long range sand pile divisible Chiari L ni DELFT IM PA TU - - - PowerPoint PPT Presentation

▶
long
SMART_READER_LITE
LIVE PREVIEW

long range sand pile divisible Chiari L ni DELFT IM PA TU - - - PowerPoint PPT Presentation

Feb 01, 2023 96 likes •580 views

The long range sand pile divisible Chiari L ni DELFT IM PA TU - M j Jara W Ruszel w w . . . . , . MPA I DELFT TU EBP 2019 - = Zn% Toros G Discrete : 2nd Mass IR Dist of So - . holes ) ( or C. 7. .

slide-1
SLIDE 1

The

long

range divisible

sand pile

L

Chiari

ni IM PA
  • TU
DELFT

j

. w . w .

M

.

Jara

,

W

.

Ruszel

I MPA TU DELFT

EBP

  • 2019
slide-2
SLIDE 2

G

= Zn% Discrete Toros

So

: 2nd
  • IR
Dist .
  • f
Mass

(

  • r

holes )

C.

.

7.

slide-3
SLIDE 3 Deterministic

diffusion

  • f
mass So ( x )

(

× . . .
slide-4
SLIDE 4 Deterministic

diffusion

  • f
mass So ( x ) 5=1

(

× . . .
slide-5
SLIDE 5 Deterministic

diffusion

  • f
mass

Spread

the excess e C x )
  • ( Stx )
  • I 'T
⇐ .

¥

.

f.

y × Z
slide-6
SLIDE 6 Deterministic

diffusion

  • f
mass

Spread

the excess e C x ) = ( Stx )
  • I 'T

according

to Pn ( x , . )
  • f
a random walk . T

elxlpfxit)

5=1

F- →

(

× . . .
slide-7
SLIDE 7 Deterministic

diffusion

  • f
mass

Spread

the excess e C x )
  • ( Stx )
  • I 'T

according

to Pn ( x , . )
  • f
a random walk .

simultaneously

.

⇐

  • '
→

I

⇒ ← ~ 5=1

(

× . . .
slide-8
SLIDE 8 We will consider

Pn ( X

, y ) =

PIX

  • y ,
  • )
and

Pnlo ,x )

=

I

,

CCDI

ZE 219303

1/-2/11+4

ZE X ( mod 2nd)
slide-9
SLIDE 9 We will consider

gNorm

. Const .

Pn ( X. y )

=

PIX

  • y ,o )
and Pn " " ' =¥¥¥%.¥d÷

IT

. . .
  • n
~ my . . .
slide-10
SLIDE 10

The

Odometer Uel

x ) Eet

( s

; ex ,
  • IT
= total mass

expelled

by

x up to time t . Sati fi es St =

so

  • (
  • A)I
" ut
slide-11
SLIDE 11

The

Odometer

yet

Cx )
  • Uel
x ) Et

(

sjcx

,
  • IT
= total mass

expelled

by

x up to time t . Sati fines

Generator

  • f
pin St =

so

  • f
  • A)I
" ut
slide-12
SLIDE 12

Explosion

vs

Stabilisation

Moo l × ) = fins

MEH

U = Too ⇒

Explosion

{

uctoo ⇒ Fixation
slide-13
SLIDE 13 For So : Ind →

IR

s 't

×Eso(

x )
  • nd
We have that nooooo and Soo = finna Stu ) =L .
slide-14
SLIDE 14 For So : Ind →

IR

St

×E

so ( x )
  • nd
We have that nooooo and Soo =fih→oo Stu ) =L .

①

What

happens if

we take

⇐txt)×e

# nd " almost " iid

?

slide-15
SLIDE 15 Let

(01×1)

, # nd iid with Var
  • a
too ? And set six I = 1+9×1
  • Id

y¥±nddY

)
slide-16
SLIDE 16 Let

(01×1)

, # nd iid with Var
  • a
too ? And set six I = 1+9×1
  • Id

Fe FLY

) In which case

C-

At Y'

ZU

= I
  • Sol
x )
  • { mini

:c

, = . = " "

*

slide-17
SLIDE 17 Let

(01×1)

, # nd iid with Var
  • a
too ? And set six I = 1+9×1
  • Id

y¥±nddY

) In which case

f.

a

iii. ee.ari.im :¥%

,

Yin

Ufo

I x ) = O F
slide-18
SLIDE 18 Then , we want to know I .

