide
is the called the dynamics for
the mean field Potts model which
means again pick a side at random and
then flip it according to the proportion
of its neighbors that have
the same state as it or to have
some other state and we're going to
be interested in is classical quantity
which is the mixing time which is.
Measuring how long it takes for a year
probability distribution starting from
a worst case starting point to
converge to equilibrium or be close to
equilibrium in total variation or
Al one distance OK and.
So the globe or dynamics if you look
at this picture as I was just saying it
should be it's a local dynamic so
it's hard to jump around you have to move
slowly and so you get stuck in medicine
able regions where you like to reside and
so for that reason one might expect
that when you're in the region where
there's a medicine to build the picture it
takes exponentially long to equilibrium
when you're in low temperature you
again have three stable points which
correspond to the three different colors
and it should take long to equilibrium
whereas when you're in a high temperature
regime it should be faster to equilibrium
just because it becomes essentially
a single well picture OK and
physicists in the Navy in one thousand
nine hundred nine Swenson and
when you introduced the Swenson Wang
dynamics which is a much faster
market chain Monte Carlo sampler for
the parts model because it's able to
jump around between the ordered
seats in the model and
the way it does that is instead of looking
at the colors it uses some sort of
correlation structure on the graph and
then reassigns colors so
it's able to just go from mostly
red to mostly blue to mostly green
very seamlessly so just to be precise
the Swenson Lang dynamics for
the mean field model is you take
the cue different color classes which
correspond to the vertices in with color
red with color blue etc and you do G.N.P.
on the subgraph given by those color
classes so first I look at all the red for
to seize and that's some subset
of all the various All the and
various Cs and that generates a sub
graph which is all edges that have to
read vertices on either end and
then I do G.
N.P. on that subgraph with the premise or
P.
calls to one minus even
the mind the speed over and
for experts here that
correspond to the Edward So
Cal coupling of the random cluster
model to the pot's model but.
What ends up happening is that if I'm in
a regime where there's one large dominant
color class then this G.N.P. will be super
critical and you'll stay in a regime
where there is one large color
class as long as beta is small for
instance or it is large so
that you're in low temperature and
then once I have this random graph that's
given by doing G.N.P. on all the Q.
different color classes I assign each
cluster there a uniform color so
you see even if I had a large giant
component in the random graph I generated
I'm giving it a uniform color between
one to Q So that's why I'm able to
jump between these ordered met a stable
states easily and so what that means is
that in the low temperature regime
where there is no maddest ability for
this middle guy the mix guy it's easy for
me to move around and I mix quickly OK so.
Summary of results for
the dynamical phase transition of the mean
field Potts model which is essentially
fully understood this point so
originally the dynamics was studied.
In pairs into papers and
then the picture was
completed with cut off at high temperature
and everything you would expect and
this corresponds for that using model so
the cube equals to two case and
there you have a continuous phase
transition so you go from fast mixing at
high temperature to exponentially slow
mixing at low temperature with polynomial
mixing time at the critical temperature I
think it's and to the one fourth this and
then the dynamics study was
extended to the Q bigger than or
equal to three regime where you have
a discontinuous phase transition in.
Pears and sly and so
the green line here indicates where
the mixing time and how the mixing time
of the Q State pops model transitions so
it goes from logging and mixing at high
temperature to exponentially slow mixing
at low temperature and
to the one third mixing time at.
This point which corresponds to
the onset of maddest ability OK.
Now the Swenson Wang dynamics we just said
should actually be fast at low temperature
because it can jump between these ordered
states and avoid those bottlenecks so
reason Cooper Dyer freeze and RW showed
some general balance on like the off
critical Swenson when dynamics and
then long enough and
Perez give the complete picture for
the cubicles to two using case where you
have fast mixing at low temperature
fast mixing at high temperature and
then at the critical point you have
polynomial mixing and it was a long
standing beliefs or maybe question of can
the Swenson Wang dynamics be slow because
if you look at low temperature bottlenecks
it's fast in with respect and so in one
thousand nine hundred nine Goren Jerram
showed that actually it can slow down and
they considered the mean field Potts
model and showed that it has mixing time
at least exponential in square root of and
at the critical point and the reason for
that is because even though it's easy to
jump between these three ordered states
it's hard to jump between the disordered
sea and the ordered state and
those four phases co-exist at
the critical temperature OK And
then that was extended by some people
in this room very recently I guess
simultaneously and independently so
I go on this to find kitchen Eric
they essentially showed sharp bounds for
all temperatures except for
this critical window here where
you have met a stability and.
