Christophe Raymond Garban: Good.
Christophe Raymond Garban: Well, everyone's environment is…
Courant Events Right: And beaches.
Courant Events Right: It's been systems with continuous symmetries.
Courant Events Right: Okay, Jill, I agree.
Courant Events Right: Okay, so, thank you very much, Rollo and Lina, for the invitation. It's a pleasure to speak in front of…
Courant Events Right: As many mathematicians and physicists.
Courant Events Right: So, as asked by Ronan and Nina, I will split the talk in two parts. The first part will be a review
Courant Events Right: Of spin systems which have a continuous symmetry.
Courant Events Right: So this will be the first part, and the second part, I will tell you, it will be a bit of rooftop.
Christophe Raymond Garban: MBL.
Courant Events Right: maybe, for the future. White, lemon, orange, and flat peach, follow different trajectories under the RG flow.
Christophe Raymond Garban: So, first part, first, so… I won't.
Courant Events Right: to tell you about, spin systems which live on a lattice, ZD.
Christophe Raymond Garban: But in this talk, almost.
Courant Events Right: all the time, dimension will be D is equal to 2, so it will be the Z2 grid.
Courant Events Right: And each point in this Z2 grid will carry a spin. Only the target space will change.
Courant Events Right: In the first picture here, the target space.
Christophe Raymond Garban: This is blessed.
Courant Events Right: minus 1. The underlying symmetry is discrete.
Courant Events Right: And it's, I think, well known to everyone in the room. It's called the easing model.
Courant Events Right: The second example here, which will be related to the beach later on.
Courant Events Right: is the XY model, so here, every point on Z2 carries a spin in the unit circle S1.
Courant Events Right: And this last example, which.
Christophe Raymond Garban: Plenty of.
Courant Events Right: One will be the orange.
Courant Events Right: It's the case where every spin carries an S2.
Christophe Raymond Garban: spin, and it's called the classical Eisenh.
Courant Events Right: their
Courant Events Right: The good news with these three models is that they share the same lattice, and they also share the same Gibbs measure.
Courant Events Right: Each configuration of spin sigma, whatever the target space is here, will be the.
Christophe Raymond Garban: Guarantee will be proportional to exponential beta.
Christophe Raymond Garban: Tony?
Courant Events Right: a sum of our edges, i next to J, so this is an edges of Z2, this is another edge of Z2, and each of these edges will carry the.
Christophe Raymond Garban: product sigma i times.
Courant Events Right: Sigma J.
Christophe Raymond Garban: So that's…
Courant Events Right: So when beta is large.
Christophe Raymond Garban: The system will be… will look very ordered.
Courant Events Right: At least in a large enough window.
Christophe Raymond Garban: And we'll see that if we maybe want to arbitrarily large windows, in this case here, the order.
Courant Events Right: Or maybe completely lost.
Courant Events Right: An equivalent way to consider these spin systems is to… the Gibbs measure is also exactly.
Christophe Raymond Garban: Exactly, experimentally.
Courant Events Right: And in short, minus beta.
Christophe Raymond Garban: or the two.
Courant Events Right: This time.
Christophe Raymond Garban: sum over N.
Courant Events Right: is not the Scala product, but an energy, it would be the D.ichlet energy along the edge.
Courant Events Right: I will sum over all the edges of the
Courant Events Right: The gradient of the spin squared.
Christophe Raymond Garban: So…
Courant Events Right: Depending on your viewpoint.
Christophe Raymond Garban: Either you won't.
Courant Events Right: to maximize the sum of scalar product.
Christophe Raymond Garban: Or, maybe.
Courant Events Right: Maybe, if you're a physicist, you want to minimize the energy, so you want to minimize the dirichlet energy you pay over each edge of your graph.
Courant Events Right: Okay, so… That's the three spin systems on the.
Christophe Raymond Garban: I mentioned, too, that I will.
Courant Events Right: Beta will always be the usual.
Christophe Raymond Garban: inverse temperature.
Courant Events Right: So, what I will do…
Christophe Raymond Garban: in this first part of the talk is, for each of.
Courant Events Right: these models, I would want to give you
Courant Events Right: The way physicists think about the spin system, or at least.
Christophe Raymond Garban: way, I think I understood what physicists.
Courant Events Right: Trying to do with this.
Christophe Raymond Garban: It's so much.
Courant Events Right: try to translate the way I understood.
Christophe Raymond Garban: Completely misunderstood.
Courant Events Right: I will also try to give you the mathematical way, and I will proceed as follows. For the easing model, I will only give you the physics way.
Christophe Raymond Garban: But.
Courant Events Right: The masro is extremely close.
Christophe Raymond Garban: Northwestern, all of what.
Courant Events Right: I'm going to do.
Christophe Raymond Garban: The boat would be 2 minutes long and almost 100 years old.
Courant Events Right: For the XY model, I insist I will give the two ways, and there, they are quite different from each other.
Christophe Raymond Garban: I will.
Courant Events Right: give you the.
Christophe Raymond Garban: The way business.
Courant Events Right: have thought about it, and then I will give you the way Frollish and Spencer have thought about it.
Courant Events Right: For the last model here, I wish I would give you a physics and a math way, but I will only be able to give you a physics way. The math way is lacking so far.
Christophe Raymond Garban: So, first.
Courant Events Right: The easing model, so that's the case where.
Christophe Raymond Garban: There is a discrete symmetry, so that's not supposed.
Courant Events Right: to be the overview Roland and Nina asked about, so there's no underlying.
Christophe Raymond Garban: buying continuous.
Courant Events Right: Symmetry in the.
Christophe Raymond Garban: target space. Still, it's, of course.
Courant Events Right: the next model, and that's a simulation stolen from the web.
Christophe Raymond Garban: Where…
Courant Events Right: I have a high temperature easing model here. I have the exactly critical easing model here with its nice fractal structures.
Christophe Raymond Garban: And when.
Courant Events Right: When beta is large, the system picks.
Christophe Raymond Garban: A color, plus or minus, black or white.
Christophe Raymond Garban: Things to eat, and there are.
Courant Events Right: Only slow, villages.
Christophe Raymond Garban: Resisting to this.
Courant Events Right: ground states, long-range order.
Christophe Raymond Garban: So, just for.
Courant Events Right: a couple of minutes. It's a little bit risky for me, but I will try to give you maybe the way physicists think about it. If you think that it's completely wrong, please tell me after the talk.
Christophe Raymond Garban: So that they keep some… some…
Courant Events Right: visibility.
Christophe Raymond Garban: So…
Courant Events Right: So,
Christophe Raymond Garban: I will hide it.
Courant Events Right: Mayor's argument with some words like excitations and things.
Christophe Raymond Garban: like this.
Courant Events Right: This is a… imagine a huge piece of 2D lattice with a way to break the symmetry outside, which would be a plus boundary condition outside.
Christophe Raymond Garban: the blue condition.
Courant Events Right: And now, we want to understand how the easing model is reacting to temperature, to the fact beta.
Christophe Raymond Garban: is not infinite temperature T.
Courant Events Right: P is non-zero.
Christophe Raymond Garban: And there are thermal fluctuations.
Courant Events Right: how to preempt this? Maybe there is this ground state, which is the blue ground state, reacting to the bonding.
Christophe Raymond Garban: And now…
Courant Events Right: There are some possible excitations which will break this, this bronze
Courant Events Right: This excitation could be represented by a loop, a closed-loop.
Christophe Raymond Garban: And we can see that one good loop has.
Courant Events Right: a certain energy associated to it, and it's essentially beta.
Christophe Raymond Garban: Sometimes the instant.
Courant Events Right: So, essentially, the length of the loop.
Courant Events Right: And so the energy of one such excitation in the Gibbs state is exponential minus beta times this length.
Courant Events Right: The entropy is the number of such excitations of length n
Courant Events Right: And there is a classical fight between energy and entropy.
Christophe Raymond Garban: I insist.
Courant Events Right: on these arguments, because I will use the same argument
Courant Events Right: for the phase transition for the X1.
Christophe Raymond Garban: Family.
Christophe Raymond Garban: So, if beta is large.
Courant Events Right: You pay a lot.
Christophe Raymond Garban: of energy.
Courant Events Right: And the effect of this.
Christophe Raymond Garban: This is dead.
Courant Events Right: You need for me in the size.
Christophe Raymond Garban: of the system.
Courant Events Right: Of course, there will be some excitations here and there, but the probability to see a massive excitation of the scale of your domain can easily be seen to be very unlikely. So you will only see rather small excitations, and in the bulk of the system.
Christophe Raymond Garban: him.
Courant Events Right: The ground state will still be there, and you will still.
Christophe Raymond Garban: bring this imagery.
Christophe Raymond Garban: As beta gets lower.
Courant Events Right: the…
Christophe Raymond Garban: Zvich is large.
Courant Events Right: and temperature gets lower.
Christophe Raymond Garban: lower,
Courant Events Right: No, the opposite way.
Christophe Raymond Garban: always confuse beta and T.
Courant Events Right: As temperature gets higher, as beta gets lower.
Christophe Raymond Garban: Excellent.
Courant Events Right: are less costly in terms of the.
