Download "Rotational invariance of planar random cluster models... and beyond?"

"videoThumbnail Rotational invariance of planar random cluster models... and beyond?

A new way of looking at high dimensional lattice models

A new way of looking at high dimensional lattice models

Channel: Fields Institute

АНГЛИЙСКИЙ ЯЗЫК ЗА 10 УРОКОВ 2 КЛАСС УРОКИ АНГЛИЙСКОГО ЯЗЫКА АНГЛИЙСКИЙ ДЛЯ НАЧИНАЮЩИХ УРОК 10 2

АНГЛИЙСКИЙ ЯЗЫК ЗА 10 УРОКОВ 2 КЛАСС УРОКИ АНГЛИЙСКОГО ЯЗЫКА АНГЛИЙСКИЙ ДЛЯ НАЧИНАЮЩИХ УРОК 10 2

Channel: АНГЛИЙСКИЙ ЯЗЫК ПО ПЛЕЙЛИСТАМ

Ruling the Countryside - Full Chapter Explanation Solutions | Class 8 History Chapter 3

Ruling the Countryside - Full Chapter Explanation Solutions | Class 8 History Chapter 3

Channel: Magnet Brains

Английский язык с нуля до продвинутого. Практический курс по приложению English Galaxy. А0. Урок 27

Английский язык с нуля до продвинутого. Практический курс по приложению English Galaxy. А0. Урок 27

Channel: АНГЛИЙСКИЙ ЯЗЫК ПО ПЛЕЙЛИСТАМ

Winderton: слив и разоблачение инфоцыгана. Кто такой EngineerSpock?

Winderton: слив и разоблачение инфоцыгана. Кто такой EngineerSpock?

Channel: EngineerSpock - IT & программирование

rotational

invariance

planar

random

cluster

models..