Again it's.
Continuous time mixing time in
high temperature order law again
in low temperature so fast in both and
then they identified the mixing time
as polynomial at this point where you
have the onset of minutes to belittle and
an exponential in square root and
lower bound in this critical window and
similar results also I guess and
I said independently were obtained by and
Tonio analysis or Sinclair.
Via their representation of
the random cluster representation of
the model which I mentioned and so
a question one might ask is well
dynamics exponentially slow in this middle
region which will call the critical window
is the Swenson Wang dynamics also
exponentially slow there and then another
question is Well if you're interested in
things like lattice is instead of just
the mean field model where you don't so in
the lattice you can't just restrict your
attention to a single order parameter like
the magnetization or the number of red and
blue vertices should such a picture
hold or does such a picture hold OK.
Purpose once and Wang green is clobber and
here there was a lower
bound of exponential in square root N and
an upper bound of exponential in and
whereas everything else
is sharp objects yes.
So that's that's exactly this
point where if you go to slightly
lower temperature you have met a stability
here and if you go to a slightly lower
temperature you don't even have a critical
point in the free energy at that So
here it's flat the free energy carrier
becomes flat where you start to
development of stability and OK.
And so I guess.
What I'd like to do is answer this
type of question and whether and
show that such a picture in fact does hold
in the two dimensional lattice where.
OK so I'll just quickly introduce
what the model is the two is so
now you have an underlying geometry
which is the integer lattice and
you put nearest neighbor edges and the
parts model is has a similar distribution
obviously and it's given by each
of the minus time some Hamiltonian
where the energy is minus of
the number of edges they agree so
the number of edges on which both
sites on the two sides have the same
spin are the same color and so that means
that low energy configurations the ones
that are favored are those where you
have more alignment except now it's more
of a local alignment because you
only see your nearest neighbors and
the beta term again corresponds to
an inverse temperature parameter.
Dynamics defined similarly So we have I D.
plus on clocks at all of the sites now and
each of them updates
its spin value to j with probability
proportional to each of the beta times
the sum of its neighbors only
now that are in stage a OK And
you see a similar phenomenon occurring
as you did for the mean field model and
this once in Wang dynamics is defined
in analogy to the mean field model so
you look at the subgraph generated
by the red avert a season now G.N.P.
doesn't make sense because you only want
to consider nearest neighbor edges but
you do bond percolation on that
subgraph with the same parameters P.
close to one beta and it again is able
to jump between the low temperature
bottlenecks and its coloring stage because
of this recall or step in step two where
you just recolor all of the clusters
a uniform color between want to keep.
So the pots model has a very rich
phase transition and Dimension two.
And as Eric said at the beginning
just seven years ago I guess and.
Show ports and result identifying
the critical point of the model with it's
self a dual point which is B.
the C.
of Q.
is this parameter here and
so now is known only for
that using model prior to this work.
Where it goes back to ON
solder forty four and so
once we identified the critical points we
can ask what happens at the critical point
is the face transition first order second
order continuous or discontinuous and so.
When Q.
is less than or equal to four the phase
transition is continuous and so I describe
here what that means it means there is
only one infinity volume gives measure or
if you want to think of it in terms of
the magnetization that I described.
No matter what boundary
conditions I take on my and
my own box as I send the box to infinity
the probability that the origin is red
is just one over Q OK so the origin does
not feel the influence of the boundary
conditions as I sent them to infinity or
if I look at the.
OK And that was known for Q equals to
two dating back to NS augur again.
Which to say that the critical using
model does not spontaneously magnetize
and a very recent work by DIMIA co-parents
vicious and tests on showed this for
all Q.
between for all Q.
less than or equal to for.
The reason throughout the talk I'm going
to have things like one less than or
equal to Q.
even though the parts model
only corresponds to integer Q.
is for ECT if you know what the random
cluster model is all of these things also
hold for the random question
model that corresponds to it.
And so you see these three dots
corresponding to Q equals to two three and
four we have a.
Phase transition which means
that there's only one phase or
one infinite volume gives measure.
At the critical points and
then immediately when you go into
the low temperature regime you get Q.
different phases corresponding to the Q.
different color classes.