Christophe Raymond Garban: energy. Entropy is still there, and so after a while, you start having.
Courant Events Right: more and more.
Christophe Raymond Garban: More excitations piling up on each other.
Christophe Raymond Garban: In such an efficient way that in the bulk of the system, you completely forgot.
Courant Events Right: good about.
Christophe Raymond Garban: the symmetry.
Courant Events Right: breaking, you applied at the.
Christophe Raymond Garban: on there.
Christophe Raymond Garban: So this is, I think.
Courant Events Right: think the way physicists may explain easing model, though I could be completely wrong here.
Courant Events Right: So, I want to move now to continuous symmetry, and this was the second example, the.
Christophe Raymond Garban: example in my talk, where each point in Z2.
Courant Events Right: So this is a 500, 500 pictures stolen from.
Christophe Raymond Garban: Give me the…
Courant Events Right: each point carries a spin in the unit circle S1, and this, pixel.
Christophe Raymond Garban: That is colored, according to the angle here.
Courant Events Right: If the temperature t is very large, if beta is very small, this is a very busy.
Christophe Raymond Garban: problems being true.
Christophe Raymond Garban: Well, if the temperature.
Courant Events Right: is low. This looks like we have more structure coming into the picture, but still, there is no dominant.
Christophe Raymond Garban: There's no domin.
Courant Events Right: England.
Christophe Raymond Garban: angle with.
Courant Events Right: small fluctuation.
Christophe Raymond Garban: around the.
Courant Events Right: And this is for a well-known reason. This goes back to a theorem by Mermin and Wagner in the 60s, and what they've shown is that in demonstrating.
Christophe Raymond Garban: is equal to.
Courant Events Right: If you take any spin system which has a continuous symmetry, it could be XY, it could be.
Christophe Raymond Garban: Hi, man.
Courant Events Right: Or it could be a Lee group-valued spin system with a continuous symmetry there.
Christophe Raymond Garban: Like.
Courant Events Right: SO3, for example.
Christophe Raymond Garban: Then.
Courant Events Right: They've shown that it's impossible to break the symmetry.
Courant Events Right: And the main argument they relied on is called an harmonic map.
Christophe Raymond Garban: It's appropriate.
Courant Events Right: he was.
Christophe Raymond Garban: in,
Courant Events Right: Harmonic analysis, which is such that if you.
Christophe Raymond Garban: I'm trying to break the symmetry outside.
Courant Events Right: For example, here with the east pointing direction.
Christophe Raymond Garban: And you.
Courant Events Right: Ask yourself, how is it energetically costly to point in some other direction in the bulk of the system?
Courant Events Right: I found out.
Christophe Raymond Garban: That there is a deterministic.
Courant Events Right: of angles in this end by inbox, which is such that.
Christophe Raymond Garban: You can…
Courant Events Right: slowly interpolate from one to the other in such a way that you only pay a dirichlet energy, which vanishes to zero as the.
Christophe Raymond Garban: Because of this domain.
Courant Events Right: It grows to infinity.
Christophe Raymond Garban: If you want to.
Courant Events Right: to do the exercise and get bored at some point.
Courant Events Right: If you linearly interpolate at linear speed.
Christophe Raymond Garban: on the Pandailles.
Courant Events Right: to the middle, this will not go to zero. So you need to do something a little bit.
Christophe Raymond Garban: Smarter.
Courant Events Right: Thank you.
Courant Events Right: And since it costs nearly zero energy to forget about the Pandari direction.
Christophe Raymond Garban: They, could show that there is no phase… no symmetry breaking for.
Courant Events Right: these systems in Dimension 2.
Courant Events Right: And I think I was not today.
Christophe Raymond Garban: In the 60s, but.
Courant Events Right: I've heard some videos At the time.
Christophe Raymond Garban: There's also.
Courant Events Right: okay, probably XY model, classical Eisenberg, all of those in dimension 2, they're pretty.
Christophe Raymond Garban: boring, because there's.
Courant Events Right: know.
Christophe Raymond Garban: bring.
Courant Events Right: of symmetry And it was maybe considered a little bit like a one-dimensional easing model.
Courant Events Right: That was until the early 70s, when Berzynski, Kostoletz, and Taurus Phone that's, basically.
Christophe Raymond Garban: sites, this serum by Mamine and Wagner. Still.
Courant Events Right: Those systems are very interesting.
Christophe Raymond Garban: They, they have a…
Courant Events Right: More subtle phase transition, at least if the symmetry is a billion.
Christophe Raymond Garban: And they noticed that.
Courant Events Right: to… Atlanta.
Christophe Raymond Garban: such temperature, the…
Courant Events Right: coupon function, which measures.
Christophe Raymond Garban: How the angle at that part of the.
Courant Events Right: domain correlated.
Christophe Raymond Garban: Thanks for dinner.
Courant Events Right: at that point.
Christophe Raymond Garban: part of the domain.
Courant Events Right: They noticed that this must decay explod.
Christophe Raymond Garban: Eventually, quickly, and this is quite.
Courant Events Right: Easy to show.
Christophe Raymond Garban: with two days' techniques when the temperature is large, so there's no surprise here.
Courant Events Right: The surprise was that if the.
Christophe Raymond Garban: Temperature is low, certainly the system.
Courant Events Right: Enter the phase where the decay here.
Christophe Raymond Garban: becomes collar.
Christophe Raymond Garban: And that's.
Courant Events Right: at least in this U-shaped.
Christophe Raymond Garban: We should all agree that.
Courant Events Right: Ovalu is now.
Christophe Raymond Garban: Right? This is maybe.
Courant Events Right: Maybe a sign of conformal field theory, of large-scale structures, of lots of good mathematics.
Courant Events Right: And here, we have an antagonist.
Christophe Raymond Garban: line of, of models, when I vary the time.
Courant Events Right: temperature below the face foundation.
Courant Events Right: And each of those will be associated to some critical exponents which encode this decay.
Courant Events Right: If we go back to the easing model, we would have.
Christophe Raymond Garban: One point, on the other point, maybe, of the.
Courant Events Right: your oppon Only at that point, you would have power load decay.
Christophe Raymond Garban: He is.
Courant Events Right: It's very different. At that point, included and below, we have power low.
Courant Events Right: So, what I want to do
Courant Events Right: in the next few slides is to give you the physics way of understanding this phase transition.
Courant Events Right: Which is beautiful, I find.
Christophe Raymond Garban: And then I will give you the mass way. It's not orthogonal to the physics way, but I claim it.
Courant Events Right: It's a little bit different.
Courant Events Right: So, how did Berzinski, Kostoletz Staus.
Christophe Raymond Garban: explained this phase from the They started with the Gibbs measure.
Christophe Raymond Garban: which I had written before, so it was.
Courant Events Right: financial beta Some over edges of my graph, which.
Christophe Raymond Garban: So the nearest neighbor ages.
Courant Events Right: Scala product.
Christophe Raymond Garban: of the S1 spin at I with the S1 spin at the point just next to it, which is J.
Christophe Raymond Garban: The scalar product of S1 value spins, if you.
Courant Events Right: You know about.
Christophe Raymond Garban: This is just a cosine of the difference of angles.
Christophe Raymond Garban: So, a good idea is to reparametrize the S1 value.
Courant Events Right: spin system by 0 to 2 power.
Christophe Raymond Garban: angles theta i, where.
Courant Events Right: Why are still the vertices.
Christophe Raymond Garban: of Minecraft.
Courant Events Right: So you can write down the Gibbs measure this way.
Christophe Raymond Garban: And now you… Try to tailor-expand this cosine.
Courant Events Right: With the idea that if beta is very large, maybe the gradient will tend to be typically very small.
Courant Events Right: So if you do this, you, at large scales, you may understand your Gibbs.
Christophe Raymond Garban: measure as exponential minus beta over 2.
Courant Events Right: And the energy that you are analyzing.
Christophe Raymond Garban: I think it's a Dirichlet Energy.
Courant Events Right: on the field.
Christophe Raymond Garban: of angers.
Courant Events Right: And now comes May.
Christophe Raymond Garban: Maybe the main, observation.
Courant Events Right: This field of angles in this two-dimensional plane, you should not think of it as a.
Christophe Raymond Garban: 0 form as a function, but instead as a one form.
Christophe Raymond Garban: Why that?
Courant Events Right: But, if I take a point here.
Christophe Raymond Garban: And the point here?
Courant Events Right: And now I take any pass which goes from this first point to the second point.
Courant Events Right: If beta is very large, if the temperature is very small, the gradients of angles are going to be very, very tiny.
Christophe Raymond Garban: So.
Courant Events Right: It will be possible to make.
Christophe Raymond Garban: So, how does the angle move along.
Courant Events Right: this path? How do I move continuously along the S1.
Christophe Raymond Garban: Let's see here.
Courant Events Right: Okay.
Christophe Raymond Garban: But if I were to take a completely different path here, it will still be very well defined, but it might.
Courant Events Right: Maybe that when.
Christophe Raymond Garban: And reach the same final point.
Courant Events Right: The angle that I will have computed will be the same modulo and monodromy of plus 2 pi plus 4 pi minus 2 pi.
Courant Events Right: So, all this to say.
