Quenched Limit Laws for Transient, One-Dimensional Random Walk in Random Environment

Transcription

1 Quenched Limit Laws for Transient, One-Dimensional Random Walk in Random Environment Jonathon Peterson School of Mathematics University of Minnesota April 8, 2008 Jonathon Peterson 4/8/ / 19

2 Background RWRE in Z with i.i.d. environment An environment ω = {ω x } x Z Ω = [0, 1] Z. P a product measure on Ω. Quenched law P ω : fix an environment. X n a random walk: X 0 = x, and P ω X n+1 = y + 1 X n = y := ω y Annealed law P: average over environments. PG := P ω GdPω Ω Jonathon Peterson 4/8/ / 19

3 Background RWRE in Z with i.i.d. environment An environment ω = {ω x } x Z Ω = [0, 1] Z. P a product measure on Ω. Quenched law P ω : fix an environment. X n a random walk: X 0 = x, and P ω X n+1 = y + 1 X n = y := ω y Annealed law P: average over environments. PG := P ω GdPω Ω Jonathon Peterson 4/8/ / 19

4 Background RWRE in Z with i.i.d. environment An environment ω = {ω x } x Z Ω = [0, 1] Z. P a product measure on Ω. Quenched law P ω : fix an environment. X n a random walk: X 0 = x, and P ω X n+1 = y + 1 X n = y := ω y Annealed law P: average over environments. PG := P ω GdPω Ω Jonathon Peterson 4/8/ / 19

5 Recurrence / Transience Background A crucial statistic is: ρ x := 1 ω x ω x Theorem Solomon 75 Transience or recurrence is determined by E P log ρ 0 : a E P log ρ 0 < 0 lim X n = +, n P a.s. b E P log ρ 0 > 0 lim X n =, n P a.s. c E P log ρ 0 = 0 X n is recurrent, P a.s. Jonathon Peterson 4/8/ / 19

6 Law of Large Numbers Background Assume E P log ρ < 0 transience to the right. Assume E P ρ s = 1 for some s > 0. Theorem LLN, Solomon 75 P a.s.: X n a s > 1 E P ρ < 1 lim n n = 1 E Pρ 1 + E P ρ > 0 X n b s 1 E P ρ 1 lim n n = 0 Denote lim n X n n =: v P. Jonathon Peterson 4/8/ / 19

7 Annealed Limit Laws Annealed Limit Laws Theorem Kesten, Kozlov, Spitzer 75 There exists a constant b such that Xn a s 0, 1 lim P n n s x = 1 L s,b x 1/s Xn nv b s 1, 2 lim P P n n 1/s x = 1 L s,b x Xn nv c s > 2 lim P P n b x = Φx n where L s,b is an s-stable distribution function. Proof: First prove stable limit laws for hitting times T n := inf{k 0 : X k = n} Jonathon Peterson 4/8/ / 19

8 Annealed Limit Laws Annealed Limit Laws Theorem Kesten, Kozlov, Spitzer 75 There exists a constant b such that Xn a s 0, 1 lim P n n s x = 1 L s,b x 1/s Xn nv b s 1, 2 lim P P n n 1/s x = 1 L s,b x Xn nv c s > 2 lim P P n b x = Φx n where L s,b is an s-stable distribution function. Proof: First prove stable limit laws for hitting times T n := inf{k 0 : X k = n} Jonathon Peterson 4/8/ / 19

9 Annealed Limit Laws Annealed Limit Laws Theorem Kesten, Kozlov, Spitzer 75 There exists a constant b such that a s 0, 1 lim b s 1, 2 lim n P c s > 2 lim n P Tn P n Tn nv 1 P n 1/s Tn nv 1 P b n where L s,b is an s-stable distribution function. n 1/s x = L s,b x x x = L s,b x = Φx Jonathon Peterson 4/8/ / 19

10 Annealed Limit Laws Characteristic Function of L s,b : exp { b t s 1 i } t t tanπs/2 Jonathon Peterson 4/8/ / 19

11 Quenched Limit Laws Quenched Limit Laws Gaussian Regime Theorem Goldsheid 06, P. 06 If s > 2 then lim n P ω Tn E ω T n σ n x = Φx, P a.s. where σ 2 = E P Var ω T 1, and lim n P ω X n nv P + Z n ω v 3/2 P σ n x where Z n ω depends only on the environment. Question: What happens when s < 2? Do we get quenched stable laws? = Φx, P a.s. Jonathon Peterson 4/8/ / 19