00:00:01

yes thank you very much for coming

00:00:03

back so today it's for

00:00:06

probabilist um yeah so what I was

00:00:11

thinking uh to present is some ongoing

00:00:13

work so you're going to see I mean I'm

00:00:15

going to be a little bit careful about

00:00:17

stating theorems or not

00:00:20

theorems uh I mean it's a work that in

00:00:23

full honesty should have been already

00:00:24

completed a long time ago but uh between

00:00:28

several things including many kids among

00:00:30

the authors young kids so it's it's uh

00:00:35

yeah it is taking more time than

00:00:37

expected but I hope that this year will

00:00:39

be the right one so so what's the idea

00:00:43

um I want to tell you about two

00:00:46

models and tell you about the fact that

00:00:48

again it's a little bit the same story

00:00:50

as yesterday if you jump from one model

00:00:52

to the other one you manage to go much

00:00:54

farther than if you stay with one of the

00:00:55

models so the two models so let's me I

00:00:58

mean introduce the model

00:01:01

I should immediately

00:01:05

there so the motors they will

00:01:10

be so let's say we we work on a Taurus I

00:01:13

mean in fact we will work in infinite

00:01:15

volume but let's say it's a n by m

00:01:22

torus and on this uh on this Taurus I'm

00:01:26

going to Define two uh things so let's

00:01:28

say

00:01:30

the ltis is like that so the first model

00:01:33

is going to be a percolation

00:01:36

model but a little bit strange so the

00:01:39

percolation model is going to be not on

00:01:41

the original latis but on the ltis that

00:01:45

you get by coloring in a chessboard

00:01:49

fashion your lce and putting edges like

00:01:52

that so it's a rotated and rescale

00:01:55

version of the square latice okay and so

00:01:59

on this model we are going to exactly do

00:02:01

FK

00:02:02

percolation I mean like like yesterday

00:02:05

so we are going to sample at

00:02:09

random let's let me drop the n m but

00:02:13

there will be two parameter p and Q so a

00:02:16

percolation configuration so every edge

00:02:18

here will be retained in my Omega so

00:02:21

Omega maybe is going to be 01 to the

00:02:25

edges of this uh violet

00:02:28

lce so one would mean that it's an edge

00:02:32

of Omega zero would mean it's not an

00:02:34

edge of Omega so the probability of

00:02:37

Omega would be proportional to P to the

00:02:40

number of open

00:02:42

edges in Omega or if you prefer I'm

00:02:45

going to write it like that so this

00:02:48

is sum of the Omega e so it's just

00:02:51

counting the number of open

00:02:53

edges 1 minus P to the number of close

00:02:57

edges so the this is the the total

00:03:00

number of

00:03:03

edges minus the number of open edges if

00:03:07

I would stay here you agree that this

00:03:09

would exactly be bar percolation I'm

00:03:12

just sampling independently edges so

00:03:15

what I'm just going to do but if you

00:03:16

don't like it think of B percolation

00:03:18

it's totally fine it's already you will

00:03:20

see the results are already interesting

00:03:22

for B percolation but let me add so look

00:03:26

at what we call FK percolation meaning

00:03:29

that I'm going to add a q to the number

00:03:31

of connected components so K of Omega

00:03:34

will be the number of connected

00:03:38

components in

00:03:44

Omega and if you do that just in order

00:03:46

to have a probability measure you need

00:03:49

to

00:03:51

divide

00:03:54

by a renormalization factor that I would

00:03:57

did not like that okay so for Q equal 1

00:04:00

it's barly percolation and in fact for

00:04:02

every Q larger than zero at least you

00:04:05

get a Hest percolation model okay so

00:04:08

that's called FK

00:04:16

percolation now there is a second

00:04:20

player which has nothing to do with

00:04:22

percolation a

00:04:24

Priory and which is going to be defined

00:04:26

on the edges of the white lattice okay

00:04:29

so on the white

00:04:33

latice I'm going to define the Omega

00:04:37

Arrow okay Omega arrow is going to be an

00:04:41

assignment of orientation to every edge

00:04:45

of the white

00:04:46

latice except there is one rule which is

00:04:49

that there should be exactly two edges

00:04:52

incoming and two edges outgoing for

00:04:54

every vertex so Omega

00:04:57

is an orientation

00:05:02

of

00:05:04

edges in this lce with the constraints I

00:05:09

mean satisfying what we

00:05:13

call the ice

00:05:16

rule which means that you have as many

00:05:18

so two

00:05:19

incoming and two

00:05:22

outgoing

00:05:27

edges at every vex

00:05:33

okay and there again I'm going to sample

00:05:39

at random this type of

00:05:43

object there will be a parameter this

00:05:45

time it's going to be C for people who

00:05:47

know this it's a six vertex model I will

00:05:49

come back to it later and here I'm going

00:05:51

to do something or let let me put ABC so

00:05:55

I'm going to take three parameter a b

00:05:57

and c all positive and basically the

00:06:00

probability of Omega is going to be

00:06:02

a to the number of so so you agree Let

00:06:06

Me Maybe do it here you agree that the

00:06:10

ice rule is forcing that at every vertex

00:06:14

there are six possible six possibilities

00:06:17

for the orientations right so let me

00:06:20

just draw them once what I will do is

00:06:24

that I will only

00:06:27

draw the let's say out going edges the

00:06:30

two others will be incoming

00:06:33

so there are this two and this

00:06:36

two there are this

00:06:38

two and this two and then you have this

00:06:44

two and this okay these are the six

00:06:47

possibilities that every vertex so again

00:06:49

I only drew the orientation of the

00:06:51

outgoing the other ones are in I mean

00:06:54

coming in and here what I'm going to do

00:06:56

is I'm going to call this so type one

00:07:00

vertices type two types three types four

00:07:02

five and six and here I'm going to just

00:07:05

say that it's a to the N1 plus N2 which

00:07:08

is number of type one plus number of

00:07:11

type two vertices B to the N3 plus N4

00:07:16

and C to the

00:07:17

N5 plus n6 and

00:07:20

again here I need to

00:07:23

renormalize if I

00:07:26

want to have a probability measure

00:07:30

so again here it's only orientation

00:07:32

satisfying the ice schol okay so it's

00:07:35

not uh should not be forgotten so this

00:07:38

is called the six Vex

00:07:42

model it's a very very classical model

00:07:46

in particular because it has

00:07:47

integrability properties and I will come

00:07:49

back to that later

00:07:51

maybe and the thing the the the way I

00:07:56

mean the the bottom line of the talk is

00:07:58

going to be that

00:08:00

by a back and forth between these two

00:08:03

models you can prove something very cool

00:08:06

on six vertex model and on FK per

00:08:08

colation so in order to be able to tell

00:08:11

you a little bit about I mean why at

00:08:13

least to make you feel why this uh this

00:08:17

back and forth is going to be useful

00:08:18

what I should tell you is what's the

00:08:19

connection between the two models

00:08:21

because at this stage it's not that

00:08:23

clear so it's a connection that is due

00:08:26

to

00:08:27

backer Kellen and V

00:08:38

and I will not I mean I will not

00:08:40

describe it fully but uh at least I will

00:08:42

try to give you a hint so what happen is

00:08:45

the following so on one side you have FK

00:08:48

perol by the way don't hesitate to ask

00:08:51

questions I mean that's um so on one

00:08:53

side you have FK

00:08:58

percolation

00:09:00

on the other side you have six vertex

00:09:03

model so here you have omega so Omega is

00:09:06

really a subgraph of here you have omega

00:09:10

oriented it's an orientation of edges

00:09:13

but what you can do is the

00:09:15

following if you look here at a

00:09:17

percolation configuration there is a

00:09:19

natural families of Loops that you can

00:09:22

draw on the white lce maybe let me draw

00:09:25

it here so let's say

00:09:28

that

00:09:35

it's let's say that this EDG

00:09:42

is are the edges in my uh in My Graph

00:09:46

Omega and the light one are not in Omega

00:09:49

what you can do okay I did something

00:09:51

that I will regret I think so uh let's

00:09:54

see do I manage to so it should be the

00:09:57

medial

00:09:58

graph

00:10:05

whoa if you find it stressful it's much