Courant Events Right: a path, I can associate a gradient of angle, but different paths may not agree, and they may accumulate monodromies in between.
Courant Events Right: This meant that these.
Christophe Raymond Garban: Random, one form that we're trying to understand.
Christophe Raymond Garban: Its randomness may arise from two things.
Courant Events Right: arise from a zero form, on the one hand.
Christophe Raymond Garban: And it may also arise from a.
Courant Events Right: Two forms if this one form happened to be non-exact.
Courant Events Right: So, that's the basis of their analysis, and if the model is slightly different from the XY model, it's called the VIN model, I will not define it, but it's a very close cuisine.
Christophe Raymond Garban: It's good.
Courant Events Right: It can be rigorously proved that the zero form.
Christophe Raymond Garban: So…
Courant Events Right: of randomness is actually independent from the two-form source of.
Christophe Raymond Garban: random.
Courant Events Right: This one is known as.
Christophe Raymond Garban: Gaussian-free field, and this gives rise to the.
Courant Events Right: complex, exponential.
Christophe Raymond Garban: GFF, which is a very nice.
Courant Events Right: objective.
Christophe Raymond Garban: its own, as we've seen in previous talks.
Courant Events Right: But if you take the complex exponential of a GFF, you will never see any BKD transition.
Courant Events Right: The only mechanism to break power law into exponential low will come from the coolant gas, which I will not define.
Christophe Raymond Garban: I haven't took.
Courant Events Right: about in the next slide.
Christophe Raymond Garban: So, what is…
Courant Events Right: this cooling gaps.
Christophe Raymond Garban: F?
Christophe Raymond Garban: The source of randomness coming from the two forms describes.
Courant Events Right: Places where we do accumulate monodromies around these plackets.
Courant Events Right: And the argument of Berzinski, Kosolitz, and Taurus is that every time.
Christophe Raymond Garban: you have.
Courant Events Right: Monodromy, such a defect that will do topological defects.
Courant Events Right: You will need to find somewhere else an anti-vortex, so that if you take the integral around the big box, you don't accumulate too much angles around the box.
Courant Events Right: So typically, a vortex will come.
Christophe Raymond Garban: With an anti-vortex.
Courant Events Right: At a certain distance.
Courant Events Right: And now.
Christophe Raymond Garban: a similar argument as I did for the.
Courant Events Right: easy model. It was targeted to do it now for the XY model.
Courant Events Right: You can ask yourself what's.
Christophe Raymond Garban: It's the energy that you.
Courant Events Right: Pay if you want to produce an isolated pair of vortex and anti-vortex.
Courant Events Right: It's easy to check that if you try to compute the sum.
Christophe Raymond Garban: over the edge.
Courant Events Right: of the gradient squared here.
Christophe Raymond Garban: What you will…
Courant Events Right: And the quiz isn't.
Christophe Raymond Garban: constant times the log of the Euclidean.
Courant Events Right: The distance between the vortex and the antivortex.
Courant Events Right: So each isolated pair of vortex and antivortex will come with a log n energy price.
Courant Events Right: In such a way that if beta is small, if the temperature is high.
Christophe Raymond Garban: Every pair of vortex will.
Courant Events Right: Pay this polynomial cost.
Courant Events Right: But the…
Christophe Raymond Garban: attribute.
Courant Events Right: way of choosing a vortex and an antivortex is n to the 4, basically, so the entropy at large temperature may easily.
Christophe Raymond Garban: in over.
Courant Events Right: the energy.
Courant Events Right: So…
Christophe Raymond Garban: No.
Courant Events Right: As beta grows, at some point, this energy term will start to be too much compared to the entropy, and that's basically where the BKT transition will happen.
Courant Events Right: So there is an analog situation, it's called.
Christophe Raymond Garban: two components.
Courant Events Right: coolant gas in R2.
Christophe Raymond Garban: And there's a…
Courant Events Right: Very nice recent work.
Christophe Raymond Garban: Jean Bourussier.
Courant Events Right: Sylvia Serati.
Christophe Raymond Garban: Where the…
Courant Events Right: analyze how the.
Christophe Raymond Garban: is, voice.
Courant Events Right: Vortices and antivortices cluster together.
Christophe Raymond Garban: all the way to the BKT transition.
Christophe Raymond Garban: So back…
Courant Events Right: to the lattice?
Christophe Raymond Garban: We've beaten.
Courant Events Right: is now very large.
Christophe Raymond Garban: Nope.
Courant Events Right: What happens is that you will still have vortices and antivortices everywhere.
Christophe Raymond Garban: revenue.
Courant Events Right: Huti domain But it would be very hard for them to be separated from each other at macroscopic distance.
Courant Events Right: Yes, the easy computation here is for one isolated pair. If you start having a typical configuration, which is a
Courant Events Right: positive density of vortices, it's much harder to make sense of this.
Courant Events Right: And that's why I…
Christophe Raymond Garban: Right.
Courant Events Right: It's hard to make these, heuristics by Berzinski Cosarest.
Christophe Raymond Garban: a rigorous.
Courant Events Right: Okay. But if you just had one pair of vortices on two vertices, then.
Christophe Raymond Garban: You can compute the optimal way of achieving this, it's like a harmonic map, and this costs log N.
Courant Events Right: And so this argument is for comparing
Courant Events Right: Very sparse collection of pairs with having nothing at all.
Christophe Raymond Garban: Exactly. So, maybe, compared to the easing model, I should say that.
Courant Events Right: The XY model has this GFF fluctuation, which… which will.
Christophe Raymond Garban: always be power low. And on top of this GFF, you have these vortices which will.
Courant Events Right: even more disorder.
Christophe Raymond Garban: And when you have tons of those, it far only.
Courant Events Right: from each other, it actually produces exponential decay.
Courant Events Right: So this transition is also called the vertical.
Christophe Raymond Garban: on.
Courant Events Right: Vortex binding, unbinding phenomenon.
Christophe Raymond Garban: And that's the physics way.
Courant Events Right: Unless there are questions, I want you to go to the mathematical.
Christophe Raymond Garban: right now.
Christophe Raymond Garban: And this is the thing.
Courant Events Right: also stolen from the web managers, tried to indicate the vortices.
Christophe Raymond Garban: Versus the online.
Courant Events Right: offices, hopefully.
Christophe Raymond Garban: There's nothing communicate?
Courant Events Right: So, I will move now to the way.
Christophe Raymond Garban: Matthew, mate.
Courant Events Right: Politicians have managed to… Inc.
Christophe Raymond Garban: prove these, predictions from BK teams? Yep.
Courant Events Right: There is some… can you come up with some artificial model.
Christophe Raymond Garban: Where you could tune the progress into vortices so that.
Courant Events Right: To be clear that.
Christophe Raymond Garban: The signal bear domain.
Courant Events Right: It's a very pillow.
Christophe Raymond Garban: facility.
Courant Events Right: religion.
Christophe Raymond Garban: I wore the cool guys.
Christophe Raymond Garban: So, for the particular latest model you are considering, of course, you don't have the.
Courant Events Right: motion.
Courant Events Right: You guys can do speed.
Christophe Raymond Garban: So I don't know if this answers.
Courant Events Right: the question, but you can, on the lattice, you can make a Coulomb gas which has a Z to the power, the number of… and if you do this, then there is a very nice diagram which captures Z and beta simultaneously.
Christophe Raymond Garban: And along this.
Courant Events Right: diagram, you can see the RG flow.
Christophe Raymond Garban: And…
Courant Events Right: There is a very nice work.
Christophe Raymond Garban: I wouldn't, I wouldn't.
Courant Events Right: state this work in 5 minutes, where in some sense, they use this
Courant Events Right: small fugacity explanation. And where are you in better shape with respect to this experiment? So, as far as I understand, if your fugacity is extremely small.
Christophe Raymond Garban: then you.
Courant Events Right: clean them.
Courant Events Right: shape, you can actually even compute exponents.
Christophe Raymond Garban: So there is a… Pierre Lucie Falcobe.
Courant Events Right: There is also a work I will mention.
Christophe Raymond Garban: of.
Courant Events Right: Borisin Park in Rodriguez Apottis.
Courant Events Right: Could you repeat the questions?
Christophe Raymond Garban: It's hard to…
Courant Events Right: Oh, yeah, so the question was.
Christophe Raymond Garban: here, I will rephrase it here. If I look at.
Courant Events Right: the Villan model, or the XY model,
Christophe Raymond Garban: They are what.
Courant Events Right: Vortices and antivortices all over the place.
Courant Events Right: try to generalize this model by adding augacity which maybe forces more vortices or less vortices, so it's another parameter which does not exist in this model.
Courant Events Right: And if you play with this fugacity, when the fugacity is small, you can reach more phenom energy than.
Christophe Raymond Garban: In these material elections.
Courant Events Right: more exponents.
Courant Events Right: It's additional modified the line.
Christophe Raymond Garban: The field which is part of the plaquettes, you treat as a background field, I mean.
Courant Events Right: Right, so, so I think, I will try to rephrase. So, is it a general… the question is, is it, like, in the general.
Christophe Raymond Garban: horizon.
Courant Events Right: Pilot.
