A hierarchical renormalization model on trees Yueyun Hu (Paris 13) - PowerPoint PPT Presentation

Motivation Results : rate of convergence Proofs A hierarchical renormalization model on trees Yueyun Hu (Paris 13) The 3rd conference Probability and Analysis 15–19 May 2017, Be ¸dlewo, Poland Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Overview Model (Derrida and Retaux) : Let X 0 be random variable taking values in { 0 , 1 , 2 , ... } . Define law = ( X (1) + X (2) − 1) + , X n +1 ∀ n ≥ 0 , n n with two independent copies X (1) n , X (2) of X n . n Question : What can we say about the asymptotic behaviors of X n as n → ∞ ? Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Overview Model (Derrida and Retaux) : Let X 0 be random variable taking values in { 0 , 1 , 2 , ... } . Define law = ( X (1) + X (2) − 1) + , X n +1 ∀ n ≥ 0 , n n with two independent copies X (1) n , X (2) of X n . n Question : What can we say about the asymptotic behaviors of X n as n → ∞ ? Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Results : rate of convergence Proofs Outline 1 Motivation Pinning model on a hierarchical structure Derrida and Retaux’ model 2 Results 3 Proofs Proof of the lower bound in Theorem 2 Proof of the upper bounds in Theorems 1 and 2 Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs The pining model on Z ( ω i ) 0 ≤ i ≤ N i.i.d., independent of ( S n ); ( S n ) 0 ≤ n ≤ N , S 0 = S N = 0 . ω 0 ω 1 ω 2 ω N ω i WALL WALL WALL Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs The pining model on Z The measure of polymer of length N : � N P N ,ω ( dS ) := 1 � � exp ω i 1 { S i =0 } , Z N i =1 Z N is called the partition function [ ω i > 0 ( i attractive) ; ω i < 0 ( i repulsive)]. See Giacomin’s book (Random Polymer Models, 2007). Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs The pining model on Z The measure of polymer of length N : � N P N ,ω ( dS ) := 1 � � exp ω i 1 { S i =0 } , Z N i =1 Z N is called the partition function [ ω i > 0 ( i attractive) ; ω i < 0 ( i repulsive)]. See Giacomin’s book (Random Polymer Models, 2007). Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs The pining model on Z The measure of polymer of length N : � N P N ,ω ( dS ) := 1 � � exp ω i 1 { S i =0 } , Z N i =1 Z N is called the partition function [ ω i > 0 ( i attractive) ; ω i < 0 ( i repulsive)]. See Giacomin’s book (Random Polymer Models, 2007). Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Pinning model on a hierarchical lattice Derrida, Hakim and Vannimenus (1992) 1 At level 0, there is a unique bond. 2 Fix an integer B ≥ 2 (for e.g. B = 3) 3 Rule : Each bond gives B branches consisting of 2 bonds each. Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Case B = 3 Level n = 0 Level n = 1 d 0 d 0 ω 1 WALL WALL d 1 ω 2 WALL d 1 d 2 Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Case B = 3 level n = 2 d 0 Wall ω 1 d 1 ω 2 d 2 Wall ω 3 d 3 Wall ω 2 n d 2 n Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Pinning model on a hierarchical lattice At level n , each (direct) trajectory ( S i ) 1 ≤ i ≤ 2 n (from d 0 to d 2 n ) contains 2 n bonds. Choose the uniform measure P B , n on all possible trajectories [simple random walk ( S i ) 1 ≤ i ≤ 2 n on the hierarchical lattice]. Let ( ω i ) 1 ≤ i ≤ 2 n be i.i.d. and independent of ( S n ). The partition function � 2 n � � Z n := E B , n exp ω i 1 { S i − 1 = d i − 1 , S i = d i } , i =1 where the expectation is only taken with respect to ( S n ). Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Pinning model on a hierarchical lattice Let N n := number of trajectories γ from d 0 to d 2 n and � 2 n � � � R n := exp ω i 1 { γ i − 1 = d i − 1 ,γ i = d i } . γ : γ 0 = d 0 ,γ 2 n = d 2 n i =1 Then Z n = R n N n . Easy to see that B N 2 = n , N n +1 R (1) n R (2) + ( B − 1) N 2 R n +1 = n , n with two independent copies R (1) n , R (2) of R n . n Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Pinning model on a hierarchical lattice Then = Z (1) n Z (2) + B − 1 Z n +1 = R n +1 n , N n +1 B with two independent copies Z (1) n , Z (2) of Z n . n Stability : ( B , Z ) �→ ( B ′ , Z ′ ) with B ′ := B − 1 , Z ′ := B Z B − 1 . See Monthus and Garet (2008), Derrida, Giacomin, Lacoin and Toninelli (2009), Lacoin and Toninelli (2009), Giacomin, Lacoin and Toninelli (2010, 2011), Berger and Toninelli (2013) for the studies of this model [disorder relevance, critical line...] Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Pinning model on a hierarchical lattice Let X n := log Z n . Then X n +1 = log Z n +1 log e ( X (1) n + X (2) n ) + B − 1 = B + log 1 + ( B − 1) e − ( X (1) n + X (2) n ) X (1) + X (2) = . n n B Let 1 < B < 2 and define a := − log( B − 1) > 0. If X n ≥ − a , a.s., then X n +1 ≥ − a a.s. Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Derrida and Retaux (2014)’s model Fix a > 0. For any n ≥ 0, law = max( X (1) + X (2) n , − a ) , X n +1 n with two independent copies X (1) n , X (2) of X n . n Replacing X n by X n + a and taking a = 1, the recursive equation becomes law = ( X (1) + X (2) − 1) + , ∀ n ≥ 0 , X n +1 n n with two independent copies X (1) n , X (2) of X n . n Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Derrida and Retaux (2014)’s model X (5) X (6) X (1) X (2) X (3) X (4) b a level 0 0 0 0 0 0 0 X (1) X (2) X (3) ( a + b − 1) + level 1 1 1 1 X (1) X (2) level 2 n − 1 n − 1 X n level n Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Derrida and Retaux (2014)’s model E ( X n ) 1 Free energy : F ∞ := lim n →∞ ∈ [0 , ∞ ) exists. 2 n law = ( X (1) n − 1 + X (2) n − 1 − 1) + , Proof : As X n 2 E ( X n − 1 ) ≥ E ( X n ) ≥ 2 E ( X n − 1 ) − 1 , implying that n →∞ ↓ E ( X n ) n →∞ ↑ E ( X n ) − 1 F ∞ := lim = lim . 2 n 2 n 2 "Percolation on tree" : Let law X 0 = (1 − p ) δ { 0 } + p δ { Y } , with 0 ≤ p ≤ 1 and Y > 0 a positive random variable. Define p c := sup { 0 ≤ p ≤ 1 : F ∞ ( p ) = 0 } . Yueyun Hu (Paris 13) A hierarchical renormalization model on trees

Motivation Pinning model on a hierarchical structure Results : rate of convergence Derrida and Retaux’ model Proofs Derrida and Retaux (2014)’s model E ( X n ) 1 Free energy : F ∞ := lim n →∞ ∈ [0 , ∞ ) exists. 2 n law = ( X (1) n − 1 + X (2) n − 1 − 1) + , Proof : As X n 2 E ( X n − 1 ) ≥ E ( X n ) ≥ 2 E ( X n − 1 ) − 1 , implying that n →∞ ↓ E ( X n ) n →∞ ↑ E ( X n ) − 1 F ∞ := lim = lim . 2 n 2 n 2 "Percolation on tree" : Let law X 0 = (1 − p ) δ { 0 } + p δ { Y } , with 0 ≤ p ≤ 1 and Y > 0 a positive random variable. Define p c := sup { 0 ≤ p ≤ 1 : F ∞ ( p ) = 0 } . Yueyun Hu (Paris 13) A hierarchical renormalization model on trees