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

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. .

Deterministic of mass diffusion So ( x ) ( × . . .

Deterministic of mass diffusion So ( x ) 5=1 ( × . . .

Deterministic of mass - I 'T - ( Stx ) e C x ) Spread the excess - diffusion ¥ . ⇐ . f. × Z y

Deterministic of mass - I 'T = ( Stx ) e C x ) Spread diffusion the excess . ) walk to of according ( x random Pn a , . elxlpfxit ) T F- → 5=1 ( × . . .

Deterministic of mass - I 'T - ( Stx ) e C x ) Spread diffusion the excess - . ) walk to of according ( x random Pn a , . simultaneously ⇐ . ' - I → ⇒ ~ ← 5=1 ( × . . .

We will consider PIX o ) , y ) Pn ( X - y , and = CCDI I Pnlo ,x ) = , 1/-2/11+4 ZE 219303 mod 2nd ) ZE ( X

We will consider Const . . gNorm PIX Pn ( X. y ) - y ,o ) and = = ¥¥¥%.¥d÷ Pn " " ' IT ~ n my - . . . . . .

The Odometer - IT x ) Eet ( s Uel ex , ; expelled total = mass t to time by x up . Sati fi es - ( " - A) I St so ut =

The Odometer Cx ) yet - IT - x ) Et sjcx ( Uel , expelled total = mass t to time by x up . of pin Sati fines Generator - f " - A) I St so ut =

Explosion Stabilisation vs = fins × ) MEH Moo l U ⇒ Explosion = Too { Fixation ⇒ uctoo

: Ind - nd IR For 't ×Eso( x ) → s So - have that and We nooooo = finna Stu ) =L Soo .

: Ind - nd IR For St ×E so ( x ) → So - have that and We nooooo Stu ) =L Soo =fih→oo . ① What if happens take we ⇐ txt )×e " " ? almost iid # nd

with iid Let ? (01×1) Var o too a # nd , And set ) six 1+9×1 I Id = - y¥±nddY

with iid Let ? (01×1) Var o too a # nd , And set ) six 1+9×1 I Fe FLY Id = - In which case ZU At Y' * C- x ) Sol " " I { mini = o = :c - - , . =

with iid Let ? (01×1) Var o too a # nd , And set ) six 1+9×1 I Id = - y¥±nddY In which case ee.ari.im :¥% iii. f. a , F Ufo I x ) O Yin =

Then want to know we , of Elmo Asymptotic . ] the in I case . N 10,1 ) that on . field the un Scaling of 2. . ) Ink axlntz.I.fndwooln-IIBqz.IN = )

Then want to know we , of Ecus Asymptotic . ] the in I case . N 10,1 ) that on . c- 1127323 min { 2,4 ) for nor %gn let f- a , > Iz dlz r r - , = OId.ly/=1ogn,8=d1z n E CUT ] " ) req , ( 15 ) Levine , Peres n . n : , Ugurcan can u - . , ' ) - range ( C Ruszel long Jara 18 - . . , ,

Then want to know we , of Ecus Asymptotic . ] the in I case . N 10,1 ) that on . c- 1127323 min { 2,4 ) for nor %gn let f- a , > Iz dlz r r - , = OId.ly/=1ogn,8=d1z n E CUT ] " ) req , ( 15 ) Levine , Peres n . n : , Ugurcan can u - . , ' ) - range ( C Ruszel long Jara 18 - . . , ,

EID " " Proof Chaining Tal Inequality agram ④ of Convergence Rate of eigenvalues

. Scaling field the un of 2 . ) Ink adntz.I.fndunoln-ZIIB.cz = , In ) on :÷ .int a . . I Then In - the zr , ( FGF Fractional ) the G Field 28 - . 17 ) : ( Ruszel Hazra Cipriani . n n - , , - long ( C 18 ) - range Ruszel Jara . , - ,

Coo ( IT ) f is all That for 't E s ↳ f dx=o Ear Hflhzr ) NCO , f I 7 ~ , with Htt .zr=¥z¥ttI÷ , tatty it = .