Christophe Raymond Garban: So I think…
Courant Events Right: Indeed, the VLAN model is something you could see as a model which leads along the edges of the graph.
Courant Events Right: And you could ask for each placket, you could penalize that it's actually trying to make a monodromy there. And then, yes, indeed, it would be a bit like.
Christophe Raymond Garban: having a small figure.
Courant Events Right: CTZ. That's correct.
Christophe Raymond Garban: Okay, so in…
Courant Events Right: in, in Nikola's, Right.
Christophe Raymond Garban: Very balanced so far.
Courant Events Right: Nope.
Christophe Raymond Garban: Okay, so…
Courant Events Right: A few words about the…
Christophe Raymond Garban: The way it has been understood.
Courant Events Right: in math, so the.
Christophe Raymond Garban: the brick…
Courant Events Right: Through work here was.
Christophe Raymond Garban: in…
Courant Events Right: in the early 80s by Frolis and Spencer.
Christophe Raymond Garban: And, to make the story short.
Courant Events Right: I could maybe say the following. If you want to prove that a phase transition holds for a S1 value.
Christophe Raymond Garban: a good model, which…
Courant Events Right: Goes from exponential to quasi-long-range order.
Christophe Raymond Garban: There is a difficulty, which is that.
Courant Events Right: You don't have any order parameters.
Christophe Raymond Garban: to play with a pre-ring.
Courant Events Right: the idea.
Christophe Raymond Garban: That's too…
Courant Events Right: Go to a different.
Christophe Raymond Garban: where the order parameter was easier to find, and what they've done is that.
Courant Events Right: They used that the four-year.
Christophe Raymond Garban: dual of S1 is Z. So they basically wrote this model in the Fourier,
Courant Events Right: side, so they Fourier transform the XY or the VIN model, and they end up with a dual model, which is now living on the plackets of Z2, but the.
Christophe Raymond Garban: C is equal.
Christophe Raymond Garban: And each…
Courant Events Right: Each such placket will now carry not a S1-valued spin, but a Z-valued spin.
Christophe Raymond Garban: So this is called a…
Courant Events Right: Z valued height function.
Courant Events Right: And the easy part of BKT, at least on the lattice, is sent to the easy part of the dual model, which is a low temperature.
Christophe Raymond Garban: Z-valued, head function.
Courant Events Right: model, which.
Christophe Raymond Garban: Locally…
Courant Events Right: At least at.
Christophe Raymond Garban: Very high temperature, yeah, very low, the dual temperature is the inverse.
Christophe Raymond Garban: Payroll's argument comes back, excitations, as I did before, comes back.
Courant Events Right: The model will typically be very flat.
Courant Events Right: The difficult part was here.
Christophe Raymond Garban: Which is incorrect.
Courant Events Right: with the difficult part of BKT, and there…
Christophe Raymond Garban: They were able to prove that.
Courant Events Right: the Z-valued Gaussian-free field, or the dual of the XY model, which is a bit more complicated to write down.
Christophe Raymond Garban: They look…
Courant Events Right: localizes at,
Christophe Raymond Garban: low temperature for XY, which is close to high temperature for the dual model.
Christophe Raymond Garban: This delocalization property was the hard part.
Courant Events Right: Oof!
Courant Events Right: And,
Christophe Raymond Garban: In fact.
Courant Events Right: The way that the proof uses a little bit intuition.
Christophe Raymond Garban: the Coulomb gas, a little bit of intuition from binding on binding, but not.
Courant Events Right: Exactly.
Christophe Raymond Garban: bet.
Courant Events Right: Is this C-valid height function same?
Christophe Raymond Garban: So the Z value, the height function, it does.
Courant Events Right: It doesn't have monotone.
Christophe Raymond Garban: It's a really Oh, good question. It… so…
Courant Events Right: If I take a VLAN.
Christophe Raymond Garban: and I look at the fields of.
Courant Events Right: monodrome is. This is a very nice poeticistic object, it's exactly a cooling gas, so it's a.
Christophe Raymond Garban: integer-valued Gaussian.
Courant Events Right: This model is exactly the information to.
Christophe Raymond Garban: early, so… Not a monotopy, but very related.
Christophe Raymond Garban: Isn't it true that in this…
Christophe Raymond Garban: mathematics Pro, there's also an interesting The education group under.
Courant Events Right: interpretation on the hard part, where you're trying to.
Christophe Raymond Garban: to start.
Christophe Raymond Garban: You don't.
Courant Events Right: won't.
Christophe Raymond Garban: which has a uniform Distribution fields, and then it is determined by some periodic potential.
Christophe Raymond Garban: Right, so I think these…
Courant Events Right: hard refloat.
Courant Events Right: That you are mentioning is exactly.
Christophe Raymond Garban: I will mention it in this slide.
Courant Events Right: I think visiting.
Christophe Raymond Garban: Not even that physics is not… physics, people think in terms of both of them, especially nowadays.
Christophe Raymond Garban: Okay, make me know what it is.
Courant Events Right: I read Blazinski Costco Restore as the compute login.
Christophe Raymond Garban: Again, we say expansion.
Courant Events Right: tremendous detail.
Christophe Raymond Garban: It didn't find until…
Christophe Raymond Garban: But, okay, I think what you have in mind, maybe I will switch to this one. If you take this dual model.
Courant Events Right: measure.
Christophe Raymond Garban: So, beta is large.
Courant Events Right: is… this is a…
Christophe Raymond Garban: I inverted the usual temperature.
Christophe Raymond Garban: So, in…
Courant Events Right: In this case.
Christophe Raymond Garban: case,
Courant Events Right: If you are at a… Very large beta.
Christophe Raymond Garban: Yeah, which is, like.
Courant Events Right: very high temperature Z-valued GFF.
Christophe Raymond Garban: Bojmines Paprodeguise, in a very important work, have shown that.
Courant Events Right: At large scale.
Christophe Raymond Garban: This will look like a GFF.
Christophe Raymond Garban: But the temperature may be…
Courant Events Right: that it's.
Christophe Raymond Garban: Please.
Christophe Raymond Garban: And…
Courant Events Right: Here, indeed, it's.
Christophe Raymond Garban: So, in.
Courant Events Right: in the realm of sign.
Christophe Raymond Garban: And there is an activity, and you run it slow.
Courant Events Right: And this also follows a physics argument, I fully agree with this, but the 70s, I'm not.
Christophe Raymond Garban: As sure as that.
Christophe Raymond Garban: So there's…
Courant Events Right: risky work?
Courant Events Right: Proof of the…
Christophe Raymond Garban: Here, as I mentioned, was by Fish and Spencer. There is a…
Courant Events Right: And more…
Christophe Raymond Garban: taught by people.
Courant Events Right: So, his work is based more on the techniques developed by.
Christophe Raymond Garban: Good Sheffield.
Christophe Raymond Garban: Based on this new proof by Pete Lovers, there are now new proofs of the BKTS condition. All of these proofs booked.
Courant Events Right: through the.
Christophe Raymond Garban: roulette compound?
Christophe Raymond Garban: by von England Poglis and Eisenman, Ariel, and Shapiro, both works, give a very nice new point of view.
Christophe Raymond Garban: More modest, but with…
Courant Events Right: We could show that.
Christophe Raymond Garban: If you look at this effective temperature, the hard part is the existence of it. We could show that it must deviate from what the GFF would do, so that's the contribution of the vortices.
Christophe Raymond Garban: And…
Christophe Raymond Garban: Last item, maybe.
Courant Events Right: You don't.
Christophe Raymond Garban: many directions. Last year, Earth and.
Courant Events Right: and Schweiger, they could extend the delocalization to non-communic.
Christophe Raymond Garban: insurance.
Courant Events Right: And, today on Africa, you would have.
Christophe Raymond Garban: A different proof of this.
Courant Events Right: per W.
Christophe Raymond Garban: And delivering them.
Courant Events Right: Angular.
Christophe Raymond Garban: Maybe one interesting aspect of this is that if you take the two.
Courant Events Right: mad.
Christophe Raymond Garban: of these. There are more general S1-valued models. They include the Villan, EXY, and many others.
Christophe Raymond Garban: And one consequence of this paper is that all of you also have a BKT transformation, so it extends the possible models which you have PK.
Courant Events Right: A last sentence about BKT, which.
Christophe Raymond Garban: There were some, talks of.
Courant Events Right: followed by.
Christophe Raymond Garban: what's going on in hyperbolic geometry. So, I…
Courant Events Right: I have a…
Christophe Raymond Garban: A work in progress with.
Courant Events Right: Do you think?
Christophe Raymond Garban: So we identify a BK transition on the H2, which exists on H3.
Christophe Raymond Garban: And just to explain.
Courant Events Right: in one sitting where this comes from, there is the same battle.
Christophe Raymond Garban: energy.
Courant Events Right: entropy, then we have.
Christophe Raymond Garban: invaluable.
Courant Events Right: Why that? If I were to do an XY model.
Christophe Raymond Garban: along this hyperbolic plane. Maybe I will also put.
Courant Events Right: Spotify.
Christophe Raymond Garban: vortices and antivortices, and as I did before, it's not very hard to.
Courant Events Right: See that if I have a plus and a minus.
