Introduction Main Results Proof Sketch and Main Ingredients Summary Discrete Fractal Dimensions of the Ranges of Random Walks Associate with Random Conductances Xinghua Zheng Department of ISOM, HKUST http://ihome.ust.hk/ ∼ xhzheng/ International Conference on Advances on Fractals and Related Topics, Dec 2012 Based on Joint Work with Yimin Xiao 1 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary Outline Introduction The Random Conductance Model Discrete Fractal Dimensions Main Results Proof Sketch and Main Ingredients Summary 2 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary The Random Conductance Model Discrete Fractal Dimensions The Random Conductance Model • Z d = d -dimension integer lattice; E d = {non-oriented nearest neighbor bonds} • Environment : for a given distribution Q on [ 0 , ∞ ) , µ e ∼ i . i . d . Q , for all e ∈ E d ; • Given a realization ω = { µ e : e ∈ E d } , two random walks: 1. Variable speed random walk (VSRW), ( X t ) , waits at x for an exponential time with mean 1 /µ x ; 2. Constant speed random walk (CSRW), ( Y t ) , waits at x for an exponential time with mean 1; and then jumps to a neighboring site y with probability P xy ( ω ) = µ xy � where µ x = µ xy . µ x y ∼ x 3 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary The Random Conductance Model Discrete Fractal Dimensions Transition Probabilities 4 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary The Random Conductance Model Discrete Fractal Dimensions Examples Eg 1 : • Q = δ { 1 } , then µ e are constantly 1, and Y t is just the usual nearest neighbor random walk • Functional CLT (FCLT): Y nt √ n ⇒ B t . Eg 2 : • Q = Bernoulli ( p ) , then Y t is a simple random walk on the connected component of percolation Eg 3 : • Q supported on [ 1 , ∞ ) – what we shall focus on 5 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary The Random Conductance Model Discrete Fractal Dimensions Two laws • Two laws: 1. Quenched Law : For any given realization ω , study the law P ω of ( X t ) / ( Y t ) under this realization 2. Averaged (or Annealed) Law : the law by taking expectation of the quenched law P ω w.r.t. P • Focus on quenched law P ω • Basic Questions: the long run behavior of ( X t ) / ( Y t ) , e.g., 1. does the quenched FCLT (QFCLT) hold? 2. What about the fractal properties of the sample paths of ( X t ) / ( Y t ) ? 6 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary The Random Conductance Model Discrete Fractal Dimensions QFCLT • [Barlow and Deuschel(2010)] For the VSRW X , when d ≥ 2, for P -a.a. ω, under P ω 0 , X n 2 t / n ⇒ σ V B t , where σ V is non-random, and B t is a standard d -dimensional Brownian-motion. • [Barlow and Deuschel(2010)] For the CSRW Y , when d ≥ 2, for P -a.a. ω, under P ω 0 , Y n 2 t / n ⇒ σ C B t , � � σ V / 2 d E µ e , if E µ e < ∞ , where σ C = 0 , if E µ e = ∞ . • [Barlow and ˇ Cerný(2011)], [ ˇ Cerný(2011)] For the CSRW Y , when d ≥ 2 and Q ( µ e ≥ u ) ∼ C / u α for some α ∈ ( 0 , 1 ) , then for P -a.a. ω , under P ω 0 , Y n 2 /α t / n converges to a multiple of the fractional kinetics process; • [Barlow and Zheng(2010)] For the CSRW Y , when d ≥ 3 and Q is Cauchy tailed, then for P -a.a. ω , under P ω 0 , Y n 2 ( log n ) t / n converges to a multiple of a d -dimensional Brownian-motion. 7 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary The Random Conductance Model Discrete Fractal Dimensions Discrete Hausdorff Dimension • For any n ∈ N , let V n = V ( 0 , 2 n ) be the cube of side length 2 n centered at 0 ∈ Z d , and S n := V n \ V n − 1 • For any set B ⊆ Z d , let s ( B ) be its side length • [Barlow and Taylor(1992)] For any measure function h and any set A ⊆ Z d , the discrete Hausdorff measure of A w.r.t h is ∞ � m h ( A ) = ν h ( A , S n ) . n = 1 where k k � � � s ( B i ) � � � ν h ( A , S n ) = min h : A ∩ S n ⊂ B i . 