12 Quenched Limit Laws Quenched Limit Laws Gaussian Regime Theorem Goldsheid 06, P. 06 If s > 2 then lim n P ω Tn E ω T n σ n x = Φx, P a.s. where σ 2 = E P Var ω T 1, and lim n P ω X n nv P + Z n ω v 3/2 P σ n x where Z n ω depends only on the environment. Question: What happens when s < 2? Do we get quenched stable laws? = Φx, P a.s. Jonathon Peterson 4/8/ / 19

13 Traps Traps Define the potential of the environmnet i 1 k=0 log ρ k, i > 0 V i := 0, i = 0 1 k=i log ρ k, i < 0 Trap: An atypical section of environment where the potential is increasing. Time to cross a trap is exponential in the height of the uphill. Largest uphill of V in [0, n] is 1 s log n Erdös & Renyi 70. scaling of n 1/s in annealed limit laws of T n. Jonathon Peterson 4/8/ / 19

14 Traps Traps Define the potential of the environmnet i 1 k=0 log ρ k, i > 0 V i := 0, i = 0 1 k=i log ρ k, i < 0 Trap: An atypical section of environment where the potential is increasing. Time to cross a trap is exponential in the height of the uphill. Largest uphill of V in [0, n] is 1 s log n Erdös & Renyi 70. scaling of n 1/s in annealed limit laws of T n. Jonathon Peterson 4/8/ / 19

15 Blocks of the environment Traps Ladder locations {ν n } defined by ν 0 = 0, ν n := inf{i > ν n 1 : V i < V ν n 1 } ν n := sup{j < ν n+1 : V k > V j k < j} ν 3 ν 2 ν 1 ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 ν 5 ν ν 0 4 Define a new measure on environments Q = P {V i > 0, i < 0} Under Q, the environment is stationary under shifts of the ν i. Jonathon Peterson 4/8/ / 19

16 Blocks of the environment Traps Ladder locations {ν n } defined by ν 0 = 0, ν n := inf{i > ν n 1 : V i < V ν n 1 } ν n := sup{j < ν n+1 : V k > V j k < j} ν 3 ν 2 ν 1 ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 ν 5 ν ν 0 4 Define a new measure on environments Q = P {V i > 0, i < 0} Under Q, the environment is stationary under shifts of the ν i. Jonathon Peterson 4/8/ / 19

17 Quenched Analysis of T νn Heuristics of Quenched Limit Laws T νn = n T νi T νi 1 Law i=1 where µ i,ω = E ω T νi T νi 1 Quenched CLT? Only if n expµ i,ω i=1 Var ω T νi T νi 1. lim max µ 2 i,ω = 0, P a.s. n i n Var ω T νn Exponential limit if lim max µ 2 i,ω = 1, P a.s. n i n Var ω T νn Jonathon Peterson 4/8/ / 19

18 Quenched Analysis of T νn Heuristics of Quenched Limit Laws T νn = n T νi T νi 1 Law i=1 where µ i,ω = E ω T νi T νi 1 Quenched CLT? Only if n expµ i,ω i=1 Var ω T νi T νi 1. lim max µ 2 i,ω = 0, P a.s. n i n Var ω T νn Exponential limit if lim max µ 2 i,ω = 1, P a.s. n i n Var ω T νn Jonathon Peterson 4/8/ / 19

19 Quenched Analysis of T νn Heuristics of Quenched Limit Laws T νn = n T νi T νi 1 Law i=1 where µ i,ω = E ω T νi T νi 1 Quenched CLT? Only if n expµ i,ω i=1 Var ω T νi T νi 1. lim max µ 2 i,ω = 0, P a.s. n i n Var ω T νn Exponential limit if lim max µ 2 i,ω = 1, P a.s. n i n Var ω T νn Jonathon Peterson 4/8/ / 19

20 Quenched Analysis of T νn Heuristics of Quenched Limit Laws T νn = n T νi T νi 1 Law i=1 where µ i,ω = E ω T νi T νi 1 Quenched CLT? Only if n expµ i,ω i=1 Var ω T νi T νi 1. lim max µ 2 i,ω = 0, P a.s. n i n Var ω T νn Exponential limit if lim max µ 2 i,ω = 1, P a.s. n i n Var ω T νn Jonathon Peterson 4/8/ / 19

21 Quenched Analysis of T νn Theorem P. 07 Assume s < 2. Then b > 0 s.t. lim Q Varω T νn n n 2/s x = L s,bx. 2 α-stable process with α < 1 has jumps. This hints that when s < 2 lim inf n Q max i n µ 2 i,ω Var ω T νn < δ > 0 and lim inf n Q max i n µ 2 i,ω Var ω T νn > 1 δ > 0 Jonathon Peterson 4/8/ / 19