Christophe Raymond Garban: I don't…
Courant Events Right: symbolic distance and from its.
Christophe Raymond Garban: This will cause a little bit of energy and distance. Linar in the hyperbolic distance, while in two dimensions can cause logic in the distance.
Christophe Raymond Garban: But now, the battle is as in dimension 2 at exponential minus beta times.
Courant Events Right: by February.
Christophe Raymond Garban: Jasons?
Courant Events Right: But the volume in hyperbolic graph itself is also exponential.
Courant Events Right: So when you look at it, you feel that depending.
Christophe Raymond Garban: On the value of beta, you may have.
Courant Events Right: either go pick binding.
Christophe Raymond Garban: All of this combined, and that's what we do in this workplace.
Christophe Raymond Garban: Shouldn't just follow from following barriers for the Facebook.
Courant Events Right: But the first transaction here is performing better in both.
Christophe Raymond Garban: distinguishing.
Christophe Raymond Garban: Blessed.
Christophe Raymond Garban: I think…
Courant Events Right: Here, it's a bit different, because the S1
Courant Events Right: very different face transition. It will have, actually.
Christophe Raymond Garban: intercom.
Courant Events Right: No, no, H3 is the basic place.
Christophe Raymond Garban: I mean.
Courant Events Right: is from the S1.
Christophe Raymond Garban: unvalued system.
Courant Events Right: This is good.
Christophe Raymond Garban: Thank you.
Christophe Raymond Garban: a long range of face. Very different from the poisoning other.
Courant Events Right: Unituent.
Courant Events Right: So, we can discuss later, but I think it's quite different with energy. But the entropy energy, I agree, is fairly simple.
Christophe Raymond Garban: Yep.
Courant Events Right: Okay, that's a good point.
Christophe Raymond Garban: Maybe I can…
Courant Events Right: transfer, saying that we first tried to do the Frawley-Spencer argument.
Christophe Raymond Garban: But…
Courant Events Right: Push it further to try to have the.
Christophe Raymond Garban: Binding, binding there.
Christophe Raymond Garban: But because of the volume growing this fast.
Courant Events Right: Combinator.
Christophe Raymond Garban: it's collapsed. So, at least we are not a.
Courant Events Right: But we did face an issue here.
Christophe Raymond Garban: I'm sorry.
Courant Events Right: Anything else?
Christophe Raymond Garban: Yes?
Christophe Raymond Garban: What might be?
Christophe Raymond Garban: But indeed…
Courant Events Right: So, if you are in Z2, you have BKT, Z3, you
Courant Events Right: D4 you don't, and so on, and then H2 strikes back, and then H3 is lost again. So, H2 has a very…
Christophe Raymond Garban: Guys.
Courant Events Right: economic analysis.
Christophe Raymond Garban: Particularly.
Courant Events Right: reality that, there are…
Christophe Raymond Garban: many ways.
Courant Events Right: to virtual harmonic functions.
Christophe Raymond Garban: Real quick.
Christophe Raymond Garban: Go ahead.
Courant Events Right: Beta effective is equal to beta in this case.
Christophe Raymond Garban: Guess.
Courant Events Right: Oh, that's a good question. I should know, I don't…
Christophe Raymond Garban: I don't, I don't…
Courant Events Right: So, but it's a good question.
Courant Events Right: So.
Christophe Raymond Garban: So, the question…
Courant Events Right: by Renault is, is the beta effective the same as the beta here, as opposed to two dimensions?
Christophe Raymond Garban: So.
Courant Events Right: what do we mean by beta effective? So there will be a… a G.
Christophe Raymond Garban: That's true.
Courant Events Right: indexed by.
Christophe Raymond Garban: Yeah, the vertex.
Courant Events Right: of the hyperbolic graph. I still think restricting it to the.
Christophe Raymond Garban: The integers will be lower.
Christophe Raymond Garban: it's…
Courant Events Right: We've been discussing a lot this year together.
Christophe Raymond Garban: and every time you were right, so I don't want to…
Christophe Raymond Garban: Wants to go too far.
Christophe Raymond Garban: So, I want to.
Courant Events Right: go to, the classical Eisenberg now for.
Christophe Raymond Garban: a short time.
Christophe Raymond Garban: And…
Courant Events Right: So now, every spin carries a.
Christophe Raymond Garban: as.
Courant Events Right: too.
Christophe Raymond Garban: I use PIN, and for Yakov.
Courant Events Right: predicted in 75 that.
Christophe Raymond Garban: the…
Courant Events Right: Should have exponential decay.
Christophe Raymond Garban: At arbitrarily low temperature.
Courant Events Right: In other words, you know.
Christophe Raymond Garban: What's clean.
Courant Events Right: Polyakov predicted that this BKT phenomenon that I just explained for the S1.
Christophe Raymond Garban: unvalued spin system does not.
Courant Events Right: when you go to S2-valued spin system.
Courant Events Right: You might say that if you believe this prediction, maybe this spin system is as boring as the one-dimensional easing model, because easing model also has exponential decay.
Christophe Raymond Garban: At all temperature. It's not…
Courant Events Right: really boring.
Christophe Raymond Garban: Because if you solve this.
Courant Events Right: It's considered for many good reasons.
Christophe Raymond Garban: A great story.
Courant Events Right: model to understand confinement.
Christophe Raymond Garban: input.
Courant Events Right: for SU3.
Christophe Raymond Garban: It's still here.
Courant Events Right: Z4, which.
Christophe Raymond Garban: Many people in the audience are more exposed than I am.
Christophe Raymond Garban: So, I want to focus…
Courant Events Right: very shortly on the physics argument here. I will go quickly for two reasons. I'm running a bit off time for what I want to say next.
Christophe Raymond Garban: And also, there are really…
Courant Events Right: bigger experts than.
Christophe Raymond Garban: So, how did Paul Yakov, Understood, that's a…
Courant Events Right: So there should be exponential decay at whole temperatures.
Courant Events Right: So, it goes as follows. So, you sample the… imagine you sample the S2-valued spin system
Courant Events Right: zit graph, that's the original model I started with.
Christophe Raymond Garban: And maybe you want to.
Courant Events Right: grain this a little bit. So maybe you want to represent, that's called the background field method.
Christophe Raymond Garban: You want to represent.
Courant Events Right: represent in which, average direction this box is pointing to, in which average direction this other box is pointing to.
Christophe Raymond Garban: And you…
Courant Events Right: end up with the signal, a coarse-grained signal, which is living at.
Christophe Raymond Garban: That's pretty…
Courant Events Right: low frequencies.
Courant Events Right: The actual spin system in S2.
Christophe Raymond Garban: Who is, in some sense.
Courant Events Right: It's a combination of these.
Christophe Raymond Garban: low-frequency signal, this false grain S2 value, with a higher frequency
Christophe Raymond Garban: which tells you the details of where the.
Courant Events Right: S2 spin system lives near the.
Christophe Raymond Garban: is, slow mode, S2 valued.
Courant Events Right: And now you need this,
Christophe Raymond Garban: business of RGO in this background filters, you try to.
Courant Events Right: average out what the small scales are doing, you try to average out what the.
Christophe Raymond Garban: Higher frequency mode, army.
Courant Events Right: And try to see what you end up with on the,
Christophe Raymond Garban: post-grained L2.
Courant Events Right: consistent.
Christophe Raymond Garban: And what…
Courant Events Right: As you notice, I always try.
Christophe Raymond Garban: But what you eventually notice is that, if you remove lots of.
Courant Events Right: higher order.
Christophe Raymond Garban: After one step of the algae flow.
Courant Events Right: putting aside these higher-order terms, the beta that you have in front of this coarse-grained S2-valued spin system is slightly smaller than the one you thought.
Christophe Raymond Garban: Technically.
Christophe Raymond Garban: And that's how, I think, for Yakos.
Courant Events Right: predicted that if you keep doing.
Christophe Raymond Garban: this RG fluid.
Courant Events Right: To go to higher and higher scale.
Courant Events Right: Eventually, this effective beta will become smaller and smaller.
Courant Events Right: And we'll enter the regime where it's easy to find exponential decay.
Courant Events Right: As I said, I don't have a mass counterpart.
Christophe Raymond Garban: After this prediction.
Christophe Raymond Garban: It's my favorite conjecture.
Courant Events Right: true.
Christophe Raymond Garban: It'.
Courant Events Right: Where's…
Christophe Raymond Garban: conjecturing the field.
Courant Events Right: But I want to go to the seventh part.
Christophe Raymond Garban: where I want to discuss the following question. If you take the polyapos model, what happened?
Courant Events Right: If you slightly nurture the Polyakov portal. So, the Polyakov will be the.
Christophe Raymond Garban: range. I mean, the range will never appear again in the.
Courant Events Right: Right?
Courant Events Right: But the perturbation will either go to the lemon or to the flat pitch.
Christophe Raymond Garban: So… Before we go…
Courant Events Right: the lemon and the flat peach. I want to…
Christophe Raymond Garban: described two, I think, natural perturbations of the classical Eisenberg model. The first natural perturbation, which I think would be the first that.
Courant Events Right: in the room would agree with, would be to say that we could slightly protrude the Romanian metric that we equip the sphere with.