00:10:07

more stressful for okay good so on this

00:10:12

thing we are missing colors ah no we are

00:10:15

not missing colors at

00:10:17

all

00:10:20

um what you can do is that you can

00:10:25

draw loops on the on the white lce by

00:10:29

this side iding that you draw for every

00:10:32

Edge which is not present in my original

00:10:34

thing you draw the maybe I'm going to

00:10:36

use another color just that we see so

00:10:39

these are the

00:10:41

edges of

00:10:43

Omega so what I can do is that for every

00:10:46

Edge which is not present in Omega I can

00:10:49

draw the Dual Edge what we call so just

00:10:52

the edge of the Dual

00:10:54

latis uh intersecting it in the middle

00:10:57

so I can KN that half both the Primal

00:11:01

and dual configuration and once I have

00:11:04

this there is a natural families of

00:11:05

Loops I can draw which are Loops that

00:11:08

are bouncing every time they hit either

00:11:12

a primal or dual

00:11:14

Edge so you see it's going to look like

00:11:18

that on the other side here there is one

00:11:20

that is bouncing like that so it's you

00:11:23

know they don't touch they really think

00:11:25

of them as as bouncing

00:11:28

really Etc here you have one from for

00:11:32

instance etc

00:11:34

etc is it clear for everybody what I

00:11:37

did well I hope because I cannot redraw

00:11:40

this a second time without and but okay

00:11:43

so this Omega from now

00:11:46

on I'm just going to think of it as this

00:11:48

family of Loops right I mean this is not

00:11:51

a very big uh difference it's in

00:11:54

bijection so I just think of it either

00:11:57

instead of thinking of it as a subgraph

00:11:59

I think of it as a familiar of loops and

00:12:01

in fact I can redraw the graph inside

00:12:03

easily but the advantage of seeing it

00:12:05

like that is that there is a

00:12:07

natural oriented Loop so let me maybe do

00:12:11

it like that Omega Loop there is a

00:12:14

natural family of oriented Loops I can

00:12:16

create which is the following what I'm

00:12:18

going to Simply do is I'm going to say

00:12:21

that I'm oriented each Loop in one way

00:12:23

or the other one direction or the other

00:12:28

okay okay and what I'm going to do is

00:12:31

that the probability of this uh oriented

00:12:33

Loop

00:12:34

model so it's not a it's not F so so

00:12:38

from yeah it's not FK and it's not six

00:12:40

vertex it's something different the

00:12:43

probability of an oriented Loop I'm

00:12:45

going to set it to be one

00:12:48

over a certain

00:12:50

constant times mu to the number of left

00:12:57

turns mu bar to the

00:12:59

number of right

00:13:02

turns I'm just stating this like that

00:13:05

okay I'm introducing the thing like

00:13:08

that and what I

00:13:11

claim is that if I choose p q a b c and

00:13:15

mu properly I'm going to be able to do

00:13:18

the

00:13:19

following if I sample something like

00:13:22

that and I forget about the

00:13:25

orientations I would end up with FK

00:13:28

percolation

00:13:30

if I sample something like that and I

00:13:32

forget the loops I just keep the

00:13:34

orientation of every Edge I will end up

00:13:37

with six vertex model

00:13:40

okay there will be a

00:13:43

cost is that it's not going to be a

00:13:45

probability measure mu will in fact not

00:13:47

be

00:13:49

positive that's why there is Mu and new

00:13:51

bar but you are going to see it's not

00:13:53

that problem so let's try to do

00:13:57

that so

00:14:00

maybe let's try to see how we would go

00:14:02

first from uh the oriented Loop model to

00:14:06

to FK per

00:14:08

cation so the the the

00:14:11

observation is that this oriented Loop

00:14:18

model new to the number of left terms

00:14:21

minus number of right terms because mu

00:14:23

and mu bar will be conjugate so it's

00:14:25

really me to number of left term minus

00:14:27

right terms the have another way of

00:14:29

writing it is that any

00:14:31

Loop oriented Loop clockwise is going to

00:14:34

have exactly four left terms more than

00:14:37

right terms right on the other hand any

00:14:41

Loop oriented counterclockwise will have

00:14:44

four this was quter clockwise for

00:14:48

you we'll have four right term more than

00:14:51

left terms so this is going to be in

00:14:53

fact proportional to Mu to

00:14:56

4times number of Loops oriented that

00:15:00

way times I mean minus number of Loops

00:15:04

oriented the other

00:15:07

way

00:15:08

okay but now if I forget about the

00:15:12

orientations I have exactly two

00:15:14

orientation for every Loop so this thing

00:15:18

now if I look at

00:15:19

P of a configuration Omega which would

00:15:23

be the sum of the Omega oriented

00:15:26

compatible with Omega so if I sum on all

00:15:29

those that are compatible with

00:15:34

Omega then what I'm going to end up with

00:15:38

is exactly mu to the 4+ MU bar to the 4

00:15:42

which corresponds for each Loop that I

00:15:44

have two orientation possible to just

00:15:46

the number of

00:15:49

Loops

00:15:51

right okay so let's say that I fix this

00:15:55

to be equal to square root Q I don't

00:15:58

know I'm crazy I do that so I choose mu

00:16:00

in such a way that mu to 4 plus mu to

00:16:03

the 4 is otk Q I get that the prity of

00:16:07

Omega in this model where I'm projecting

00:16:09

like that is proportional to square root

00:16:12

Q to the number of

00:16:14

Loops in fact sare root 2 Q to the

00:16:16

number of Loops by using just

00:16:20

Elementary

00:16:21

manipulations it's exactly equal to this

00:16:24

when p is equal to otk Q over 1 +un Q so

00:16:28

this in fact

00:16:29

fact I can uh sorry so this thing sorry

00:16:33

so

00:16:34

it's yes it's proportional to this and

00:16:38

so this thing when I'm in proportional

00:16:39

just up to a universal constant that

00:16:41

depends on nothing and so this square

00:16:43

root Q to the number of Loops what I

00:16:45

claim is that it's exactly equal

00:16:49

to root Q over 1 + root Q to the number

00:16:53

of edges in Omega 1 / 1 + otk Q to uh

00:16:58

the constant minus number of edges and Q

00:17:03

to the number of

00:17:04

clusters I claim

00:17:07

that it's really an elementary exercise

00:17:10

to check it something like the number of

00:17:14

open edges person I mean what it's it's

00:17:17

really easy to uh to

00:17:19

do except it's not true but uh because

00:17:23

I'm on the Taurus but uh I mean on the

00:17:25

Taurus you need to modify accordingly

00:17:27

but but it's easy

00:17:29

and there there is something good

00:17:31

because this thing here square root of Q

00:17:33

over 1+ root Q it's not a random number

00:17:36

it's actually exactly the value of the

00:17:38

critical point for FK

00:17:41

percolation lucky

00:17:43

us so this thing is actually

00:17:46

PC so what we are claiming is that the

00:17:49

projection like that if you choose so

00:17:54

if mu to 4 + mu bar to 4 is OT cu

00:18:00

then this is exactly FK

00:18:02

percolation for PC and Q so add

00:18:08

criticality okay and here really I'm

00:18:11

basically not uh hiding anything

00:18:15

basically okay now let's look at the

00:18:19

other

00:18:26

projection so the O projection

00:18:30

you you forget about the orientation uh

00:18:33

you forget about the loops so what

00:18:36

happens is that if you had two turns

00:18:39

like that I mean so let let's look at

00:18:42

how you can get this thing for instance

00:18:45

a Vertex like that you agree that there

00:18:47

is only one

00:18:49

possible family of uh I mean one

00:18:52

possible local Arrangement that gives

00:18:54

you that is if the original Loops were

00:18:56

doing like that that's the only way

00:18:59

same thing for the others let me still

00:19:01

draw them this will be like

00:19:04

that then

00:19:07

this would be this

00:19:10

one this would be this one you start to

00:19:14

understand and these two well these two

00:19:18

are a little bit different because they

00:19:20

can come from two potential Arrangement

00:19:23

right this one could be this but it

00:19:26

could also be this

00:19:30

and same thing here this one could

00:19:34

be well

00:19:36

this all

00:19:40

this I'm always

00:19:43

wrong what did I do this was like that

00:19:46

sorry and this one is like

00:19:50

that

00:19:52

okay so let's look at the contribution

00:19:55

there so here there is one left term one

00:19:58