Asymptotic

  • f

Elmo

. ] in the case that
  • n
N 10,1 ) . 2.

Scaling

  • f
the

field

un .

Ink

) =

axlntz.I.fndwooln-IIBqz.IN

)

slide-19
SLIDE 19 Then , we want to know I .

Asymptotic

  • f

Ecus

. ] in the case that
  • n
N 10,1 ) .

for

a c- 1127323 , let f- min { 2,4 ) n r
  • dlz
, r > Iz E CUT ] = OId.ly/=1ogn,8=d1z

(

Levine , nor %gn

)

" ,

req

u can , Peres , Ugurcan
  • 15 )
: n . n . ( C . , Jara , Ruszel
  • 18
' ) . long
  • range
slide-20
SLIDE 20 Then , we want to know I .

Asymptotic

  • f

Ecus

. ] in the case that
  • n
N 10,1 ) .

for

a c- 1127323 , let f- min { 2,4 ) n r
  • dlz
, r > Iz E CUT ] = OId.ly/=1ogn,8=d1z

(

Levine , nor %gn

)

" ,

req

u can , Peres , Ugurcan
  • 15 )
: n . n . ( C . , Jara , Ruszel
  • 18
' ) . long
  • range
slide-21
SLIDE 21

EID

"

Proof

" Tal

agram

Chaining

Inequality

④

Rate
  • f Convergence
  • f

eigenvalues

slide-22
SLIDE 22 2 . Scaling
  • f
the

field

un .

Ink

) =

adntz.I.fndunoln-ZIIB.cz

, In ) a

.int

.
  • n :÷
. Then ,

In

  • I
zr the the 28
  • FGF
( Fractional G . Field )

(

Cipriani , Hazra , Ruszel
  • 17 ) :
n . n

( C

. , Jara , Ruszel
  • 18 )
  • long
  • range
slide-23
SLIDE 23 That is

for

all

f

E Coo ( IT ) s 't

↳

f

dx=o

I

Ear

, f 7 ~

NCO

,

Hflhzr )

with

Htt .zr=¥z¥ttI÷

=

it

, tatty .
slide-24
SLIDE 24
slide-25
SLIDE 25

EID

"

Proof

"

Tightness

: Sobolev + Chebchev

Finite

dim : Moment methods t Eigenvalues

convergence

slide-26
SLIDE 26

thanks

slide-27
SLIDE 27

EID

"

Proof

"

Tightness

: Sobolev + Chebchev

Finite

dim : Moment methods t Eigenvalues

convergence

slide-28
SLIDE 28
slide-29
SLIDE 29 That is

for

all

f

E Coo ( IT ) s 't

↳

f

dx=o

(

Ear

, f 7 ~

NCO

,

Hflhzr )

with

Htt .zr=¥z¥ttI÷

⇒

it

, tatty . I
slide-30
SLIDE 30 2 . Scaling
  • f
the

field

un .

Ink

) =

adntz.I.fndunoln-ZIIB.cz

, In ) a

.int

.
  • n :÷
. Then ,

In

  • I
zr the the 28
  • FGF
( Fractional G . Field )

(

Cipriani , Hazra , Ruszel
  • 17 ) :
n . n

( C

. , Jara , Ruszel
  • 18 )
  • long
  • range
slide-31
SLIDE 31

EID

"

Proof

" Tal

agram

Chaining

Inequality

④

Rate
  • f Convergence
  • f

eigenvalues

slide-32
SLIDE 32 Then , we want to know I .