Courant Events Right: So this would be the wrong metric.
Christophe Raymond Garban: I would still…
Courant Events Right: view my spin system as an S2 valued spin system.
Courant Events Right: But the metric I would use to build a Gibbs measure would be the push forward.
Christophe Raymond Garban: from a slightly padded.
Courant Events Right: ape.
Christophe Raymond Garban: So, if you slightly contribute.
Courant Events Right: The metric with epsilon, and then it sends beta to infinity.
Christophe Raymond Garban: We conjecture with.
Courant Events Right: my course.
Christophe Raymond Garban: That's…
Courant Events Right: At least if you have enough symmetry, but we believe maybe this is not even.
Christophe Raymond Garban: This is so…
Courant Events Right: But just to make the conjecture a bit more natural.
Christophe Raymond Garban: We conjecture that it's the.
Courant Events Right: The phenomenology of Polyakov still prevailed, you would still have exponential decay.
Christophe Raymond Garban: At arbitrarily large bids.
Courant Events Right: Yep.
Courant Events Right: Why do we conjecture that?
Christophe Raymond Garban: F?
Courant Events Right: This comes from the fact that there is a work after Polyakov, which is very beautiful. I don't do so much of it even, though I asked Slava to.
Christophe Raymond Garban: Help.
Courant Events Right: me, and this was a huge help, but I still struggle very much. So Dan Friedan, in 1980, he came up with an understanding of the.
Courant Events Right: renormalization group flow of more general spin systems than classical Eisenberg.
Christophe Raymond Garban: Okay?
Courant Events Right: Precisely of those spin systems here.
Christophe Raymond Garban: And he found out that along the RG flow.
Courant Events Right: The relevant thing to follow is the rich sea flow of this,
Christophe Raymond Garban: Romanian metric.
Courant Events Right: And if I slightly perturb the round metric, it will flow to the wrong sphere quicker.
Christophe Raymond Garban: than.
Courant Events Right: and the way beta is evolving.
Christophe Raymond Garban: So, if one takes that, I think Slava agreed, but, but…
Courant Events Right: as, Schenko, disagreed, so I cannot blame either of the two. It was short discussions.
Courant Events Right: But we conjecture.
Christophe Raymond Garban: the Richie Flow gives the right.
Courant Events Right: Beautiful, and that's… The Polyakov phenomenology is exactly the same for this perturbation.
Courant Events Right: The second perturbation I will now, explain to you is more this lemon and the flat peach.
Courant Events Right: I should have changed my slide.
Christophe Raymond Garban: So, another natural.
Courant Events Right: observation that you could come up with, is instead of changing the metric.
Christophe Raymond Garban: Thank you.
Courant Events Right: Maybe you want to change the Scala product that I used before to define the Gibbs measure.
Christophe Raymond Garban: So here, the…
Courant Events Right: The Sigmas.
Christophe Raymond Garban: attached.
Courant Events Right: to each vertex of Z2. I will write those in.
Christophe Raymond Garban: in the coordinates of R3.
Courant Events Right: Sigma 1, Sigma 2, Sigma 3.
Christophe Raymond Garban: If London…
Courant Events Right: is equal to 1, this is just exactly the classical Eisenberg model I played with up to now.
Christophe Raymond Garban: But I want to slightly.
Courant Events Right: change the strength of the third company.
Christophe Raymond Garban: Which…
Courant Events Right: In my pictures will be the vertical component.
Christophe Raymond Garban: So if lambda is larger, I'm basically,
Courant Events Right: a Lemon-valued spin system, which puts more.
Christophe Raymond Garban: Cool!
Courant Events Right: strength to spins pointing in the north or south.
Christophe Raymond Garban: direction.
Christophe Raymond Garban: Longdae is equal.
Courant Events Right: One is the perfect orange, and this is the.
Christophe Raymond Garban: the flat speech.
Courant Events Right: So, what are the known results here? If lambda is.
Christophe Raymond Garban: is,
Courant Events Right: large.
Courant Events Right: Boats and griffids have shown in the 70s.
Christophe Raymond Garban: That at low enough temperature, the spin system will carry.
Courant Events Right: such an easing phase.
Christophe Raymond Garban: It didn't true.
Courant Events Right: Be either.
Christophe Raymond Garban: very close to the North Pole, or very close to the South Pole.
Christophe Raymond Garban: The gap was…
Courant Events Right: filled by Mali Sheff and Froli Schlieb in the late 70s, where they proved that this is still true.
Christophe Raymond Garban: For one letter.
Courant Events Right: I mean, arbitrary growth to the.
Christophe Raymond Garban: Jewish.
Christophe Raymond Garban: And in this work by Felicia.
Courant Events Right: They introduced this,
Christophe Raymond Garban: chessboard estimate, which happened.
Courant Events Right: Could be very useful since then.
Courant Events Right: So now, I will cross over to the flat pitch, and brickman Levovitzpister, they conjecture that if you go to the flat pitch.
Courant Events Right: At any longer less than what?
Christophe Raymond Garban: If you slightly perturb the.
Courant Events Right: cool as in the.
Christophe Raymond Garban: Now, a BKT phase should, should erupt.
Courant Events Right: In fact, while they conjectured this, they actually proved the conjecture was wrong.
Christophe Raymond Garban: 1 octa is exactly equal to zero, which doesn't give any.
Courant Events Right: strengths to the.
Christophe Raymond Garban: critical direction.
Courant Events Right: Soon after, Dunlop proved that if.
Christophe Raymond Garban: The longer is small enough, if you take a very flat pitch, if you just push hard on your pitch.
Courant Events Right: Then, indeed, at very low temperature, the spin system will experience a power loop decay phase along the equator here.
Christophe Raymond Garban: So…
Courant Events Right: Now comes our main CRM with Nathan de Montgolfie. What we can show is that.
Christophe Raymond Garban: For arbitrarily small perturbation of the classical eigen parameter in this subconscious.
Courant Events Right: sense.
Christophe Raymond Garban: Because in the first.
Courant Events Right: In other sense, we conjecture it's a different phenomenology.
Courant Events Right: Then, yes, a decade phase.
Christophe Raymond Garban: is…
Courant Events Right: 16.
Courant Events Right: Here is the second theorem, which has some.
Christophe Raymond Garban: some,
Courant Events Right: quotation marks?
Christophe Raymond Garban: I will explain what's.
Courant Events Right: going on here.
Christophe Raymond Garban: So…
Courant Events Right: Since we have many physicists in the room.
Christophe Raymond Garban: It…
Courant Events Right: It may be interesting to ask what the RG flow is doing on this perturbation.
Courant Events Right: So maybe it's natural to try to plot in the plane where I have the temperature here, and I have the amount of perturbation in the.
Christophe Raymond Garban: vertical.
Courant Events Right: excuse?
Christophe Raymond Garban: What's going on?
Courant Events Right: What is the phase which would be exponentially decaying? What is the phase which would be long-range order and easing light? And what is the phase which would be deciding?
Christophe Raymond Garban: Of course, we do not.
Courant Events Right: Prove where the lines are exactly.
Christophe Raymond Garban: But if we were to prove this, it would be more difficult than polyaco.
Courant Events Right: conjunctional.
Christophe Raymond Garban: for you, because.
Courant Events Right: predictions.
Christophe Raymond Garban: But the first part of.
Courant Events Right: these.
Christophe Raymond Garban: to M is actually not our contribution, it's,
Courant Events Right: Slava during a lunch at IHCS on pieces of paper this big, where he helped us understanding what.
Christophe Raymond Garban: could be the curve here, along the RG flow.
Courant Events Right: So, we'll try to explain this in the paper, as best as we can.
Christophe Raymond Garban: But where we.
Courant Events Right: We do have a CRM, he's in the… so that's…
Christophe Raymond Garban: supposedly the RG flow, What we are doing.
Courant Events Right: I mean, the second part.
Christophe Raymond Garban: of this paper.
Courant Events Right: We take,
Christophe Raymond Garban: like a space where exposition.
Courant Events Right: financial decay.
Christophe Raymond Garban: is actually…
Courant Events Right: be known rigorously.
Christophe Raymond Garban: And this is in a work by.
Courant Events Right: Boris Schmidt, Enmuss, and Schwab.
Christophe Raymond Garban: Where they.
Courant Events Right: show that if you take the hemisphere, so there's no SO3 symmetry, but still you take the hemisphere.
Christophe Raymond Garban: There is exponential.
Courant Events Right: ridiculous.
Christophe Raymond Garban: population. In the two core.
Courant Events Right: Coordinates that are along the plane here.
Courant Events Right: And so this is known, this is their CRM.
Christophe Raymond Garban: And what we.
Courant Events Right: do with Netflix.
Christophe Raymond Garban: We try to understand up to which deformation of the atmosphere we can push.
Courant Events Right: the exponential decay phase, and the good news We can reach the,
Christophe Raymond Garban: the line that's lava and Earth.
Courant Events Right: to understand with the RG flow.
Courant Events Right: So, at least…
Christophe Raymond Garban: At least for the.
Courant Events Right: Hemisphere, we do have a regular.
Christophe Raymond Garban: explanation of these are Gila.