22 Quenched Analysis of T νn Theorem P. 07 Assume s < 2. Then b > 0 s.t. lim Q Varω T νn n n 2/s x = L s,bx. 2 α-stable process with α < 1 has jumps. This hints that when s < 2 lim inf n Q max i n µ 2 i,ω Var ω T νn < δ > 0 and lim inf n Q max i n µ 2 i,ω Var ω T νn > 1 δ > 0 Jonathon Peterson 4/8/ / 19

23 Quenched Limit Laws Quenched Limit Laws sub-gaussian regime Theorem P. 07 If s < 2 then P a.s. there exist random subsequences n k = n k ω, and m k = m k ω such that T nk E ω T nk a lim P ω x = Φx k Varω T nk b lim k P ω T mk E ω T mk Varω T mk Contrast with the annealed results: s 0, 1 lim n P s 1, 2 lim n P x Tn = { 0 if x < 1 1 e x 1 if x 1 n 1/s x = L s,b x Tn nv 1 P n 1/s x = L s,b x Jonathon Peterson 4/8/ / 19

24 Quenched Limit Laws Quenched Limit Laws sub-gaussian regime Theorem P. 07 If s < 2 then P a.s. there exist random subsequences n k = n k ω, and m k = m k ω such that T nk E ω T nk a lim P ω x = Φx k Varω T nk b lim k P ω T mk E ω T mk Varω T mk Contrast with the annealed results: s 0, 1 lim n P s 1, 2 lim n P x Tn = { 0 if x < 1 1 e x 1 if x 1 n 1/s x = L s,b x Tn nv 1 P n 1/s x = L s,b x Jonathon Peterson 4/8/ / 19

25 Quenched Limit Laws Quenched Limit Laws ballistic, sub-gaussian regime Theorem P. 07 If s 1, 2 then P a.s. there exist random subsequences n k = n k ω and m k = m k ω such that X tk n k a lim P ω x = Φx k v P Varω T nk Xt b lim P m { k e x 1 if x < 1 k ω < x = k v P Varω T mk 1 if x 1, where t k = E ω T nk and t k = E ωt mk. Contrast with lim P Xn nv P n n 1/s x = 1 L s,b x Jonathon Peterson 4/8/ / 19

26 Quenched Limit Laws Quenched Limit Laws ballistic, sub-gaussian regime Theorem P. 07 If s 1, 2 then P a.s. there exist random subsequences n k = n k ω and m k = m k ω such that X tk n k a lim P ω x = Φx k v P Varω T nk Xt b lim P m { k e x 1 if x < 1 k ω < x = k v P Varω T mk 1 if x 1, where t k = E ω T nk and t k = E ωt mk. Contrast with lim P Xn nv P n n 1/s x = 1 L s,b x Jonathon Peterson 4/8/ / 19

27 Quenched Limit Laws Quenched Limit Laws Zero-Speed Regime Theorem P., Zeitouni 07 If s 0, 1, then P a.s. there exist random subsequences n k = n k ω, m k = m k ω, t k = t k ω, and u k = u k ω s.t. { Xnk 0 x 0 a lim P ω x = k m 1 k 2 0 < x < and b lim k P ω log m lim k = s k log n k Xtk u k log 2 t k [ δ, δ] = 1, δ > 0. Contrast with lim P Xn n n s x = 1 L s,b x 1/s Jonathon Peterson 4/8/ / 19

28 Quenched Limit Laws Quenched Limit Laws Zero-Speed Regime Theorem P., Zeitouni 07 If s 0, 1, then P a.s. there exist random subsequences n k = n k ω, m k = m k ω, t k = t k ω, and u k = u k ω s.t. { Xnk 0 x 0 a lim P ω x = k m 1 k 2 0 < x < and b lim k P ω log m lim k = s k log n k Xtk u k log 2 t k [ δ, δ] = 1, δ > 0. Contrast with lim P Xn n n s x = 1 L s,b x 1/s Jonathon Peterson 4/8/ / 19

29 Proofs Proof of Stable Limits for the Quenched Variance Theorem P., Zeitouni 07 lim Q Varω T νn n n 2/s x = L s,bx. 2 Assume that E P log ρ < 0. For any s > 0, there exists a constant K > 0 s.t. QVar ω T ν > x QE ω T ν 2 > x Kx s/2. That PE ω T 1 > x K x s follows from a result of Kesten 73. The proof of the above mimics the proof of Kesten, Kozlov, and Spitzer 75. Jonathon Peterson 4/8/ / 19