right term so the contribution of the MU

00:20:00

and mu bar cancel I get weight one here

00:20:04

I get weight

00:20:06

one one one here the two contributions

00:20:12

give you actually two left terms or two

00:20:14

right terms so I'm going to get mu^ 2

00:20:17

plus mu bar squ right and here same

00:20:24

thing and of course the fact that I was

00:20:27

Loops I mean I was made of Loops

00:20:29

guarantees that I have the ice schol so

00:20:32

what do I get I get an FK a six vertex

00:20:36

model with a = b = 1 okay and C = mu bar

00:20:43

s + mu squared and let me I mean do not

00:20:47

ask me how it Expresses in terms of

00:20:49

square root Q because I'm always

00:20:50

confused but yeah there is a simple way

00:20:53

of of getting it in terms of SC so with

00:20:57

this kind of M model above you get

00:21:01

two uh your two models as projection of

00:21:05

this so here you notice that for

00:21:07

instance here looking at this thing you

00:21:10

notice that except when Q is larger than

00:21:13

four larger or equal to

00:21:16

four mu is not real right I mean I

00:21:20

should have said that here I could have

00:21:22

put a mu minus one maybe instead of new

00:21:24

bar but I mean my point is that for Q

00:21:27

smaller equal to 4 I get two complex

00:21:30

conjugates when mu when Q is larger than

00:21:34

four I get just a real and one of so for

00:21:37

Q larger than four it is actually a real

00:21:41

it transfers into a real

00:21:42

coupling Hess

00:21:44

coupling the thing is that for Q larger

00:21:47

than

00:21:48

four um well the models do not undergo

00:21:53

the FK percolation is undergoing a

00:21:55

discontinuous pH transition so in

00:21:58

particular

00:21:59

you don't expect anything super uh

00:22:04

amazing at PC and on the side of the six

00:22:07

Vex

00:22:08

model well it's a first order phe

00:22:11

transition if I mean depends exactly how

00:22:13

you but I mean in some sense for

00:22:15

instance one thing we could say is that

00:22:17

it's a Frozen I mean there there is a

00:22:19

localization of the height function I

00:22:20

will come back to this

00:22:22

later so we will be interested in the

00:22:27

Q

00:22:29

smaller equal to four and in order for

00:22:31

FK to be a little bit nice I'm going to

00:22:34

also ask that one is smaller or equal to

00:22:37

Q for a reason that I mean that's where

00:22:40

you want I mean that's when your your

00:22:43

model is kind of nice it has fkg

00:22:45

properties and like

00:22:46

that okay so at this stage maybe you

00:22:50

already completely lost I'm

00:22:53

sorry and but but at least there are two

00:22:57

I mean I think you can split in two uh

00:22:59

groups for those that are not

00:23:02

lost either your favorite model is B

00:23:05

percolation in this case you can think Q

00:23:07

equal

00:23:08

1 then uh mu is a little bit weird it's

00:23:12

like e to the I pi over 12 or something

00:23:15

like that I mean something

00:23:17

weird the other option which I am

00:23:21

totally happy with as well I like both

00:23:23

models is Q equal

00:23:25

4 because for Q equal 4 what is very

00:23:27

good

00:23:28

is that it is still a continuous SP

00:23:31

transition but mu is just one what does

00:23:35

it mean it means that there it's a fully

00:23:37

like honest coupling and basically what

00:23:41

it gives you is the following you just

00:23:44

Orient Loops one half one half that's

00:23:47

your model so in some sense this is

00:23:49

maybe what is the simplest in order to

00:23:51

understand the

00:23:52

coupling right that's the only case

00:23:54

where it's a

00:23:55

copy okay so now what are we going to do

00:23:58

with this

00:24:01

connections so by the way for if you are

00:24:04

afraid about uh the fact that it's not a

00:24:07

coupling you know it's not because it's

00:24:10

not a coupling that if you take a

00:24:12

natural observable on this side it's not

00:24:14

going to translate into something

00:24:16

natural on this side right it could be

00:24:18

that just it goes through complex number

00:24:20

but goes back to something isable and

00:24:21

you're going to see that's what will

00:24:23

happen Okay so now what do we know on

00:24:26

these models so let's start on the FK

00:24:28

percolation side on the FK percolation

00:24:31

side a few years

00:24:35

ago we Prov something that we we got

00:24:38

very surprised about I'm must say this

00:24:40

is this kind of things where you don't

00:24:42

expect that you are going to get this

00:24:44

and what we did is the following so it

00:24:46

was a work with the

00:24:50

so klovski

00:24:53

this is the hardest part of my talk

00:24:55

every single time it's not confusing the

00:24:57

order so

00:25:01

because

00:25:04

manes and mesar

00:25:07

so both kashun and andar were my my PhD

00:25:11

students and Manu and kosi are

00:25:13

colleagues so what we prove is the

00:25:16

following we prove that if you think so

00:25:20

if you take Q between one and

00:25:24

four and you take P equal PC so if you

00:25:28

look at the critical FK percolation

00:25:30

between one and four you have rotation

00:25:33

in

00:25:34

variance so what does it mean it's going

00:25:37

to mean for instance something like if I

00:25:39

look at

00:25:41

probability and I take X1

00:25:44

X2 X3 X4 Etc like a certain number of

00:25:49

points and I look at it on the on a ltis

00:25:53

so I mean I'm going to try to be

00:25:55

consistent I'm look at it on this ltis

00:26:02

well this is

00:26:03

equivalent as the mesh size of the ltis

00:26:06

TS to zero let's say or if you prefer

00:26:08

when you take points farther and farther

00:26:10

away but I mean both things are it's

00:26:12

going to be equivalent to exactly the

00:26:13

same thing except you look at row of

00:26:17

X1 row of X2 I mean row

00:26:20

Theta for any so you rotate all

00:26:27

the all the things and you stay on the

00:26:30

same ltis or if you prefer you don't

00:26:32

move the points but you rotate the

00:26:34

lce so it's something a little bit this

00:26:37

is exactly an example for people that

00:26:39

were here yesterday it's an example of

00:26:42

emergent property when you take larger

00:26:45

and larger distances you get a rotation

00:26:48

inv variance that was absolutely not

00:26:50

present in the original model and this

00:26:52

is typical from uh so why why do I I

00:26:57

mean why did I say that we were

00:26:58

surprised about this this

00:27:00

result is because there is this kind of

00:27:04

saying in

00:27:06

physics that in particular percolation

00:27:10

models but maybe other models should be

00:27:12

conformally invariant meaning that what

00:27:15

I just stated here so so

00:27:18

if for every

00:27:22

SATA this becomes equivalent when Delta

00:27:25

say so if this is Delta meas size when

00:27:29

Delta goes to zero

00:27:33

okay and whatever the number of points

00:27:35

Etc so so yeah what is predicted is that

00:27:38

there is even

00:27:42

yes oh yes maybe I should uh any

00:27:46

anything you want so basically you could

00:27:48

ask X1 is equal to X2 I mean it's

00:27:52

connected to X2 X3 is not connected to

00:27:54

X4 basically all the microscopic

00:27:56

properties are fine but you are right I

00:27:59

should have a so for instance yeah let's

00:28:01

say

00:28:02

that yeah it's a catastrophically poorly

00:28:06

stated theorem but it's a cool theorem

00:28:09

it's just me who is stating it very

00:28:10

poorly that's uh yeah and so so why do

00:28:14

people believe in conformal invariance

00:28:15

because the first thing is that you

00:28:17

believe in scal invariance because scal

00:28:19

invariance will correspond to assuming

00:28:21

that this is going to converge when

00:28:24

Delta tends to

00:28:25

zero because it's the scal of the ltis

00:28:28

is really what you expect to be the

00:28:31

scaling maybe it's converging when you

00:28:33

properly rescale it because this is

00:28:35

tending to zero but I mean maybe by

00:28:37

putting Delta to some power for instance

00:28:40

it will converge and this this is a

00:28:43

fairly reasonable assumption in some

00:28:45

sense to believe that when Delta tends

00:28:47

to zero the things when you are at

00:28:49

criticality converge to something it's

00:28:51

very difficult to justify but this one I

00:28:52

can

00:28:54

buy then physicists argue through the

00:28:58

renormalization