What happens when Q.
is bigger than four is that actually
the phase transition becomes discontinuous
which means that at the critical point
you can have spontaneous magnetization
which means if I take a bound if I
take red boundary conditions and
send them to infinity the probability that
the origin is red is strictly bigger than
one over Q OK And so in the low
temperature phase there are the Q.
ordered phases and
in the high temperature phase there is
the one disordered phase and
just like the mean field Fox model.
Once Q.
is bigger than four you have Q.
plus one different med a stable
phases which correspond to the Q.
ordered ones and
the one disordered one and
this was originally shown using
cluster expansion techniques for Q.
sufficiently large and now is improved
to Q bigger than twenty four and
protects us min then
recently very recently for
all Q bigger than four which is
the true thing by Dominican.
Test So that's the equilibrium picture and
you see that it corresponds very
well to the picture that we had for
the Q State mean field Potts model except
now the discontinuity starts when Q.
is bigger than four instead of one Q.
is bigger than two.
So the natural question well
before we get to the dynamics
here are some pictures of what the model
looks like at the critical point.
When Q.
is equal to three four and five on and
by and torso the Taurus doesn't.
Doesn't favor any of the different phases
just because it's been seen that trick and
so when you're looking at
cubicles to three and four you
see that there are macroscopic clusters
of all the different color classes and
then once you go into Q.
equals five where the face transition is
Can discontinuous you've picked out one
of the phases to be with the dominant one
and that's the great color class here OK.
And so the continuity of the phase
transition corresponds to some very.
Popular things in the last
twenty fifteen or
so years which is that it's
believed to be conformal invariant
have critical exponent corresponding to
like the probe will be that the origin and
some site far away is the same
color decays polynomial in their
distance with some exponent and these
critical exponent can formally invariance
correspond to scaling limits described by
a sally if you know that what that is and
these interfaces converging to random
fractals that are also can formally
invariant things like and
that's only been established for the Q.
equals two case and it's a big
open problem to establish it for
the cubicles three and four bottles Yeah
yeah exactly so the Q.
ordered phases in the one disordered
phase are all of the same magnitude.
In the mean field you also had a.
So you just jump from nothing no
maddest ability to this picture and
then to know maddest ability again there
is no window in which you met a stability
arises so this beta little ass
corresponds to a bit of capital S.
corresponds to be critical.
And so when Q.
is bigger than for all of this universe
ality in conformal invariance breaks down
and the picture you should get is
something more like a low temperature
picture because you have phase coexistence
there should be what's called Positive
surface tension between the different
phases and if you look at their interfaces
they should look like Brownian bridges
with large deviation estimates and
everything and that's only known for
two sufficiently large.
OK.
So then one might ask how
does this all correspond
to the dynamical picture do you see
the same thing that you saw in field and
then answer is pretty much yes so
in the high temperature regime
it's known that you have in continuous
time log and mixing for the dynamics.
And that's due to Mark's Nellie Oliveri
and Alexander and Martinelli.
Think dating back to ninety's which showed
that things like weak spatial mixing or
exponential decay of correlation imply
through spawn strong spatial mixing in
order one spectral gap and then if you
combine that with the result of the pan
which identified the face transition that
tells you that haven't you have an order
one spectral gap in the high temperature
regime and you have logon except.
On the other hand in the low temperature
regime it's believed they of course all
the way up to the critical point you
should have exponentially slow mixing but
that's only known along
the line here where Q.
equals to two.
Cheese Shay's Schoenman and says need
Martinelli and Schwartzman and for a Q.
sufficiently large dood.
Borg Che's freeze Kim Perseid Eric and
the famous six elf their paper and
later the seven
seven author paper and later that was
refined inboards Chaison to tally so
that slow mixing actually holds for all
dimensions bigger than or equal to two and
they showed that you have exponentially
slow mixing up to the critical points for
all Q sufficiently large.
And so I say this low temperature
exponential and expected that's for
the dynamics of course the Swenson Wang
dynamics it's able to jump between
the different color classes so it should
be fast in the low temperature regime and
so that was shown by all rich thirteen and
twenty thirteen and
twenty fourteen by comparison estimates
of the dynamics and by Antonio and
Alistair via the F.K.
representation of the model.
And as well as.
The fast mixing for
the Swenson Wang dynamics and
the high temperature regime was
also shouldn't by them and.
