Large deviations of SLE and Weil PeterssonTeichmillerspaceLecture 3 Yilin Wang MT

Loewher energy a

SLE A Weil PeterssonTeichmiller spanSchramin Leewner

tevolutions

ADeterministicProbabilistic

Stochastic analysis quasicircles

Conformally invariant quasiconformal mapping2D Statistical mechanics

Geometricfunction theorymodelComplex structures

Random planar maps on Riemannsurfaces

2D quantum gravity Kahler geometryConformal field theory String theory

Monday

1 Brownian Motion and Dirichlet energySchilder's theorem

I SLE and Loewner energyF RB Icr I wait dt

4 SLE large deviations

Energy reversibility from SLE reversibility

Tuesday

2 Loop energyGeneralizes chordal energy

I Weil Petersson Teichmiller space248 a J 8 is Weil Peterssonand 25 other equivalent definitions

Today1 Radial SLE a large deviations

1 Foliations by Weil Petersson quasicircles

Motivation SLE duality

For K 78Dubédat Ethan Miller Sheffield

ylocally a SLE a

www.odftfo gEk

Asymptoticbehavior of Slee

dualShea

WPquasicircle Leewner energy c

V W LaettnerKufuor energy

Q What happens when we let k o

For SLEK.KZ EH2

9114 Gees fBy gotZ

Ect maximal solution time

I toys

td.ttHt 9 Z E1H TA t 3

domainof definitionof go 1418 to t

Can show Gt itk

y z

Not so interesting

Issue normalized at a boundarypoint too

1 Radial SLE Large deviations

1 Loewner Kufra equationNy Pelt o Pt EProbis

measurable in t

Loewner Kufarev equation ZED

4ft'D Ztily two II jPDE

folz Z TIE 0

ft ID Dt with ft lo o ft cos et

A not ft too Evolution family Dt

The normalvelocity of 2Dt at file is

27Pelo file if Prado Pt

Stift satisfies

CODE at9 12 g iz fe 9 14e geez

Pt'do

Examples

Ptldo I do f t yo

yDtt

of

De é'D

f Dirac measure

ft Ido I Sei Bt Brownian motion on R

MI ack28Radial

ni i t.ggSEKp Dt Dt

so

2 K soo limit

Let K 70 in ft Ido e Sei But

Is trot

09th Jett gigs eh Seibu idol ds

I 9 12 gots gaz

Littorio IO do

occupation

T t go.izfs.eeog I y measure up

to time t

at Getz

2 9 12 9 12 Gold Z

gait et z De ID

DtD

Illustration of occupation measure

to

tangoEmoteuponto

S x iRt 3h10 t

Seibaldodt dy It

3 Radial SLE LDP

Thm Ang Park W Io

Radial stem process satisfies the LDP as k so

with rate function St CLeewner Kufarer energy

IP I SLEK Dit o expL KSelps as k a

whereSeip J Lepe dt

Lepe Ifs Ive cost do

if Pt Dt lo do and left o otherwise

Followsfrom Donsker Vardhan theorem on

the LDP of occupation measures

Stlp so only for a c measures

on S x RtMore regular than Dirac masses

Q What are the families Dt generated byPt where Step co

I Foliations by Weil Petersson quasicircles

Whole plane Loewnerkafana equation

Pt Pt lean Dt teaand ft D De withft't o ftlie e

t

such that

chain generated

is the Leever tape

0 by Pas thoID

Sip r f Lepe de

Claim LoewnerKufuor chain in ID is

a special case of whole plan L K chain

Given Pt ex

Set Pt do for all to

Do ID

fDe et'D for tooDotty is the family generated

by Pt too

Then Vikland W

If Sep co then

2Dt is aWeil Petersson quasicircle ft er

U JDt 61103t t 2Dt is continuous in the sup norm

Foliation of Ello by Weil Petersson quasicircles

a Non smoothMonotone but not strictly monotone2Dt is called a leaf

Finite L K energy foliation has finite Leewner

energy leaves

Wewill prove it by showing aquantitative result

Recall g t ft De D

Define Y iz arg giltst

go12

if Z t ODt

z

Y is the winding function ofthe foliation generated by Pt tar

Than V W

ab Stp Jeter day Die

ab is consistent with SLE duality

K as In

Example

Pt do I sin E do for te to

t I do otherwise

foliation 4Rt

Corollary Sips es Ein

xD

ID feces LP

Pr generates the foliation formedby equipotentialab Sip7 142 21g I

Y is harmonic in 618

We recall I 8 If pargf dat fye Tarp Fda

41 14 1 I

fkn o

Question from T Amaba

16 from SLE duality

IP SLEa loop stays close to 8

a

exp l ÉSI

ko

Pld SLE It stays close too

SI

exp I k If sips11 IDE 8

expl I Sip's

Cor W Definition 27AJordan curve 8 separating o and a is Weil Petersson

8 can be realized as a leaf inthe foliation generated by a measure

with Stp r

Proof follows from previouscorollary

248 set Sept em Sip

y

inA

winding function

yry a y

harmonic in 618since leg 51 no true for all Jordan

curvesseparating

o and a

Reversibility of Loaner kafana energy

p y5 I

Sip SipProof Yes yes DII

Remark

Reversibility of radial slew forK 8 is not known

This resultsuggests

it to be true

Proof sketch of 4 1681ps Dce

Dirichlet energy is conformally invariant

jLoenner chain Explore a conformallyinvariant object layerbylayer

Assume Pete generates a foliation of ID

of a function on ID sit D O a

op o few0 any

ahit

e poet4 Aw Dtharmonicin De

o outsideofDt

p Pt idol dt

Thin VW Disintegration isometry

WED D SxiR 28

u Ian tfis an bijective isometry with inverse operator

in

harmonic function in Dt

Aconsequence GEF Whitenoise decamp generalizes

IHedenmalm Nieminen

Proof of 4

If 9 a winding function

p Eco do dt2WEshow v4 ont ut

DIY44 Yup Jfs 4kt4 2 24 IO do It

161 Lapeldt lbStp

Conclusion

StenKoot

Largedeviations

Loewner energyI'in

t

SLE

duality conformal frappeLDP Diy duality byWeil

geometryK or

Petersson

sips

quasicircles

largeammoLeewner

knfarer energySLEWox

Wang 2021