00:29:00

group that okay if it converges it

00:29:03

converges to a fixed point if the fix

00:29:06

point is unique then if you start from a

00:29:08

ltis or a rotation of the ltis you

00:29:10

should converge to the same fixed Point

00:29:12

hence rotation in

00:29:13

variance as

00:29:15

mathematicians I mean we have to believe

00:29:18

them in the sense that they are right

00:29:19

it's going to be rotation in Varan

00:29:21

that's actually but there is really I

00:29:23

mean we do know a few dynamical system

00:29:26

that don't have a unique fix point right

00:29:28

it happens I was told so uh you need to

00:29:32

be very careful with this argument and

00:29:33

in some sense there are no very

00:29:36

strong argument for the fact that there

00:29:39

should be a unique fix point so we all

00:29:42

we really believe that this step we had

00:29:44

no clue how to do it was kind of a

00:29:46

wishful thinking and then once you have

00:29:49

scale invariance and rotation invariance

00:29:52

there is this kind of bootstrap I mean

00:29:55

this way of kind of merging the two to

00:29:57

get for more inv variance using

00:29:59

basically locality of of certain

00:30:01

quantities but this one I could

00:30:04

definitely believe that this was doable

00:30:06

actually we almost have a proof for ver

00:30:08

peration that if you have translation I

00:30:10

mean scale invariance and rotation

00:30:12

invariance you must have conformal

00:30:14

inance there is no

00:30:15

choice so this third step looked

00:30:19

reasonable Second Step looked completely

00:30:22

hopeless so we thought okay let's start

00:30:25

with the first step try to prove scale

00:30:27

invariance by trying to prove scale in

00:30:29

variance we understood that we were

00:30:30

would never do it basically but we felt

00:30:34

almost by mistake on rotation in so I I

00:30:38

don't I cannot really say why it's this

00:30:42

one that felt first but this step works

00:30:45

well problem is that then we really

00:30:47

don't know how to do first step or third

00:30:50

step actually except in special cases

00:30:52

for step so we were stuck with this

00:30:55

thing until we realize

00:30:59

and that's where uh I mean I'm not going

00:31:01

to prove Conant I should say but um

00:31:05

until I mean not now at least I'm hoping

00:31:09

to do it one day but

00:31:12

um so until we realize something very

00:31:17

strange which is that I mean first if

00:31:21

you look add the mapping

00:31:24

here in fact if you have rotation inv

00:31:26

variance here it's it's really not so

00:31:29

difficult to prove that you have

00:31:30

rotation in variant here as well I mean

00:31:33

not too difficult it's not easy but this

00:31:36

is more like a PhD dis kind of work it's

00:31:40

not uh I mean this is difficult but it's

00:31:43

in the real of what you think you can do

00:31:46

uh once you got to this stage so from

00:31:50

that maybe that's uh so we got as a

00:31:55

corollary in some sense that six vertx

00:31:57

model

00:31:59

is

00:32:03

rotationally

00:32:05

invariance when so a = b = 1

00:32:12

and C so I think it's between Square <

00:32:16

TK 3 and two something like that when you

00:32:19

look Q between one and four it gives you

00:32:24

that don't don't take I mean it's maybe

00:32:26

not Square three but

00:32:28

it is something between one and

00:32:32

two but what is surprising is that when

00:32:36

you are on the six vertex model this

00:32:38

rotation invariance gives you much more

00:32:40

information that for

00:32:42

FK I mean at least we believe that

00:32:45

that's what we are trying to write and

00:32:46

for now the proof didn't collapse but

00:32:49

I'm not going to State it as a theorem

00:32:51

but basically what it tells you is the

00:32:53

following when you look at the six

00:32:55

vertex

00:32:56

model there there is a natural way so

00:32:58

let's say you look at

00:33:01

expectation so for the six vertex model

00:33:05

of something like h of so maybe let me

00:33:08

not use X1 but um U1 minus h of

00:33:14

U2 h of u3 minus h of

00:33:17

u4 Etc some kind of product of

00:33:22

differences so you want you where H is

00:33:25

the model is the height function that

00:33:27

you define as

00:33:28

follows because you have the ice hole

00:33:32

every single time you jump through an

00:33:34

edge if you cross the edge that way you

00:33:38

increase so you are going to define the

00:33:39

height function

00:33:42

on the vertices by saying if you have an

00:33:47

edge like that and here you put H as a

00:33:50

height you are going to put H + one here

00:33:53

okay and if it's in the other

00:33:56

direction

00:33:58

you go from H to H minus one okay

00:34:01

because you have the ice rule it's

00:34:05

consistent you define the height

00:34:07

function up to MTI I mean addition but

00:34:11

as long as you take differences the

00:34:13

thing is perfectly well defined so for

00:34:16

instance when I mean rotation in

00:34:18

variance I mean that this is going to be

00:34:20

invariant when I rotate the U1 U2 u3 u4

00:34:25

okay but on the six Vex side side there

00:34:28

is something that doesn't happen on the

00:34:30

on the FK side is that these kind of

00:34:33

things are naturally expressed in terms

00:34:35

of uh what we call transfer

00:34:39

matrices and the rotation in variance

00:34:42

seems to be giving in fact very strong

00:34:45

connection between two matrices that are

00:34:47

used in order to express this thing so

00:34:49

the standard transfer Matrix and a

00:34:52

matrix which correspond to Shifting the

00:34:54

configuration by one on the right let's

00:34:56

call it the shift MX matx so these are

00:34:59

two natural matrices they Co I mean they

00:35:01

commute for instance so they can be co-

00:35:03

diagonalizable and what happens is that

00:35:06

the rotation invariance basically allows

00:35:08

you to relate Theon values of both

00:35:10

things in this in this bases in a very

00:35:14

efficient way we don't really understand

00:35:16

it's due probably to the locality of uh

00:35:20

of the whole um of the six vertex model

00:35:23

it's much more local in in in this

00:35:25

expression than in FK percolation so in

00:35:28

particular this connection between these

00:35:30

matrices it allows you to say something

00:35:32

pretty cool which is that this quantity

00:35:37

is harmonic in each one of the

00:35:40

coordinates until I mean except when you

00:35:42

merge

00:35:44

them and once you have that well you are

00:35:48

basically done there is only one kind of

00:35:50

height function that has his property

00:35:53

and it's the G fre

00:35:54

field so from that we believe that here

00:35:58

again now we are at the at the level

00:36:00

where we are writing the paper but we

00:36:02

already discovered two very big mistakes

00:36:04

that we each one I mean we managed to

00:36:07

fix both of them but uh since we found

00:36:11

them in the two first pages of the

00:36:14

paper I'm not sure we can claim this but

00:36:17

I don't resist any way I find the the

00:36:20

thing interesting without even having an

00:36:22

actual theorem so let me um keep going

00:36:26

but

00:36:28

so theorem with a equation

00:36:33

mark if I take this model so if I take

00:36:38

uh H six vertex H Delta which is sample

00:36:43

to according to a six vertex on Delta Z2

00:36:49

so I started with a with a with a Taurus

00:36:52

but you can really take an infinite

00:36:55

volume limmit there is no problem so

00:36:57

then you are on the six Vex on Delta Z2

00:37:00

so what we believe is that H Delta

00:37:03

converges in

00:37:06

low to um the

00:37:10

GFF with a certain variance that depends

00:37:14

on on

00:37:15

C and we also believe we can compute the

00:37:19

variance so what do I mean by that just

00:37:21

to be certain I mean as a distribution

00:37:24

meaning that what I'm going to do is I'm

00:37:26

going to take for instance

00:37:28

uh

00:37:29

f a compact is supported smooth function

00:37:33

and I'm going to test it against Edge

00:37:36

Delta and what it's going to convert to

00:37:39

so this is going to be something like

00:37:41

sum of

00:37:43

X of f

00:37:45

ofx h Delta of

00:37:48

X okay and maybe with a Delta squar just

00:37:51

to to make it like an

00:37:54

integral and this is going to converge

00:37:56

to a normal G random

00:37:58

variable with a certain variance which