So on and then what happens at the
critical temperature well when you have
a continuous phase transition from reasons
of physics and critical exponent or
from the mean field picture you believe
that you have a critical slowdown but
only to a polynomial when the face
transitions continuous with some special
exponent and when the phase transition
is discontinuous you would believe that
you have exponentially slow
mixing both for the dynamics and
the Swenson Wang dynamics because
it's just hard to jump between
the ordered phases and the disordered
phase so just to show that picture.
Quick simulations this is
the pots model with three colors
started from completely random at its
critical temperature for glob or dynamics
you see OK you start to see macroscopic
clusters fairly quickly and now you
start to see that type of picture that you
believe after you like a few minutes in.
So this is periodic down visions.
So this now it looks like a picture
from the equilibrium for the possible
on the other hand I'm still looking
at the critical parts model now but
now with ten colors and I still have one
I still have periodic boundary conditions
OK and I start from the all free or
the all white phase where
white white is a color and
OK we could wait a million years and.
Nothing would change for
you see that you're always in the phase
where the white color cost on minutes and
you just have like exponentially small or
small clusters of other colors
to have exponential tails and
this these were both at
the critical temperature so.
That's what I see here is
the expected picture for
the very end Swenson line dynamics.
Yeah.
So because the critical temperature is
not known for a larger dimension and
so you don't know exponential decay of
correlations up to the critical point.
So all of this is after the work of you
know cope and that makes it possible.
So both clobber and Swensen Wang dynamics
at the critical point should have
polynomial mixing time for Q.
equals to two three and four and
maybe you believe that this
is he is a dynamical critical exponent for
the model and
it's independent of the boundary
conditions whereas when you go to Q.
bigger than for where you have
a discontinuous a strength the mixing time
should be exponentially slow for both club
or dynamics and Swensen like dynamics and
so at the critical point there's two
results in this regime that for Q.
equals to two that using model
I was shown that you have X.
polynomial mixing time both a lower
bound of end to the seven fourth and
upper bound event to the C.
by Lubetkin sly and
this should be something like two and
to the two point one five
something I don't know.
But this upper bound is like and
to the ten million who knows.
And by the comparison results the upper
bound carries over the Swenson Lang
dynamics but the lower bound does not
even though it should also be polynomial.
And as mentioned the seven
author paper and
then Maurice Chase to tally also showed
that at the critical point for Q.
sufficiently large you have
exponentially slow mixing for
both dynamics and Swensen length dynamics
and then again this result was also for D.
bigger than or equal to two and Q.
sufficiently large in the dimension but
there's this big gap between Q.
equals to two and Q.
sufficiently large.
And so.
Talked about the recent results of.
Bets in itself so when Q.
is less than or equal to four and
you have a continuous phase transition
the Garber dynamics on and by N.
Taurus has polynomial mixing time if Q.
equals to three so
you have a polynomial lower bound and
Per bound and it's continuous time so
the lower bound is like so low.
On the gap.
And when Q.
equals to four you have a polynomial lower
bound in a quasi polynomial upper bound
and on the other hand
immediately when you go to Q.
bigger than four you have exponentially
slow mixing for both clobber dynamics and
Swensen length dynamics.
So for all Q.
bigger than for mixing time of
quadrants once and rank on then by N.
Taurus is exponentially slower so
the reason this
one is on then by in Taurus is because
it's symmetric with respect to the Q.
plus one or phases and all mention
more about boundary conditions in
this regime later whereas the continuous
phase transition case if I took arbitrary
boundary conditions both of these
estimates would hold and upper bounds here
hold for this once in mind dynamics
also because of these comparison results
that say Swenson Wang dynamics is
always faster than Glover dynamics OK.
Yes so I should have mentioned that so
this should be polynomial and
there should be polynomial lower bounds
on Swenson Wang dynamics that we do not
know it should not that much should match
once here bigger than dimension five
maybe for for whatever the critical
dimension is for the Q State pots model.
But here I think once and
rank should have different X.
one and seven.
Is predicted probably by like.
These type of things but.
But no so lower bounds on twenty and
Wang aren't known for any Q.
polynomial lower bound and
the upper bounds are like horrendous.
So.
I want to focus on give a short
proof of that is the second result.
