Mini-course part 1: Hipster random walks and their ilk Louigi - PowerPoint PPT Presentation

Mini-course part 1: Hipster random walks and their ilk Louigi Addario-Berry, Luc Devroye, Celine Kerriou, Rivka Maclaine Mitchell 12’th MSJ-SI August 5, 2019

✓ 3 infinite binary T - way infinite path ← = vo.hr one o Y . & . . . . to • node or of root voor the complete tree is rn a canopy o tree of binary depth leaves of n L T - - at rooted subtree • Tn Vn = . o ↳ of leaves Tn Ln = i yo or i. ffvca , b) - input T from children ④ Functions : on at combination function nodes at fb - - output D to D parents function ( fu ve Tm Ln ) ; this Tn functions turns into Choose a . , = ( output at Ln ) x ) root Tn ( input xv.VE x 1- ← un on x , . and ( fu VE Tm Ln ) of random both be Either can x or ,

Examples with prob b ( a. b) that p iiiiiiiiisiitiiiiiis ④ f. =/ . wit . . . . . . . . . . . :p Then # nodes at le ve Tn ( I ) - Watson =D Galton tree in en a , b) + Dr Ca max - - Figg . { I E :b :b titty :-p ' with dist fifth offspring Hdmi : mdiaspf , = b an o = + Dr ② T ) law let be with fu Ca , b) la , b) Let ( Dv HD max ve = µ - ' {}{u} / , , ! Then Tricot position in maximum n 's generation , aniefnenjdisbinmar / with branching walk random vertices ) at ( displacements a Dvotb Dnf ③ a Duo + b. Dr , VET ) law let fuca be with , b) Let ( Dv HD µ = , , . or This studied ; fixed transform points by smoothing is a ay qb 's Liggett ( 1983 ) , Durrett others many Duo Dw Dv , ] " . from these equations have studied fact been all the perspective In , of fixed - point equations shift ) wish ( sometimes introduce rescaling to a or .

- point without fixed Examples theory a / - Retauxmc.de//' fla.bt-maxfatb-l.co ) ④ on trees " Derrida . Here ' Parking - ( Xv ,VELn ) of Tn( X ) : Large behaviour Question where X with HD n - law µ µ some of depends ( u ) on answer course [ Refs , Shi ' Goldschmidt , Przykucki 1610.08786 Malkin , Pain Hu 1705.03792 Hu i 1811.08749v2 ; : , , , , . ] , Hu , Lifshitz Shi , Derrida Chen , Dugard 1907.01601 , Parallel connection ⑤ lattice hierarchical connection Resistance Series Random atb → Resistance aatbb . → ( a. b) atb with f. ={ prob BMNH ' → P • . * • . . ( a. b) l→aa¥b , - p with prob a I . THI [ Ref . ] pa 's TNCI ) Jordan ) exponentially 's Hambly 2004 decays → → ps grows ; exp : - . ⑥ tree Pemantle 's Min - Plus GET ( a. b) atb with prob → p fu ={ ( a. b) . min ( a. b) - p with prob Theorem lgg.tn#Jd-sBeta( → I - C ) ( A 2,1 ) . o , : Auf finger n conjectured that Fast [ Ref Cable pemantle 1709.07849 : . - ' ) ( Open : universality to-9nT.nl#d-sBetaC2 inputs ? ) what question for happens other . - .

Aside from Ivan Example Corwin 's course : , p ) Ca Beta ( - Bt - 2- ( t Z ( t Z ( t.nl th n ) Bin - D n ) I = I n . - - , , . In n ) 2- ( o , = > i r of for Recurrence fn polymer / - to - line Beta partition point of random Beta RWRE CDF - with E ( 2- ( t , na ) ] o 2- Ct → Md ) ult , ) , Cn . . : = , n . - - - - then - j K j jtYM-iuct.cn#.n.nIi.Tn-iD Htt . .nl ) in ( u , n = , . . . ( Lt P ) k

新しがり屋. ショップにいる⼈亻々 中⽬盯黒や下北磻沢にあるような サードウェーブコーヒー walk model New Hipster random is hipper hipper is Vo Vi defined ( Dv vet . Let ) fu be by Fix IID f btDIa=bf , if at Dutta - b ③ I at Duda with ③ ( a. b) prob V > V - b . - . 1¥ btDrIa=b a) ca . b) fb af fb with prob I . DW D " DW D " of Think time the Idea tree running as up the other ( chosen randomly ) of hipper than 1 One is vo v1 , child off If another takes particle shows hipper 2 up , . Prob 't 1 " will We random walk Dv={ symmetric simple hipster study - • . I w/ prob -1 SSHRW . { 1- Prob P random walk " simple hipster . → Dr lazy . ) peco . • totally asymmetric . - p prob . O w/ I TALSHRW Theorem Hipster : Thos - z d For SSHRW Betaczz , → A ) , . ( 36N ) 's B) d- For TALSHRW Bethel ) - pin ) " ( 44