00:38:01

will be given by double integral of f

00:38:04

ofx f of

00:38:05

Y something like well K of

00:38:09

XY or let's say G of XY DX Dy and this

00:38:14

in order to get this you need actually F

00:38:16

to have zero average

00:38:18

but so if f is zero average this thing

00:38:22

converges in

00:38:23

low to this quantity which is exactly

00:38:26

what you get for for for the sorry for

00:38:29

um so you take a you test your height

00:38:32

fun height function against a smooth

00:38:35

compactly supported function it should

00:38:37

converge to a g r variable with the

00:38:39

right

00:38:41

variance so here there is a sigma

00:38:44

squared of

00:38:46

C okay so this yeah this is a little bit

00:38:49

surprising because in some sense this is

00:38:51

a much stronger result than this in the

00:38:54

sense that it tells you what is the

00:38:55

scaling limit of six VX model at least

00:38:58

of all this kind of observable of the

00:38:59

six VX

00:39:01

model so it's a little bit surprising

00:39:03

that you can

00:39:04

get kind of the scaling full scaling

00:39:07

limit in one of the models while it was

00:39:10

only rotation in variance on the other

00:39:11

model so here I was careful that's the

00:39:14

only thing I did well when I stated this

00:39:16

thing I was careful to put an equivalent

00:39:19

meaning I have no clue I don't know that

00:39:20

this

00:39:21

converges I have zero clue I just know

00:39:24

that these two things the ratio tends to

00:39:26

one

00:39:28

okay here I have really

00:39:32

limit okay I have no clue what time it

00:39:36

is sorry oh yes I could

00:39:39

just okay

00:39:42

so so it's like there was if if we

00:39:46

summarize in uh I mean maybe I'm going

00:39:48

to keep this here but if we

00:39:56

summarize

00:39:58

by going from

00:40:00

FK to six vertex you go from rotation in

00:40:05

variance to full scaling limit

00:40:11

again of this type of observable you

00:40:14

could imagine to look at other

00:40:15

observable like maybe correlations

00:40:17

between being in some direction that we

00:40:20

don't know how to reach for now but for

00:40:22

these observables we have so now the

00:40:25

question I mean it's like now that I

00:40:26

have the full scaling limit what can I

00:40:28

hope to

00:40:29

get on the FK side back

00:40:33

right and so it depends a little bit how

00:40:36

optimistic you

00:40:37

are I'm very optimistic but I'm also

00:40:40

very wrong quite often so

00:40:44

um of course if you have a full scaling

00:40:47

limit of something you may hope to get

00:40:49

the full scaling limit on the other side

00:40:52

so potentially but this we know they are

00:40:56

difficult questions

00:40:58

you could hope that it characterizes so

00:41:01

the the convergence on this side

00:41:03

characterizes the convergence on this

00:41:05

side and then if this was the

00:41:08

case then it would tell you so there

00:41:10

will be go back to FK it would tell you

00:41:14

that FK converges to Cle Kappa with a

00:41:18

certain Kappa so conformal Loop and

00:41:20

symbol with a right

00:41:24

Kappa so this is a little bit too

00:41:28

optimistic let's face it um it's really

00:41:32

unclear that you have sufficient

00:41:34

information on the right so what the the

00:41:36

the information you have for the six

00:41:38

vertex model is really the convergence

00:41:39

of all these

00:41:42

observables so you could ask okay what

00:41:44

is the kind of information it gives me

00:41:46

on the FK side and I will maybe give you

00:41:48

one example just

00:41:50

after but a Priory yeah it should tell

00:41:53

you that certain quantity on the left

00:41:55

side on FK side converges converge right

00:42:00

and if there are sufficiently many of

00:42:01

those maybe it characterizes the

00:42:04

limit I would say to be conservative I

00:42:08

would actually kind of believe that for

00:42:10

Q equal 4 it's

00:42:12

true so if I would have to guess really

00:42:15

like in front of

00:42:18

God uh God being stas of know I

00:42:22

don't no no but um so if you would have

00:42:25

to guess maybe maybe for qal 4 I would

00:42:28

say that it characterizes AIT under some

00:42:31

mild thing that you want to add I it's

00:42:35

kind of you get a lot of information the

00:42:38

loops for Q equal 4 they don't touch

00:42:40

each other the coupling is fairly act

00:42:44

explicit so I could believe that Q equal

00:42:46

4 is going to work to get full conformal

00:42:48

inves because if you get Cappa then you

00:42:50

have ways to even get back to the finite

00:42:52

volume so uh I mean in finite domain so

00:42:56

so that I could be for Q strictly

00:42:58

smaller than four I'm a little bit less

00:43:01

optimistic I mean for for people who

00:43:03

know for instance there are all these

00:43:04

question about double

00:43:06

diers where there are certain

00:43:08

observables by Kenyon and then by dueda

00:43:11

that were obtained and there was this

00:43:12

question whether they characterize the

00:43:15

loops and these observable they really

00:43:17

seem much more General than those and

00:43:20

still it was quite difficult to prove

00:43:22

that they characterize the loops so okay

00:43:26

I would maybe not

00:43:27

I mean I would not know what to answer

00:43:29

if God ask me for Q smaller than

00:43:31

four but what is very surprising is that

00:43:34

you can still get very interesting

00:43:37

information and let me try to give you a

00:43:40

glimpse at this with one example so let

00:43:45

me put it

00:43:47

here so maybe not full comir or

00:43:52

invariance but uh at least for the

00:43:55

statistical physicist that I am already

00:43:58

very interesting information and let me

00:44:00

not tell you which which information let

00:44:02

me first try to to make a small

00:44:05

computation with you so what this means

00:44:07

is what this means that if I take

00:44:10

expectation for

00:44:13

instance of

00:44:16

exponential of

00:44:18

Lambda time t f of H Delta right if I

00:44:25

take the characteristic function let me

00:44:27

put an

00:44:28

i this converges to e Theus Lambda 2

00:44:32

over 2 uh time the right uh double

00:44:36

integral

00:44:45

right okay with the right Sigma squar or

00:44:49

so so my point is that this is

00:44:51

completely explicit what it converges to

00:44:54

I know the convergence of this random

00:44:56

variable I take the characteristic

00:44:57

function in

00:44:58

convergence but now I can wonder what

00:45:02

this is equal

00:45:03

to in the F so this is something that is

00:45:07

purely expressed in the six Vex

00:45:10

model but in fact it translate into

00:45:14

something that is not that bad on this

00:45:18

side because if you think about

00:45:21

it if I want to compute so so f is zero

00:45:25

average okay so and compactly supported

00:45:27

so let's say you know it's supported

00:45:30

there and I want to

00:45:33

know h of U here okay h of U minus h of

00:45:40

something very

00:45:42

far then in

00:45:45

fact it's fairly easy to see that there

00:45:48

is certain so any Loop that is not

00:45:51

surrounding one of the two points is not

00:45:52

going to contribute anything to the

00:45:55

difference between edge here and Edge

00:45:57

here right but Loops that do surround

00:46:01

one of the two for instance this

00:46:04

one they are going to make so if they

00:46:06

oriented that way it means that just on

00:46:09

the left of the loop inside I have a

00:46:11

certain height and outside I have the

00:46:14

height plus one and if it's oriented the

00:46:16

other way I have height and height minus

00:46:19

one so what I can say is that basically

00:46:22

if I say that U is here h of U minus h

00:46:26

of let's say infinity or I mean edge of

00:46:29

something very

00:46:30

far let say

00:46:32

u0 it's roughly the

00:46:35

sum of the Cai C for C which is a

00:46:39

loop

00:46:43

surrounding U or u0o but not

00:46:50

both and this is size is just the

00:46:53

orientation so it gives you plus one if

00:46:55

it's oriented in one way and minus one

00:46:57

if it's oriented the other

00:47:01

way so why am I saying that because if

00:47:05

you think of it that way and you

00:47:07

remember that the law of the oriented

00:47:10

Loop model there was also expressed as

00:47:14

Mu to the four number of Loops surround

00:47:17

in one way number of Loops in the other