Which says that when you have
a discontinuous phase transition you
have exponentially slow mixing time and
actually all stated in the form it was
originally stated which was for any Q.
bigger than one if you have that take
read boundary conditions and send them
to infinity if the probability of the
origin is read is bigger than one over Q.
So that's the same as saying that
there are multiple phases or
there are multiple infinite volume gives
measures then the parts cooperate dynamics
has exponentially slim nixing time OK And
so
that in particularly shows not only that
you have exponentially slow mixing when
you have a discontinuous phase transition
at the critical point but actually.
In this whole low temperature
regime between Q.
equals two and Q.
sufficiently large you have slow mixing
all the way up to the critical point so.
When you're in low temperature and
E.Q. bigger than one
you have slow mixing up to the critical
point for the particular dynamics and
when you're in low temperature or
the critical temperature and Q.
bigger than four where there is
a discontinuous phase transition you
have slow mixing although for
the possible dynamics and
this the results at the critical
point can be adapted or the proof can
be adapted to the Swenson Wang dynamics
using the random question model but I'm
just going to focus on giving the proof
in the pots cooperate dynamic set up OK.
Yeah because it's a particular
dynamic statement but.
First once in a lang You need to use
the bottleneck between the ordered and
disordered phases and that's how
you get that's why I first went in
lang it's only at the critical
point that you get slow mixing but
it's the same bottleneck construction.
So will quickly go over some preliminaries
that are probably familiar to you so
you define the cheater constant or
conduct and as a minimum over
sets that have mass less than or
equal to one half of the ratio between
like the edge measure of the set
to its compliment which is them
mount of mass that traverses from S.
to S.
complement in the time step to the mass of
S.
and if you just take a bound of one on
the probability of going from one state to
another you see that this
edge measure is at most
the probability of being on
the boundary of the set so
we just bound this by the privilege of the
boundary conditional on being in the set.
So we're calling the conductance the
minimum over all sets that have mass less
than or equal to one half the ratio
of the boundary mass to the set and
in order to lower bound to the mix it's
a classical result the mixing time is at
most up to constant one over this
conductance OK And so that means
that in order to lower bound the mixing
time we just need to construct a set S.
to our so that this ratio or
this probability is small.
So in order to construct the bottleneck so
we want to use the topology of the Taurus
in some way because we know that
if we don't have a tourist then it could
be for it for instance in low temperature
if I took a Taurus and had read boundary
conditions then actually the mixing time
shouldn't be exponentially slow OK so
I want to looking at a tourist.
There in the.
So you might you want to construct
a bottleneck we said that there are at
least two distinct gives measures and
in particularly that means that the red
phase and the blue phase are both
matter stable because we have at least
two colors and they're both mystical.
So I want to construct a bottleneck
between going from mostly red to mostly
blue somehow using the geometry of the
Taurus so one thing you might think of is.
If you take your Taurus.
Then if you take a loop around
the Taurus in this direction so
these two points are identified
a loop of blue sites and
you take loop of blue
sights in this direction.
Then if you translate it on your Taurus
there's a translate so that there's
some sort of macroscopic loop like
this OK And inside this loop.
You really favor blue because
it's like it's like having blue
foundry conditions at infinity but
even more blue than that so that means
that the probability of having So
that means that you're not blue clusters
should have exponential tails and.
It should be hard to go from having
such a thing to breaking it but
the problem is you need to be able to
expose this component of this loop without
revealing anything inside it and
so because you also don't know how
close this Lucas to some macroscopic
box you want to be away from and
interior to force yourself
to have long crossings or
long red slash green Questers so.
You try this and you see it doesn't
work and you try it with two loops and
you see it also doesn't work and
then you almost give up and
then you try it with three loops and
you suddenly see that it works on so.
So the bottlenecks that we're defining is
essentially what I just described but.
I split my Taurus into thirds like
this and also into thirds this way and
I ask for three let's say red
to be consistent with slides.
Three loops are all around
the Taurus of homology zero one and
three were around the Taurus and
almost a G one zero.
And.
So that's just going to be the set A star
is all configurations that have this.
On the Taurus and so we're going to see.
That happens to work.
So it suffices to show that the
probability of being in the boundary of
the segue in the year in this side is
exponentially small because that that's
what we said earlier and we know that in
order to be in the boundary of this set
you have to have some site somewhere
here that's pivotal to the.