Christophe Raymond Garban: So, in the essence of…
Courant Events Right: In this talk, what I want to do is to explain to.
Christophe Raymond Garban: you, how do we prove.
Courant Events Right: CRM1.
Courant Events Right: And I will do it in two steps.
Courant Events Right: The first step, I will tell you how to.
Christophe Raymond Garban: approve?
Courant Events Right: Why a very flattened beach must have a big ediface.
Courant Events Right: the temperature is small, so I.
Christophe Raymond Garban: improve the Lobe CRM.
Christophe Raymond Garban: And then…
Courant Events Right: I will want to explain to you why there is a barrier, why there is a long dazzero.
Christophe Raymond Garban: of which we need to do something else. And the second step will require properties of a
Christophe Raymond Garban: I want to.
Courant Events Right: maps that are a well-studied object in.
Christophe Raymond Garban: analyses.
Courant Events Right: And why do we need harmonic maps? To close the gap and go all the way.
Christophe Raymond Garban: Way to the orange.
Christophe Raymond Garban: So, the first thing I do here is I recall what.
Courant Events Right: the motor here?
Christophe Raymond Garban: My spins are.
Courant Events Right: two valued, I write those in those three coordinates of R3, and lambda tells you how much strength do I give to the third coordinate. So the Gibbs measure.
Christophe Raymond Garban: is proportional to exponential beta sums over this.
Courant Events Right: modified scalar products between neighboring sites.
Courant Events Right: Different?
Christophe Raymond Garban: First, I…
Courant Events Right: ID that we use is the following. It's a classical ID, which
Courant Events Right: Bows back, maybe at least to.
Christophe Raymond Garban: But how…
Courant Events Right: for you, Xyler, I will say a word.
Christophe Raymond Garban: about that.
Courant Events Right: So, the idea that they had is that they can.
Christophe Raymond Garban: Look at this!
Courant Events Right: two valued.
Christophe Raymond Garban: system, and they can ask about the.
Courant Events Right: brewing, natural.
Christophe Raymond Garban: polaristic question. Conditioned on the vertical direction, on the.
Courant Events Right: Sigma 3 field.
Christophe Raymond Garban: What is the law of the remaining randomness?
Courant Events Right: So, probabilistic-wise, the sample.
Christophe Raymond Garban: the third.
Courant Events Right: coordinate. This Pilometer measure is the measure we are interested in. It's a complicated measure.
Courant Events Right: And I want to understand what is the remaining randomness if you give to me the positions.
Christophe Raymond Garban: the vertical positions on the S2 field.
Christophe Raymond Garban: And the observation, which goes back to Patrasios.
Courant Events Right: Even older, I don't know.
Christophe Raymond Garban: Is that the condition of the.
Courant Events Right: Remaining randomness to sample.
Christophe Raymond Garban: It's extremely simple.
Courant Events Right: to run.
Christophe Raymond Garban: It's exactly an XY model, as we had in the first part of this talk. The only difference is that it.
Courant Events Right: It's an XY model in a random disorder of conductances.
Courant Events Right: The conductance from this point to this point.
Christophe Raymond Garban: is beta?
Courant Events Right: times the radius here for the… for the S1 spin, times the radius here.
Courant Events Right: So it's very simple to describe.
Christophe Raymond Garban: condition on this, and using pedagogy.
Courant Events Right: SUM and.
Christophe Raymond Garban: Nothing ahead.
Courant Events Right: What is the fee?
Christophe Raymond Garban: In the random conductance.
Courant Events Right: where…
Christophe Raymond Garban: the, the spin in S1 is.
Courant Events Right: It still needs to be sampled.
Christophe Raymond Garban: So, if you do Polyakov conjecture, that was what.
Courant Events Right: used to try to argue against.
Christophe Raymond Garban: Yeah, good.
Courant Events Right: A few years ago.
Christophe Raymond Garban: Who is.
Courant Events Right: You are here and have a UC pouveau.
Christophe Raymond Garban: we re-discussed Patrice Allure, and we basically gave a counterexample.
Courant Events Right: to their argument against Polyakov. Of course, we did not prove Polyakov's prediction at all.
Christophe Raymond Garban: But we.
Courant Events Right: provided a counterexample to the analysis in the 90s, which I think.
Christophe Raymond Garban: businesses…
Courant Events Right: didn't…
Christophe Raymond Garban: Bave the analysis anyway.
Courant Events Right: I really liked the idea to try to test how robust.
Christophe Raymond Garban: parallel decay could be.
Courant Events Right: in this, projected.
Christophe Raymond Garban: Thank you.
Christophe Raymond Garban: In this work, we definitely go through Patraska's ILRID, but this time we get a series.
Christophe Raymond Garban: Because we get some control on this random disorder when we project.
Courant Events Right: the, flat pitch system down to an XY.
Courant Events Right: So, in.
Christophe Raymond Garban: But find me.
Courant Events Right: it?
Courant Events Right: I think I do.
Courant Events Right: We started at 20.
Courant Events Right: It's sense.
Christophe Raymond Garban: So, according to… what?
Courant Events Right: we've been.
Christophe Raymond Garban: Right, so… No, no, no, no, I…
Courant Events Right: That's right. Yes.
Christophe Raymond Garban: Last question.
Courant Events Right: I, I think 5.
Christophe Raymond Garban: I haven't… So, the idea.
Courant Events Right: Is that this field of conductances is too hard to describe.
Christophe Raymond Garban: But…
Courant Events Right: The… the…
Christophe Raymond Garban: The proof is trying to control this disorder by comparing with a percolation disorder.
Courant Events Right: So, the idea is to try to say that maybe there is a stochastic domination between this field of conductances.
Christophe Raymond Garban: And a nicer…
Courant Events Right: field of conductances, which would be percolation disorder.
Christophe Raymond Garban: And here, we spoke.
Courant Events Right: Where are you, a few years?
Christophe Raymond Garban: years ago.
Courant Events Right: We prove that if you try to do XY.
Christophe Raymond Garban: model on a percolation.
Courant Events Right: disorder.
Christophe Raymond Garban: Then, at low enough temperature, you will still have quasi-long-range order on such percolating.
Courant Events Right: the cluster.
Christophe Raymond Garban: All the way to the population transition.
Christophe Raymond Garban: 2?
Courant Events Right: The idea with not only.
Christophe Raymond Garban: the Mughal fee is that if Longda is.
Courant Events Right: Very small.
Christophe Raymond Garban: moon?
Christophe Raymond Garban: We observed that…
Courant Events Right: And it's not too difficult to prove that we can stochastically dominate our field of.
Christophe Raymond Garban: conductances.
Courant Events Right: with a Bernoulli, with an IID field of disorders.
Courant Events Right: And the idea is simple.
Christophe Raymond Garban: I don't.
Courant Events Right: He wants to have stochastic domination.
Christophe Raymond Garban: Then you can try to condition on all the spin field outside of one point, and try to prove that whatever you see outside, you still have a high probability.
Courant Events Right: to be far from North Pole.
Christophe Raymond Garban: recalls how.
Courant Events Right: pool.
Courant Events Right: So, this doesn't work, because if you're unlucky, and the four points around you.
Christophe Raymond Garban: are close to the North Pole.
Courant Events Right: Even if you have a very flat pitch, you will still be…
Christophe Raymond Garban: We want to be very close.
Courant Events Right: to the North Pole.
Christophe Raymond Garban: at low temperature.
Christophe Raymond Garban: So this does not work, but if we make the… if we make it…
Courant Events Right: slightly bigger. If we now look at the neighborhood of just one edge.
Courant Events Right: Now we can have… we have a lemma which says that if the pinch.
Christophe Raymond Garban: It's flat enough.
Christophe Raymond Garban: Yes, with high probability, uniformly on what we see away from us.
Christophe Raymond Garban: we will have two spins which will both be far from the North Pole.
Courant Events Right: to tell you why, imagine you're very.
Christophe Raymond Garban: and a 6S2, but.
Courant Events Right: It spins around you are.
Christophe Raymond Garban: All north pointing spins.
Christophe Raymond Garban: Because of the flat speech, the system will still want to simulate.
Courant Events Right: when you sleep on.
Christophe Raymond Garban: to a direction, and be far to the North Pole. So our argument is based just on the behavior of the.
Courant Events Right: the harmonic maps, the ground states, uniformly in the neighborhood.
Courant Events Right: These are…
Christophe Raymond Garban: The argument is rather soft, but it doesn't work when.
Courant Events Right: And lambda is bigger than 1 over 100.
Christophe Raymond Garban: As lambda goes to 1.
Courant Events Right: It's easy to see that we are forced to analyze.
Christophe Raymond Garban: situations which go at farther and further radii.
Christophe Raymond Garban: So, as.
Courant Events Right: As long back goes to infinity, we need to understand uniformly in what we may see outside of.
Christophe Raymond Garban: the pool of radius are? How does the harmonic.
Courant Events Right: maps how does the least energetic pitch-valued function may look.
Courant Events Right: And being in New York was actually quite useful this year, because.
Christophe Raymond Garban: I include this.
Courant Events Right: with offenders.
Christophe Raymond Garban: and with Naton, we try.