00:47:19

way

00:47:21

basically here if you write this thing

00:47:24

as sum of

00:47:25

x there is a Delta Square F ofx and now

00:47:29

this Edge Delta of X you reexpress it as

00:47:33

sum over the

00:47:35

loops

00:47:36

surrounding X of C then by exchanging

00:47:41

the

00:47:42

two this is just the sum of every Loop

00:47:46

C of c

00:47:49

c times so this is really the

00:47:52

orientation times the integral of f

00:47:55

ofx inside

00:47:57

C DX something of this kind I'm cheating

00:48:01

I mean okay I'm I'm trying to simplify I

00:48:04

mean to just give you a gimps I totally

00:48:06

understand that you cannot follow

00:48:08

exactly what I'm doing it's not your

00:48:09

fault it's mine

00:48:12

clearly but what I want to say here is

00:48:14

that the LW I mean or the expression for

00:48:19

the weight

00:48:21

is Mu to the 4 sum of the

00:48:25

C

00:48:28

and I'm twisting it when I do

00:48:29

exponential of I Lambda blah blah I'm

00:48:31

twisting it by exponential of I Lambda

00:48:35

sum of C integral of f of x DX for X in

00:48:41

the inter so in the interior of

00:48:44

c and this is basically the weight that

00:48:46

I get so what all of this to

00:48:51

say that when I look at this exponer sh

00:48:55

here if you sit at the table and you you

00:48:58

do it

00:49:00

sufficiently

00:49:02

carefully you are going to end up with

00:49:06

what I call my magic formula it's not

00:49:08

mine so I mean that it's my favorite

00:49:10

magic formula these days and it gives

00:49:13

you that

00:49:14

exponential expectation of e to the I

00:49:17

Lambda TF of H Dela in fact it's equal

00:49:22

at it's equal to expectation for FK of

00:49:27

the product over all the

00:49:31

loops of

00:49:34

cos two I mean okay so let's say that mu

00:49:38

to the 4 is e to the i 2 pi Theta so cos

00:49:43

2 pi theta plus integral of f ofx

00:49:47

DX on the interior of

00:49:50

C divided

00:49:52

by cos 2 pi

00:49:54

Theta it's what you get just it's this

00:49:57

is really an

00:49:58

identity you get

00:50:01

this but once you see that so first it

00:50:04

tells you exactly in some sense you

00:50:06

don't have more information in the

00:50:08

convergence of the distribution of the

00:50:11

height function to GF if you don't have

00:50:13

more information than this information

00:50:16

so it tells you that the information you

00:50:18

have for FK is really that all those

00:50:21

quantities converge for any F compactly

00:50:24

supported blah blah so just to make it a

00:50:26

little bit more precise that there are

00:50:28

certain observable that converge those

00:50:30

are the

00:50:32

observables and you also see why for for

00:50:35

qal 4 it may look a little bit simpler

00:50:37

because Theta is zero because mu is one

00:50:42

so here it's like cost of the inter it's

00:50:44

like weighting your Loops by a quantity

00:50:47

that depends on F but now for

00:50:50

instance you can start

00:50:53

playing imagine that you take f

00:50:59

let's say we are for Theta equal Z so

00:51:02

for qal

00:51:04

4 take F which is basically a

00:51:07

dra a sum of two dra at zero and at

00:51:11

certain X and what you're going to do is

00:51:14

you're going to put as a as a mass pi/ 2

00:51:18

so it's a d with mass pi/

00:51:22

two so F you take this and you substract

00:51:26

the same thing

00:51:27

at

00:51:28

x what does this function so when I mean

00:51:31

the D is you know approximate by kind of

00:51:34

putting the mass pi/ two on a small bowl

00:51:36

around each

00:51:38

point when I look at this what do I

00:51:41

get so if theta equals

00:51:44

z i get C of integral of f ofx DX so in

00:51:49

particular what it gives me that if I

00:51:51

have a

00:51:52

loop that surround X for instance and

00:51:55

only X

00:51:57

then COS of minus Pi / 2 zero so all the

00:52:02

loops that surround only one point they

00:52:04

get weight

00:52:06

zero same thing for here so this

00:52:09

quantity is what it's probability that

00:52:12

there are no Loops surrounding one or

00:52:14

the other which exactly means

00:52:16

probability that the two points are

00:52:18

connected by a path actually either

00:52:21

connected in in the percolation

00:52:23

configuration or it's dual but it's

00:52:26

twice of being con so what do I mean by

00:52:29

that is that on this

00:52:31

side for f equal to this I have a pretty

00:52:34

explicit

00:52:37

computation I know how it behaves and on

00:52:40

this side it's probability that the two

00:52:42

points are

00:52:43

connected so from that you get the one

00:52:46

arm exponent what we call so you or if

00:52:48

you prefer you get just the scaling

00:52:50

limit of probability of being connected

00:52:52

to each

00:52:53

other I said I did it for this for Q

00:52:56

equal 4 but for any any other Q you you

00:53:00

you may play the same game just you need

00:53:02

this plus this to be pi over

00:53:05

two and but I mean for all the Q There

00:53:07

are difficulties because if you put this

00:53:10

plus this equal pi/ two on one one of

00:53:13

the vertices on the other vertex you

00:53:15

need to cancel the sum to get zero

00:53:17

average and it will not give you

00:53:19

something that gives you zero so you

00:53:21

need to work but

00:53:23

basically what we are yeah practically

00:53:25

certain about is that this is much more

00:53:28

safe than this theem is that from this

00:53:30

theorem you get the one arm exponent for

00:53:32

any Q so in particular for

00:53:35

percolation it gives you the 5 over 48

00:53:38

exponent for for per percolation on the

00:53:41

Square l so remember for for for B

00:53:45

percolation stas of prove conformal

00:53:47

invariance for side percolation on the

00:53:50

Triangular ltis and then from that you

00:53:52

get the critical exponent for instance

00:53:55

here it's on the square is we don't we

00:53:57

didn't know up to now how to compute

00:53:59

this thing so we would get the one arm

00:54:01

exponent for this in fact we would get

00:54:03

it for any FK

00:54:05

percolation there are other exponents

00:54:07

you want to get in particular there is

00:54:10

another one which is called the

00:54:11

influence exponent because if you get

00:54:13

this one you get all the thermodynamical

00:54:16

exponents like how the free energy is

00:54:18

behaving how the probability of being

00:54:20

connected to Infinity behave Etc ET it's

00:54:24

we we understand how you should get it

00:54:26

but for some weird reason we are missing

00:54:29

a small thing but for instance we can

00:54:30

also get the two arm exponent which I

00:54:33

don't really know why two exponent than

00:54:36

than anything so the thing that I wanted

00:54:38

to say here is that this is a big

00:54:40

question mark whether you get CLE but

00:54:44

critical

00:54:47

exponents this is pretty safe I think

00:54:50

there are at least many critical I mean

00:54:53

there are critical exponents that you

00:54:54

can get so in some sense it's also a

00:54:57

message of hope I mean of course it

00:54:58

could be that you can get critical

00:55:00

exponents and not C convergence but

00:55:03

still I find it already a little bit

00:55:04

surprising that you can get critical

00:55:07

exponents I would have guess that it was

00:55:09

a zero one low so either you really get

00:55:12

something or you don't get anything

00:55:14

interesting but uh yeah so that's uh

00:55:18

that's the end of my talk tomorrow it

00:55:20

will not look at all the same it's a

00:55:22

completely different problem so don't

00:55:24

worry if you didn't like this one

00:55:26

uh and even the style I think tomorrow

00:55:28

will be maybe a little bit more

00:55:31

rigorous well no I should not say that

00:55:33

but

00:55:34

uh but yeah so I wanted to tell you

00:55:37

about that because I find it a little

00:55:39

bit uh mesmerizing that this rotation in

00:55:42

variance get upgraded to full scaling

00:55:44

limit on the six Vex model I'm pretty

00:55:46

sure that smarter people than me would

00:55:48

understand why but immediately but in in

00:55:51

our case we are we are a little bit

00:55:53

surprised by this and then you want to

00:55:56

pull back the thing on the other side

00:55:58

just just to tell you one thing that

00:56:01

it's a question I'm asking people that

00:56:02

play with CLE more than I do which means

00:56:05