To say yes Star which means
that if that site weren't to
read then you wouldn't be in
the configuration as Star OK And
since we're looking at
dynamics which a single say.
A configuration is in the boundary if and
only if there exists some site that's
pivotal to the event tests are OK so
now we can Union bound over the N.
squared edges and squared sites and
consider just the probability that
a single fixed vertex here is
pivotal to the event star conditional
on the event I start holding OK And
the reason that this three loops
in each direction works is because
I can so let's say I was
considering like this vertex here.
So so because of this Union found it
suffices to just show that the probability
of a single vertex being pivotal
is exponentially small and
then that gives me the proof
Union down over and squared.
The three different crossings in
the vertical and horizontal and
I get an exponential tail
on the down tree mass
well so even in this picture if I'm in
a blue phase I'm going to have tons of or
if I'm in a red face I'm going to have
tons of red like alternate paths so
the reason you have to
be careful is because.
If you say I was looking at just one
crossing and one crossing OK then and
I want to look at it like so I'm extending
on like the cover of the Taurus and
that's the loop I want to reveal so say I
want to reveal like the leftmost crossing.
The rightmost crossing of red sights
from starting from here on this
like Translate of the Taurus but
then I have revealed that there
are no red crossings to the right of
it which ends up becoming interior
to the loop so I've revealed some
negative information interior to the loop
which I can't do because that'll ruin
the exponential decay of correlations and
now you'll end up with something with like
two sides red and two sides not red and
now you definitely don't have
the exponential tails that you did but
when you have three you can pick
a translator of the universal cover
depending on the site so that you can
reveal everything outside of this loop
without revealing anything
inside of the loop OK so
that's where this slide is saying so
well we know is.
From the work of do we know
consider a vicious untasty on there
having multiple phases is the same
as having exponential decay of so
here they prove things in the random
cluster representation but
to avoid the random cluster model I'm
using I'm writing things in the pots
representation and using like monotone the
city in just one color of the pots model
OK so the probability if you have that
there are multiple like the red and
blue phases for
the pots model then you have that.
In an N.
by N.
box put red boundary conditions at
infinity or far away then in an N.
by N.
box the probability that there is no red
loop around the origin decays
exponentially in the size of this thing so
that means that the probability that
having a cluster of not read sites
has exponential tails OK And
so what you do is
like I was suggesting you look at
your Taurus and you fix the site and
now you move to the universal
cover of the Taurus such that So
let's say we were looking at this
it's closer to this boundary so
I want to actually put it on my
right side nominal left side and so
we shift over on a tourist and
now we're looking at this site over here.
And now I reveal the outermost red loop
in this in this translate of the Taurus OK
either that outermost red loop goes
through this site or it's impossible for
this configuration to be pivotal
to my set so let's suppose that.
It went through this OK now I've revealed
all of this configuration and
nothing inside.
So I reveal thout our most loop
which I said is this thing and
nothing inside it and that's why I needed
three crossings to be able to do this
OK and now the privilege of this site
having been pivotal to the event
is the probability that there is
no red path like bypassing it.
But this this red loop dominates
red boundary conditions and
infinity so
the probability that there is no red by
passing thing is exponentially decaying
in this distance which is order and.
So on.
So that So using this moment in
a city in the discontinuity or
the presence of multiple gives
matter is just using the Twala G.
of the tourist to get
an exponential bottleneck so
that's why I said if this point is closer
to decide then I move it over here whereas
if it were closer to if it were so that
Sirius just focus on points in this box
if it were over here then I'd stay on this
side and reveal from that side so I always
keep it further from the inner strength
and that gives me the macroscopic amount.
And so you get an exponential tail on
the probability of any size pivotal to
this event conditional on being in the
event and the last thing to check is that
this set has mass less than or
equal to one half and
how do you check that well if you're not
in this set then you can also consider
like green crossings of the same style and
those two are complementary events and
they have the same mass so
they're mass like most water so
that I mean it's a really short
proof like three pages of.
Essentially giving us the sharp result
both in the low temperature and
at the critical point so.
That it really relies on
the topology of the Taurus and
so one question you might ask is what
it what if I'm not in a Taurus and so.
So this is how they prove the the
continuity of the phase transition when Q.
is less than or equal to four is
showing that you have polynomial decay
of correlations if and only if you
have a continuous phase transition and
that means that well that plus
either your polynomial or
exponential means that if you have
exponential decay of correlations then
you have a discontinuity of your face
transition if and only if those.