Courant Events Right: to use.
Christophe Raymond Garban: Proof techniques that were used in.
Courant Events Right: Amanda analysis of harmonic maps, and we could show that they're not as ordered, as regular.
Christophe Raymond Garban: that we were hoping for. If we had regular harmonic functions.
Courant Events Right: linear program, everything would be very.
Christophe Raymond Garban: the group.
Christophe Raymond Garban: You can…
Courant Events Right: It's less regular, but we have enough weak regularity to conclude.
Christophe Raymond Garban: And the week…
Courant Events Right: The configurality that we used is something Sylvia.
Christophe Raymond Garban: knows very well, it's a Ginsburg-Lando type.
Christophe Raymond Garban: And plugging this with, the world by.
Courant Events Right: for Dario to propagate BKT.
Christophe Raymond Garban: On the bed…
Courant Events Right: operation disorder, we could handle this arbitrary.
Christophe Raymond Garban: small percolation of S2, I will solve it.
Courant Events Right: And…
Christophe Raymond Garban: Thank you.
Courant Events Right: Alright, also, we had, already a lately of directions.
Christophe Raymond Garban: And we can continue with the introduction.
Christophe Raymond Garban: Authorities.
Courant Events Right: Do you think.
Christophe Raymond Garban: productive, so…
Courant Events Right: clients, or also to teach her.
Christophe Raymond Garban: We opened every space for the moment.
Courant Events Right: Where do you think those funds.
Christophe Raymond Garban: We don't run up.
Courant Events Right: I've been upon it that they furnished.
Courant Events Right: yes, it's a good question.
Courant Events Right: I… I… It's a good quote, I believe so.
Courant Events Right: So, indeed, if the,
Christophe Raymond Garban: gear.
Courant Events Right: Each measure is now.
Christophe Raymond Garban: Excellent.
Courant Events Right: financial beta sum of Scala product squares.
Christophe Raymond Garban: The model is a model of.
Courant Events Right: Random lines instead of random points on the sphere.
Christophe Raymond Garban: This is also a.
Courant Events Right: I believe to have an exponential decay.
Christophe Raymond Garban: habitual.
Christophe Raymond Garban: Oh, no, there's busy.
Courant Events Right: Is this point?
Christophe Raymond Garban: Secretary.
Christophe Raymond Garban: Oh, these are very interesting. If this is proof to… if it's believed to be wrong, I won'.
Courant Events Right: To go on this model and see what's going on.
Courant Events Right: Can you explain?
Christophe Raymond Garban: And can you explain this?
Courant Events Right: How many different variation in the language of the law.
Christophe Raymond Garban: What is it looking like?
Courant Events Right: Okay, so.
Christophe Raymond Garban: So the…
Courant Events Right: The question is, how does this.
Courant Events Right: Where would this appear in the language of nonlinear sigma models?
Christophe Raymond Garban: So, I think…
Courant Events Right: The first perturbation on which we don't have a result, and we believe this would go, like Paul Yakov, to exponentially get all beta, this would be
Courant Events Right: This would fit well within nonlinear Sigma.
Christophe Raymond Garban: Right.
Courant Events Right: Here, I don't change the metric, I change the scalar product.
Christophe Raymond Garban: It is also detected in the potential term.
Courant Events Right: Yes, so it's more like adding a small potential term. One way to see the difference is that a nonlinear sigma model.
Christophe Raymond Garban: The grown stage…
Courant Events Right: It could be in any directions.
Christophe Raymond Garban: Reggie's community, though.
Christophe Raymond Garban: Well, so let me, let me go…
Courant Events Right: here, if I take this perturbed, what is it?
Courant Events Right: This… this first perturbation here.
Courant Events Right: Here, even if I have this flat domain, if I don't talk about entropy, if I take any constant field.
Christophe Raymond Garban: I pay…
Courant Events Right: exponential zero. So, in that.
Christophe Raymond Garban: zones?
Courant Events Right: These perturbations will have more ground states than the one we consider with modifying scalar products.
Courant Events Right: Because this second perturbation.
Christophe Raymond Garban: Only the points.
Courant Events Right: In the equator will be…
Christophe Raymond Garban: of human departments.
Courant Events Right: Yep.
Christophe Raymond Garban: just…
Courant Events Right: to look at the RG implications involved in additional companies.
Courant Events Right: And I think that's what Slava was doing on this piece of paper in IHS.
Courant Events Right: It wasn't… you didn't…
Christophe Raymond Garban: reading.
Christophe Raymond Garban: Okay, thank you.
Christophe Raymond Garban: Can you…
Courant Events Right: Hold on.
Christophe Raymond Garban: What does Bill?
Courant Events Right: about parish people.
Courant Events Right: You won't…
Christophe Raymond Garban: from getting your…
Courant Events Right: recognized.
Christophe Raymond Garban: I agree, control.
Courant Events Right: is large, and… You mean what would be known regularly about implementing.
Christophe Raymond Garban: So…
Courant Events Right: They managed to…
Christophe Raymond Garban: over the stretch.
Courant Events Right: over the implementation of our.
Christophe Raymond Garban: So, I think maybe I'm moving one first.
Courant Events Right: difficulty.
Christophe Raymond Garban: is…
Courant Events Right: It's a background field method to even define precisely what would be this low mode background field around which we are trying to.
Christophe Raymond Garban: being spent.
Courant Events Right: So I think this is… But I'm pretty sure I've seen some papers in the view, some papers of the character and the.
Christophe Raymond Garban: You can easily.
Courant Events Right: We're all about being open.
Christophe Raymond Garban: It doesn't go as well.
Courant Events Right: savings.
Christophe Raymond Garban: So, Balaband, I have never read Balaband.
Courant Events Right: video.
Christophe Raymond Garban: But…
Courant Events Right: Maybe,
Christophe Raymond Garban: Indeed.
Courant Events Right: he implemented the.
Christophe Raymond Garban: RG flow, which gives lung.
Courant Events Right: and all the.
Christophe Raymond Garban: was.
Courant Events Right: such nonlinear sigma model when you are in dimension 3.
Christophe Raymond Garban: So, it's a…
Courant Events Right: It's an implementation of RGflow to detect that, indeed, you do bring.
Christophe Raymond Garban: Ranking the symmetry in dimension 3.
Christophe Raymond Garban: Indeed, I've never read, actually.
Courant Events Right: I should have, it's a good question to ask.
Christophe Raymond Garban: what does.
Courant Events Right: his argument.
Christophe Raymond Garban: and two in dimension.
Courant Events Right: who…
Christophe Raymond Garban: Regarding this question here.
Christophe Raymond Garban: Being in New York, I would just know.
Courant Events Right: on the Ronald's office, and…
Christophe Raymond Garban: I don't think you would know more.
Christophe Raymond Garban: Yeah.
Christophe Raymond Garban: Yeah, it's kind of weird, such as your event.
Courant Events Right: Sure, I can say in the.
Christophe Raymond Garban: sense of progress.
Courant Events Right: Everyone.
Christophe Raymond Garban: We can talk later.
Christophe Raymond Garban: Dave, one last question.
Courant Events Right: We've been lawyers.
Christophe Raymond Garban: Yes, the question mark.
Courant Events Right: your head.
Christophe Raymond Garban: Emma?
Christophe Raymond Garban: The question is, what…
Courant Events Right: we do in CRM.
Christophe Raymond Garban: them too.
Courant Events Right: So, CRM2 is a… is a…
Christophe Raymond Garban: is, first…
Courant Events Right: We'll try to explain, as best as we can, what is this conjectured line between the exponential decay and the parallel decay, using our.
Christophe Raymond Garban: flow, which probably, follows your option, that there is a.
Courant Events Right: The potential, and there is the…
Christophe Raymond Garban: the classicalizing that type.
Courant Events Right: And the theorem is, in the case of a hemisphere where it is.
Christophe Raymond Garban: It's known that there is.
Courant Events Right: exponential decay at arbitrarily low temperatures.
Christophe Raymond Garban: Here we can see.
Courant Events Right: change the scale of product, and as we said.
Christophe Raymond Garban: And when the…
Courant Events Right: KT kicks in.
Christophe Raymond Garban: And your theorem is that.
Courant Events Right: DKT does.
Christophe Raymond Garban: Risk in advance.
Courant Events Right: low enough temperature.
Christophe Raymond Garban: theorem will never go to this RG curve.
Christophe Raymond Garban: But it will go all the way.
Courant Events Right: to lambda equals 1.
Christophe Raymond Garban: And the other side of the CRM is that.
Courant Events Right: We managed to extend the proof by.
Christophe Raymond Garban: Rorschmitt and Mutenzuelan, with.
Courant Events Right: scalar.
Christophe Raymond Garban: Product preservation, all the way to this.
Courant Events Right: curve.
Courant Events Right: Right, so those blockers are exponential decay is still here, still here, still here.
Christophe Raymond Garban: Then… then we don't know, and then.
Courant Events Right: Here, we know that there is parallel.
Christophe Raymond Garban: That's what…
Courant Events Right: with your MC.
Christophe Raymond Garban: You know, people.
Courant Events Right: Yeah.
Courant Events Right: That's the other…