- cable Note for to that of Auf Result TALSHRW similar - finger very . Auf finger Recall Cable : - ( a. b) atb with prob → q P Tn ( 8) ={ ( a. b) log . d Theorem - C ) Bet ( A al ? " → min ( a. b) - p with zj → prob I . Intuition and ) ( connecting mint plus TALSHRW LR for at of of children root Tn write values . is growing ( stretched natural exponential scale its to Tnto ) ) then If on a and log R L log compare . - log Rl I log L Behaviour small when then { 1094+14*10914+1 UR 2L * - log Rko thaiwseispthsecagmor.ng.nu awed If } Hog L , log R ) log L . R ) minch L min ( log L increment ' a - log Rl I log L large Behaviour when then { is just This max ( log L 109EUR ) , log R ) max ( L , R ) - log Lt ' R f I log L = If log ( value of a random child ) - , R ) ) min ( log L , log p ) log ( mink . R ) , R ) mink mink = =

intuition should for Similar the hierarchical work lattice : , { ( a. b) with E b ' → prob at fu . ( a. b) l→aa¥b with I prob . Tricot ( stretched ) Intuition scale is exponential Suppose growing i on a . of root , R values at for children L Write . - log Rl I log L Behaviour when small Lt RI is the This value then { 2L log ( Lt R ) log ( L ) -11 s common Klus log Rl If I log L small - valued increment E- " 3 a - log ( LYFT ) LIRI log ( L ) I EL - - - log Rl I log L large Behaviour when then { YE mad 1094109 match . R ) ' - ' 10914,7 If log Rl ftohgisvaiiseiousta random child , I log L big - , R ) ) min ( log L , log R ) mink log = = Lfp - I Motivates the lattice with conjecture the random hierarchical st following Fc in p - > o : , . loqtn ) d- Beta ( 2,2 ) I > . 2 C 3 n of ( Disagrees - Jordan ) with Hambly conjecture a

Theorem ) # ( Totally asymmetric lazy SHRW d→ Beta ( 2.1 ) ( 2nF Proof Idea is hipper is hipper Vo v1 Original dynamics . - bf fat Dukat at Duda Bernoulli ( E ) Dr - - ax HQs VIO G voor vo left child is always chosen By symmetry assume can . , - Tn ( ( : Tn notation ( x ) Ln ) ) For inputs VEL ) useful Cocu , x - : xv.ve - , I Tnlxlttncx if Tncx ) ) " - { ' Tnt ' ' " - Taxi + Dr if Tn I'm Tix , # T , T T 1 - - - - - - - Toki - I - . * * * * * * *

of * Idea - pack ) ) left child - k child the Pack ) ( I , right ( Totally ← - asymmetric case ) - 1) 4- both { pn ( K K step make = I ✓ - a , = KI ( et IP ( Tn ( O ) 's Pn ( KP pack ) be = both - k ← lazy - , - I I - pack ) ) - D 't Then pntilk ) ' pack ) ( I pnlk pnlkl t = - I - IT ) n Ck ) Ipn Ck Pna ( H prickle Rearranging gives - ' equation # ucx.tk - I of the ( ucx.tt ) discretization inviscid This Burgers is a t - they initial - valued ) to so solve the value ( problem trying measure we are U U - u , = = - . . t > Ut U Ux E IR { o , x = - , = Is = focus at , understood ( Dirac now , O measure ) prob ,e=o , mass as a . * points of solved discontinuity , this by Rx lo , a) IR space-time Ui given by is → Ignoring = { E ET k¥ Yt osxa - 4 t - " t ' , " otherwise o 2E 52 not Buuntigsg ! ti on * is Note the density ultra ) of . dist Beta ( 2,1 ) is always scaled prob : a a . .

Proof Idea case ) ( Symmetric HRW - g child t.li simple left child - k gnfk ) ( i n Ck ) ) , right ← - - Dk { 9nA both K step make t I = I # - a Let , IP ( Tn ( ⑤ k ) n Ck ) q = = - I make ( k -1112 step a both =k ← , - ik I I - Gn Ck ) ) Then gntilk ) gn Ck ) ( I 9nA qnlk -1112 t = IT ) qnlklitzfqnlk-T-29.nl/rYt9nlk- gnu ( H Rearranging gives - 2 equation # ucx.tk I # ( ucx.tt ) of the discretization membrane This is porous a initial to - valued ) solve value so the ( problem trying measure we are - U2 ) t > see IR Ut { o , = xx , µ = Solos A- now , - - o ] ex - - mail # fit lol ' Ii :& : :÷:÷ : " " ; . of sealed Beta ( 2,2 ) Density a = . of talk focus TALSHRW Rest principally : on

' equation - I Inviscid luck .tt ) Burgers UH .tk Initial value problem = Jo ( x ) at Dirac Ucsc , o ) ← O mass . - grid Note ulx.tt has ucx.tt/=2ulx.tlodulx.tt--2.gIIfaF+ - ( : ¥ " " " a = Satisfies # - a fun , ,y um .tk . Special i cases of solution flattens out ¥ mixture > o ucu.tk Target 8=0 a : } I - * - p a = - . , flat line these two solutions =p 2=0 8=1 ucx.tt - a - o o - - . - . , with time : solution . ) ucx.tl steepens ( problem at t I > I a I ; a o * = p - = - - - - - - , . . claimed solution ? ( ( pnlkl n > o ) should KEE ) to the why converge , , obvious is by This means no . - { 02 If - ftp.lkkpnlk with - y ' ' solve Polk ) pm , ( H ) Warning example prickle : - = { - { (22-02)=0/1 - { ( o ' -24=211 get pill ) pnlk ) K Then 2 n 2 pilot = - - - , Ktn O