EID " " Proof Tightness Sobolev Chebchev : + Finite dim t Eigenvalues Moment methods : convergence

thanks

EID " " Proof Tightness Sobolev Chebchev : + Finite dim t Eigenvalues Moment methods : convergence

Coo ( IT ) f is all That for 't E s ↳ f dx=o Ear Hflhzr ) NCO , f ( 7 ~ , with Htt .zr=¥z¥ttI÷ ⇒ , tatty it I .

. Scaling field the un of 2 . ) Ink adntz.I.fndunoln-ZIIB.cz = , In ) on :÷ .int a . . I Then In - the zr , ( FGF Fractional ) the G Field 28 - . 17 ) : ( Ruszel Hazra Cipriani . n n - , , - long ( C 18 ) - range Ruszel Jara . , - ,

EID " " Proof Chaining Tal Inequality agram ④ of Convergence Rate of eigenvalues

Then want to know we , of Ecus Asymptotic . ] the in I case . N 10,1 ) that on . c- 1127323 min { 2,4 ) for nor %gn let f- a , > Iz dlz r r - , = OId.ly/=1ogn,8=d1z n E CUT ] " ) req , ( 15 ) Levine , Peres n . n : , Ugurcan can u - . , ' ) - range ( C Ruszel long Jara 18 - . . , ,

Then want to know we , of Ecus Asymptotic . ] the in I case . N 10,1 ) that on . c- 1127323 min { 2,4 ) for nor %gn let f- a , > Iz dlz r r - , = OId.ly/=1ogn,8=d1z n E CUT ] " ) req , ( 15 ) Levine , Peres n . n : , Ugurcan can u - . , ' ) - range ( C Ruszel long Jara 18 - . . , ,

Then want to know we , of Elmo Asymptotic . ] the in I case . N 10,1 ) that on . field the un Scaling of 2. . ) Ink axlntz.I.fndwooln-IIBqz.IN = )

with iid Let ? (01×1) Var o too a # nd , And set ) six 1+9×1 I Id = - y¥±nddY In which case ee.ari.im :¥% iii. f. a , F Ufo I x ) O Yin =

with iid Let ? (01×1) Var o too a # nd , And set ) six 1+9×1 I Fe FLY Id = - In which case ZU At Y' * C- x ) Sol " " I { mini = o = :c - - , . =

with iid Let ? (01×1) Var o too a # nd , And set ) six 1+9×1 I Id = - y¥±nddY

xD with Lol iid Let ? Var 0<+00 # nd , And set " ) ndocx ) So ( x ) ) = nasty with Cx ) > 0 s a. s .

with iid Let ? (01×1) Var 0<+00 # nd , And set " ) ndocx ) So ( x ) EH ) = with Cx ) > 0 s a. s . sealing the is same

xD with Lol Let ? Var o too a # nd , with normal and covariance And set ) six 1+9×1 I Id = - y¥±nddY

with Let ? (01×1) Var 0<+00 # nd , with normal and covariance And set ) six 1=1+9×1 Id - y¥±nddY SRW pcx , y ) for ~ fields smother scaling gives but than FGG not 4 rougher - , Graff Ruszel 18 ) ( Cipriani de - , ,

with iid Let ? (01×1) Var o too a # nd , And set ) six 1+9×1 I Id = - y¥±nddY

Var€ xD with Lol iid Let # nd , tailed with heavy - P decay Hill And set ) six 1=1+9×1 Ia - Fe¥nddY SRW p( x. y ) for ~ results Scaling fields Gaussian in non - Hf 11.9 ) Efexp exp f ( i , 18 ) ( Cipriani Ruszel Hazra - , ,

: Ind - nd IR For St ×E so ( x ) → So - have that and We nooooo Stu ) =L Soo =fih→oo . ① What if happens take we ⇐ txt )×e " " ? almost iid # nd

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