30 Proofs Proof of Stable Limits for the Quenched Variance Theorem P., Zeitouni 07 lim Q Varω T νn n n 2/s x = L s,bx. 2 Assume that E P log ρ < 0. For any s > 0, there exists a constant K > 0 s.t. QVar ω T ν > x QE ω T ν 2 > x Kx s/2. That PE ω T 1 > x K x s follows from a result of Kesten 73. The proof of the above mimics the proof of Kesten, Kozlov, and Spitzer 75. Jonathon Peterson 4/8/ / 19

31 Proofs Var ω T νi T νi 1 in domain of attraction of s 2-stable distribution. But... not independent. H i the height of the block [ν i 1, ν i. Var ω T ν E ω T ν 2 e 2H 1. Iglehart 72: Pe H i > x Cx s. Binomial counting argument: Analyzing T νn, only crossing times of blocks with e H i > n 1 ε/s matter. Blocks with e H i > n 1 ε/s are well seperated Var ω T νi T νi 1 that are large are approximately independent. Jonathon Peterson 4/8/ / 19

32 Proofs Var ω T νi T νi 1 in domain of attraction of s 2-stable distribution. But... not independent. H i the height of the block [ν i 1, ν i. Var ω T ν E ω T ν 2 e 2H 1. Iglehart 72: Pe H i > x Cx s. Binomial counting argument: Analyzing T νn, only crossing times of blocks with e H i > n 1 ε/s matter. Blocks with e H i > n 1 ε/s are well seperated Var ω T νi T νi 1 that are large are approximately independent. Jonathon Peterson 4/8/ / 19

33 Proofs Var ω T νi T νi 1 in domain of attraction of s 2-stable distribution. But... not independent. H i the height of the block [ν i 1, ν i. Var ω T ν E ω T ν 2 e 2H 1. Iglehart 72: Pe H i > x Cx s. Binomial counting argument: Analyzing T νn, only crossing times of blocks with e H i > n 1 ε/s matter. Blocks with e H i > n 1 ε/s are well seperated Var ω T νi T νi 1 that are large are approximately independent. Jonathon Peterson 4/8/ / 19

34 Proofs Var ω T νi T νi 1 in domain of attraction of s 2-stable distribution. But... not independent. H i the height of the block [ν i 1, ν i. Var ω T ν E ω T ν 2 e 2H 1. Iglehart 72: Pe H i > x Cx s. Binomial counting argument: Analyzing T νn, only crossing times of blocks with e H i > n 1 ε/s matter. Blocks with e H i > n 1 ε/s are well seperated Var ω T νi T νi 1 that are large are approximately independent. Jonathon Peterson 4/8/ / 19

35 Finding the subsequences: Proofs With strictly positive probability uniformly in n find... One Large Block: One block i n with Then Var ω Tνi T νi 1 µi,ω > Mn 2/s j i Var ω Tνj Var ω T νj 1 n 2/s. µ2 i,ω Var ωt ν n > M2 M 2 +1 = 1 1 M No Dominating Blocks: Subset J {1, 2,..., n} of size 2k with i J Var ω Tνi T νi 1 µ 2 i,ω [n 2/s, 2n 2/s ] j / J Var ω Tνj T νj 1 < n 2/s Then µ2 i,ω Var ωt ν n < 2n2/s 2kn 2/s = 1 k. Jonathon Peterson 4/8/ / 19

36 Finding the subsequences: Proofs With strictly positive probability uniformly in n find... One Large Block: One block i n with Then Var ω Tνi T νi 1 µi,ω > Mn 2/s j i Var ω Tνj Var ω T νj 1 n 2/s. µ2 i,ω Var ωt ν n > M2 M 2 +1 = 1 1 M No Dominating Blocks: Subset J {1, 2,..., n} of size 2k with i J Var ω Tνi T νi 1 µ 2 i,ω [n 2/s, 2n 2/s ] j / J Var ω Tνj T νj 1 < n 2/s Then µ2 i,ω Var ωt ν n < 2n2/s 2kn 2/s = 1 k. Jonathon Peterson 4/8/ / 19

37 References References 1. H. Kesten, M. V. Kozlov, and F. Spitzer, A limit law for random walk in a random environment, Comp. Math , pp J. Peterson and O. Zeitouni, Quenched Limits for Transient, Zero-Speed One-Dimensional Random Walk in Random Environment, preprint 2007, arxiv:math/ v1 [math.pr] 3. J. Peterson, Quenched Limits for Transient, Ballistic, Sub-Gaussian One-Dimensional Random Walk in Random Environment, preprint 2007, arxiv: [mathpr] Jonathon Peterson 4/8/ / 19