2 n i = 1 i = 1 • For α > 0, define h ( r ) = r α , and let m α ( A ) = m h ( A ) . Then the discrete Hausdorff dimension of A is given by � � dim H A = inf α > 0 : m α ( A ) < ∞ . 8 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary The Random Conductance Model Discrete Fractal Dimensions Discrete Packing Dimension • [Barlow and Taylor(1992)] For any measure function h , ε > 0, and any set A ⊆ Z d , the discrete packing measure of A w.r.t h is ∞ � p h ( A , ε ) = τ h ( A , S n , ε ) , n = 1 where � � � r i k � � : x i ∈ A ∩ S n , V ( x i , r i ) disjoint, 1 ≤ r i ≤ 2 ( 1 − ε ) n τ h ( A , S n , ε ) = max h 2 n i = 1 • Say that A ⊆ Z d is h-packing finite if p h ( A , ε ) < ∞ for all ε ∈ ( 0 , 1 ) . • The discrete packing dimension of A is defined by � � dim P A = inf α > 0 : A is r α -packing finite . 9 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary Discrete Dimensions of the Range of RCM Theorem [Xiao and Zheng(2011)] Let R = { x ∈ Z d : X t = x for some t ≥ 0 } be the range of VSRW X (as well as that of CSRW Y). Assume that d ≥ 3 and Q ( µ e ≥ 1 ) = 1 . Then for P -almost every ω ∈ Ω , dim H R = dim P R = 2 , P ω 0 -a.s. . where dim H and dim P denote respectively the discrete Hausdorff and packing dimension. 10 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary Recurrent/Transient Sets for RCM Theorem [Xiao and Zheng(2011)] Assume that d ≥ 3 and P ( µ e ≥ 1 ) = 1 . Let A ⊂ Z d be any (infinite) set. Then for P -almost every ω ∈ Ω , the following statements hold. (i) If dim H A < d − 2 , then � � P ω X t ∈ A for arbitrarily large t > 0 = 0 . 0 (ii) If dim H A > d − 2 , then � � P ω X t ∈ A for arbitrarily large t > 0 = 1 . 0 Remark Both theorems are also proven for the Bouchaud’s trap model. 11 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary Main Ingredients of Proof • Basic idea: derive various estimates for ordinary random walks used in [Barlow and Taylor(1992)], by using general Markov chain techniques • Main ingredients: 1. Gaussian heat kernel bounds for the VSRW ([Barlow and Deuschel(2010)]); 2. Hitting probability estimates; 3. Tail probability estimates of the sojourn measure for the discrete time VSRW; 4. Tail probability estimates of the maximal displacement of VSRW; 5. A SLLN for dependent events; 6. A zero-one law as a consequence of an elliptic Harnack inequality that the VSRW satisfies. 12 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary Proof Sketch for Theorem 1 • dim P R ≤ 2 P ω 0 -a.s.: first moment argument; • dim H R ≥ 2 P ω 0 -a.s.: let � R be the range of the discrete time VSRW ( � Y n ) := ( Y n ) , and show that dim H � R ≥ 2. • Let µ be the counting measure on � R . Show that � � ≤ c n 2 2 k µ Q k ( x ) for every x ∈ S n and 0 ≤ k ≤ n . • Frostman’s lemma ⇒ �� � ≥ c − 1 n − 1 2 − 2 n µ ( S n ) ν 2 R , S n • Hitting probability estimate ⇒ � � E ω ≥ c 2 2 n µ ( S n ) 0 � � and hence E ω m 2 ( � R ) = ∞ . 0 �� � = ∞ P ω • To further prove m 2 0 -a.s., let n k = ⌊ λ k log k ⌋ for λ > 0 R TBD, and define � 0 , 2 n k �� � n > 0 : � τ k = inf X n / ∈ V . Show that � X τ k − 1 | > 2 n k − 3 � | � 1. P ω ≤ c exp ( − ck ) ; and 0 � X τ k − 1 | ≤ 2 n k − 3 � | � 2. On the event , � � P ω µ ( S n k ) ≥ c 2 2 n k ≥ p . � X τ k − 1 3. The SLLN for dependent event concludes. 13 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
Introduction Main Results Proof Sketch and Main Ingredients Summary Summary 0. QFCLT for the VSRW/CSRW 1. Discrete fractal dimensions of the range of VSRW/CSRW 2. Characterization of recurrent/transient sets for VSRW/CSRW 3. Similarly for Bouchaud’s trap model. Thank you! 14 / 14 Xinghua Zheng Fractal Dimensions of Range of RCM
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