Asymptotic

  • f

Ecus

. ] in the case that
  • n
N 10,1 ) .

for

a c- 1127323 , let f- min { 2,4 ) n r
  • dlz
, r > Iz E CUT ] = OId.ly/=1ogn,8=d1z

(

Levine , nor %gn

)

" ,

req

u can , Peres , Ugurcan
  • 15 )
: n . n . ( C . , Jara , Ruszel
  • 18
' ) . long
  • range
slide-33
SLIDE 33 Then , we want to know I .

Asymptotic

  • f

Ecus

. ] in the case that
  • n
N 10,1 ) .

for

a c- 1127323 , let f- min { 2,4 ) n r
  • dlz
, r > Iz E CUT ] = OId.ly/=1ogn,8=d1z

(

Levine , nor %gn

)

" ,

req

u can , Peres , Ugurcan
  • 15 )
: n . n . ( C . , Jara , Ruszel
  • 18
' ) . long
  • range
slide-34
SLIDE 34 Then , we want to know I .

Asymptotic

  • f

Elmo

. ] in the case that
  • n
N 10,1 ) . 2.

Scaling

  • f
the

field

un .

Ink

) =

axlntz.I.fndwooln-IIBqz.IN

)

slide-35
SLIDE 35 Let

(01×1)

, # nd iid with Var
  • a
too ? And set six I = 1+9×1
  • Id

y¥±nddY

) In which case

f.

a

iii. ee.ari.im :¥%

,

Yin

Ufo

I x ) = O F
slide-36
SLIDE 36 Let

(01×1)

, # nd iid with Var
  • a
too ? And set six I = 1+9×1
  • Id

Fe FLY

) In which case

C-

At Y'

ZU

= I
  • Sol
x )
  • { mini

:c

, = . = " "

*

slide-37
SLIDE 37 Let

(01×1)

, # nd iid with Var
  • a
too ? And set six I = 1+9×1
  • Id

y¥±nddY

)
slide-38
SLIDE 38 Let

Lol

xD

, # nd iid with Var 0<+00 ? And set So ( x ) =

ndocx

)

nasty

)

)

"

with

s . Cx ) > a. s
slide-39
SLIDE 39 Let

(01×1)

, # nd iid with Var 0<+00 ? And set So ( x ) =

ndocx

)

EH

)

)

"

with

s . Cx ) > a. s

sealing

is the

same
slide-40
SLIDE 40 Let

Lol

xD

, # nd with Var
  • a
too ? normal and with covariance And set six I = 1+9×1
  • Id

y¥±nddY

)
slide-41
SLIDE 41 Let

(01×1)

, # nd with Var 0<+00 ? normal and with covariance And set six 1=1+9×1
  • Id

y¥±nddY

)

for pcx , y )

~ SRW

scaling

gives

fields

smother than 4
  • FGG
, but not

rougher

(

Cipriani

, de

Graff

, Ruszel
  • 18 )
slide-42
SLIDE 42 Let

(01×1)

, # nd iid with Var
  • a
too ? And set six I = 1+9×1
  • Id

y¥±nddY

)
slide-43
SLIDE 43 Let

Lol

xD

, # nd iid with

Var€

heavy

tailed with And set

decay

Hill
  • P
six 1=1+9×1
  • Ia

Fe¥nddY

)

for p( x. y )

~ SRW

Scaling

results in non
  • Gaussian
fields

Efexp

( i

, exp f

Hf 11.9 )

( Cipriani

, Hazra , Ruszel
  • 18 )
slide-44
SLIDE 44 For So : Ind →

IR

St

×E

so ( x )
  • nd
We have that nooooo and Soo =fih→oo Stu ) =L .

①

What

happens if

we take

⇐txt)×e

# nd " almost " iid

?

slide-45
SLIDE 45

÷

:Esi÷÷:÷÷÷

①

÷÷÷÷÷÷÷÷÷i÷÷

:

slide-46
SLIDE 46

÷÷¥÷÷s:÷÷÷÷÷÷÷÷÷⇒

①

÷÷÷÷÷÷÷÷÷s•÷i÷

:

slide-47
SLIDE 47

Thanks

Again