a lot of people um there are I mean you

00:56:09

get this formula here so this

00:56:13

formula this converges to to

00:56:16

GFF so in the scaling

00:56:19

limit this

00:56:20

is GFF and what is for

00:56:24

sure is that here you get the scaling

00:56:27

limit of

00:56:28

FK and we we all believe that you get

00:56:32

C I mean at least the reasonable people

00:56:35

believe

00:56:36

that so you should get a formula like

00:56:40

that in the

00:56:41

limit the magic formula should pass to

00:56:44

the limit in fact you can even justify

00:56:46

that it does pass to the Limit what you

00:56:48

don't know is that you get c cap and

00:56:50

very

00:56:53

surprisingly when I ask people in the

00:56:55

Continuum where whether if you start

00:56:57

with this observable on C Kappa you do

00:57:00

get GFF on this side it seems to be a

00:57:03

very difficult question so for Q equal 4

00:57:05

for for Kaa equal 4 it's completely

00:57:07

obvious because in the continum if you

00:57:09

take a CLE 4 and you Orient the loop one

00:57:12

half one2 or I mean Pi / 2 minus pi/ two

00:57:16

you end up with

00:57:18

GFF so this is like the classical

00:57:21

coupling between C4 and GFF here it

00:57:24

seems to be suggesting a very natural so

00:57:26

it's not a coupling but a very natural

00:57:29

relation between C Kappa and GFF but it

00:57:32

seems to be a little bit orthogonal to

00:57:35

the tools that were developed for C

00:57:36

Kappa so it seemed that it's not obvious

00:57:40

at all to get this thing and I find it a

00:57:42

little bit surprising because you see

00:57:44

for instance it would tell you if you

00:57:45

would prove this it would tell you that

00:57:48

if you are characterized by the scaling

00:57:51

limit then you must be

00:57:53

ca it's kind of this is a question that

00:57:56

is much simpler a prior because it's not

00:57:58

a question of measurability it's like if

00:58:00

you would be measurable then you should

00:58:03

be

00:58:05

ca okay well that's if there were people

00:58:08

that are doing C Kaa for a living don't

00:58:11

hesitate to try to prove that please

00:58:13

thank you very

00:58:16

[Applause]

00:58:19

much so what do you get in the six Vex

00:58:22

model you get I think Cal < tk3 or

00:58:25

something something like that I think so

00:58:27

it's not a special model really or no it

00:58:30

doesn't look like a special model on on

00:58:32

the C on the the six VX in fact it was

00:58:35

the reason why I started being

00:58:37

interested in the first place in looking

00:58:39

at the six Vex model because the height

00:58:42

function of the six vertex model it has

00:58:44

an fkg property so positive Association

00:58:48

property up to c

00:58:51

one so it means that it doesn't see at

00:58:54

all in terms of probabilistic properties

00:58:56

it doesn't see at all s

00:58:58

root3 while on the FK side the fkg

00:59:01

property breaks down at at qal 1 so I

00:59:05

was interested in looking at FK at six

00:59:07

Vex model because in some sense it's

00:59:09

just saying that the positive

00:59:11

Association properties of percolation

00:59:13

break down because you are looking at

00:59:15

the

00:59:16

wrong uh properties if you look at the

00:59:19

six VX model Associated to it everything

00:59:22

goes fine for Q smaller than one so in

00:59:25

particular if you fully understand what

00:59:27

happens on the six vertex side you may

00:59:30

get the critical exponent on the FK side

00:59:32

even for Q smaller than

00:59:34

one yeah on the sigex side you don't see

00:59:46

anything

00:59:49

yes that

00:59:51

um for qals 4 you um expect that you can

00:59:55

conclude the the convergence to Cle um

00:59:58

from the six vertex scaling limit um and

01:00:01

that would correspond to Cle 4 um is the

01:00:03

reason you suspect that mostly because

01:00:05

of the coupling between GFF and C4 or is

01:00:08

there an add re you look okay okay so

01:00:11

first uh I think my colleagues will kill

01:00:13

me myors no I mean I mean that's my

01:00:17

guess but it's a very faint one it's not

01:00:19

uh so it's just that you kind I mean

01:00:23

what you would like to cook up is kind

01:00:25

of observables like I mean function f

01:00:29

which are kind of going to tell you that

01:00:31

there should be a loop for instance I

01:00:34

don't know you want to prove that in FK

01:00:36

you have a loop that is uh I don't know

01:00:39

you you do a shape like that and you

01:00:42

would like to say that there is a

01:00:45

loop like that with very high

01:00:47

probability what you could like to could

01:00:50

try to do is look at a function age that

01:00:54

is going to you know Force kind of

01:00:56

exactly the right gap between the that

01:00:59

is induced by the coupling between FK

01:01:01

and uh between six vertex sorry between

01:01:05

GFF and uh and C4 and it doesn't look

01:01:08

completely unreasonable that you manage

01:01:10

to do something there not

01:01:13

completely unreasonable doesn't mean

01:01:16

it's true but at least I could I could

01:01:18

believe uh that this is

01:01:23

true that maybe not

01:01:28

the question is definitely much harder

01:01:29

for Q smaller than four that's for

01:01:40

[Applause]

01:01:42

sure

Description:

Speaker: Hugo Duminil-Copin (Institut des Hautes Études Scientifiques and Université de Genève) Thursday, April 4, 2024 http://www.fields.utoronto.ca/activities/23-24/Duminil-Copin

Preparing download options

Popular

HD video

Only sound

All

* — If the video is playing in a new tab, go to it, then right-click on the video and select "Save video as..."

** — Link intended for online playback in specialized players

Questions about downloading video

How can I download "Rotational invariance of planar random cluster models... and beyond?" video?

http://unidownloader.com/ website is the best way to download a video or a separate audio track if you want to do without installing programs and extensions.
The UDL Helper extension is a convenient button that is seamlessly integrated into YouTube, Instagram and OK.ru sites for fast content download.
UDL Client program (for Windows) is the most powerful solution that supports more than 900 websites, social networks and video hosting sites, as well as any video quality that is available in the source.
UDL Lite is a really convenient way to access a website from your mobile device. With its help, you can easily download videos directly to your smartphone.

Which format of "Rotational invariance of planar random cluster models... and beyond?" video should I choose?

The best quality formats are FullHD (1080p), 2K (1440p), 4K (2160p) and 8K (4320p). The higher the resolution of your screen, the higher the video quality should be. However, there are other factors to consider: download speed, amount of free space, and device performance during playback.

Why does my computer freeze when loading a "Rotational invariance of planar random cluster models... and beyond?" video?

The browser/computer should not freeze completely! If this happens, please report it with a link to the video. Sometimes videos cannot be downloaded directly in a suitable format, so we have added the ability to convert the file to the desired format. In some cases, this process may actively use computer resources.

How can I download "Rotational invariance of planar random cluster models... and beyond?" video to my phone?

You can download a video to your smartphone using the website or the PWA application UDL Lite. It is also possible to send a download link via QR code using the UDL Helper extension.

How can I download an audio track (music) to MP3 "Rotational invariance of planar random cluster models... and beyond?"?

The most convenient way is to use the UDL Client program, which supports converting video to MP3 format. In some cases, MP3 can also be downloaded through the UDL Helper extension.

How can I save a frame from a video "Rotational invariance of planar random cluster models... and beyond?"?

This feature is available in the UDL Helper extension. Make sure that "Show the video snapshot button" is checked in the settings. A camera icon should appear in the lower right corner of the player to the left of the "Settings" icon. When you click on it, the current frame from the video will be saved to your computer in JPEG format.

What's the price of all this stuff?

It costs nothing. Our services are absolutely free for all users. There are no PRO subscriptions, no restrictions on the number or maximum length of downloaded videos.