But without this condition you can you can
use the geometric construction as long as
you know some sort of exponential
decay on some sort of correlation or
dual correlation essentially.
And so what ends up happening is
if I start to consider different
boundary conditions now the boundary
conditions on my box can destabilize all
the other phases and
make them not mete stable anymore OK so.
In the low temperature using model this is
like a famous problem that's been studied
which is put plus boundary
conditions on an end by end box and
it should be the case that
instead of mixing time or
plus or minus obviously and
it should be the case that instead of
the mixing time being exponential
of the way it is with periodic or
free boundary conditions now it's actually
really fast it takes and squared and
governed by mean curvature flow and
there's a lot of literature trying to work
towards this originally Martinelli in
ninety four and then marginally Sonali and
Lubet skidmarks in only slight analyses so
the best thing is in all low temperatures
underpriced boundary conditions you
have and to the order a lot again.
But actually what ends up happening for
the.
So if we try to say all right
while the critical clocks model
when you have a discontinuous space
transition should look kind of in a low
temperature regime because you have
coexistence of phases so maybe I have
similar sensitivity to boundary conditions
and actually the most interesting thing is
the free boundary conditions even
though they're symmetric to all the Q.
different colors they even become
fast mixing because they do.
Stabilize all phases except for
the disordered one so
since you're at the critical point the
disordered phase is met a stable on its
own so if you put free boundary conditions
that will destabilize all of the Q.
ordered phases and you quickly will
converge to some disordered mess OK And so
that's what this picture is showing is if
I put periodic boundary conditions on this
thing there is if I put free
boundary conditions on this thing so
in the periodic case the largest
component will stay large forever for
an exponentially long whereas if I put
free boundary conditions it quickly
gets small and now you're just in a regime
where you just have small components.
We can break them and striation.
This is now same dynamics
same critical point but
I have conditions instead of periodic So
you see.
It starts to look like the picture that
you have where you have like a wolf
Crystal shape and
it shrinks down in time and
squared OK this is a big system so
it's not that small but.
You see that picture emerging.
And so.
So what we show is that if you consider
the Swenson Wang dynamics on so
we showed this once in mind dynamics at
the critical point on the Taurus is slow
using a similar proof
to the one I given so
we show that if you actually
have three boundary conditions
are red boundary conditions are like free
boundary conditions on three sides and
red on the fourth and
you have some exponential mixing time.
And again really this should be and
squared and
governed by the mean
cover curvature picture.
And the reason is because the conditions
are the red boundary conditions
destabilize all that one so then you
might ask well what if I considered other
boundary conditions like read and
read and free and free OK that slow
because it's hard to go from when you have
read up and down crossing to free us and
right crossing but you could look at it
like read on a little segment there and
a little segment there and free
elsewhere and we essentially classified
the different conditions inter
plating between the Taurus and
the all thread as either
exponential mixing time or
sub exponential mixing time for
the Swans and things.
And OK so
I guess my time's running out yeah.
So.
Large enough that cluster
expansion works so
large enough that you have
a large deviation estimates
on the interfaces are you that you
have positive surface tension.
So there's like several works
dedicated showing the same kind of
things that you have a low
temperature using hold a critical
temperature pots in a discontinuous phase
transition and that requires something
called Cluster expansion which means
I mean in several stages you say Q.
needs to be something
exponent of this something Q.
needs to be something exponential so Q.
needs to be larger than
either no one thousand.
But this picture should hold with all the
same results all the way up to Q bigger
than four and square for glob or
I don't know about this one.
Probably and
squared first ones in line also.
And also this mean curvature
picture everything.
OK so I want to finish up just by
returning to the model we started
with which was the mean field
Q State Potts model and
so we looked at what happens on
lot of says and Swensen Wang slows
down whenever you have a discontinuous
phase transition to exponential in N.
and so we showed with you offer as also
that whenever you have a discontinuous
phase transition in the mean field Potts
model this once annoying lower bound
is actually exponential and instead of
exponential in square for the mean field
model and so that matches the picture that
we got on the two dimensional Taurus.
And that proof relies on
a random graph estimate Gover.
Which might be of interest which is
just if you look at the number of
sites that are in large components but
not in the giant
component when you have a giant component
then that quantity has exponential tails
Thank you.
Thank you.