Weak quenched limiting distributions of a one-dimensional random walk in a random environment Jonathon Peterson Cornell University Department of Mathematics Joint work with Gennady Samorodnitsky September 13, 2010 Cornell University Jonathon Peterson 9/13/2010 1 / 27
Introduction RWRE in Z with i.i.d. environment An environment ω = {ω x } x Z Ω = [0, 1] Z. P an i.i.d. product measure on Ω. 1 ω x ω x x 1 x x + 1 Cornell University Jonathon Peterson 9/13/2010 2 / 27
Introduction RWRE in Z with i.i.d. environment An environment ω = {ω x } x Z Ω = [0, 1] Z. P an i.i.d. product measure on Ω. 1 ω x ω x x 1 x x + 1 Quenched law P ω : fix an environment. X n a random walk: X 0 = 0, and P ω (X n+1 = y + 1 X n = y) := ω y Cornell University Jonathon Peterson 9/13/2010 2 / 27
Introduction RWRE in Z with i.i.d. environment An environment ω = {ω x } x Z Ω = [0, 1] Z. P an i.i.d. product measure on Ω. 1 ω x ω x x 1 x x + 1 Quenched law P ω : fix an environment. X n a random walk: X 0 = 0, and P ω (X n+1 = y + 1 X n = y) := ω y Averaged law P: average over environments. P(G) := P ω (G)dP(ω) Ω Cornell University Jonathon Peterson 9/13/2010 2 / 27
Introduction Transience Criterion A crucial statistic is: ρ x := 1 ω x ω x Theorem (Solomon 75) 1 If E P (log ρ 0 ) < 0 then, lim n X n = +, P a.s. 2 If E P (log ρ 0 ) > 0 then, lim n X n =, P a.s. 3 If E P (log ρ 0 ) = 0 then, X n is recurrent. Cornell University Jonathon Peterson 9/13/2010 3 / 27
Introduction Transience Criterion A crucial statistic is: ρ x := 1 ω x ω x Theorem (Solomon 75) 1 If E P (log ρ 0 ) < 0 then, lim n X n = +, P a.s. 2 If E P (log ρ 0 ) > 0 then, lim n X n =, P a.s. 3 If E P (log ρ 0 ) = 0 then, X n is recurrent. Moreover, X n lim n n = v P > 0 E P ρ 0 < 1. Cornell University Jonathon Peterson 9/13/2010 3 / 27
Introduction Scale Parameter κ(p) for Transient RWRE Assume E P (log ρ) < 0 Define κ = κ(p) by (transience to the right). E P ρ κ = 1. E P [ρ p ] 1 κ p κ is related to the strength of the traps. Cornell University Jonathon Peterson 9/13/2010 4 / 27
Averaged limiting distribution Introduction Theorem (Kesten, Kozlov, Spitzer 75) There exists a constant b such that ( ) Xn (a) κ (0, 1) lim P n n κ x = 1 L κ,b (x 1/κ ) ( ) Xn nv (b) κ (1, 2) lim P P n n 1/κ x = 1 L κ,b ( x) ( ) Xn nv (c) κ > 2 lim P P n b x = Φ(x) n where L κ,b is an κ-stable distribution function. Remark: Also limiting distributions for κ = 1, 2 with log corrections to centering or scaling. Cornell University Jonathon Peterson 9/13/2010 5 / 27
Introduction Averaged Limiting Distribution (Hitting Times) T n := inf{k 0 : X k = n} Theorem (Kesten, Kozlov, Spitzer 75) There exists a constant b such that (a) κ (0, 1) lim (b) κ (1, 2) lim n P (c) κ > 2 lim n P ( Tn P n ( Tn nv 1 P n 1/κ ( Tn nv 1 P b n where L κ,b is an s-stable distribution function. ) n 1/κ x = L κ,b (x) ) x x ) = L κ,b (x) = Φ(x) Cornell University Jonathon Peterson 9/13/2010 6 / 27
Introduction Totally Asymmetric κ-stable Distribution Characteristic Function of L κ,b : ( exp { b t κ 1 i )} t t tan(πκ/2) Cornell University Jonathon Peterson 9/13/2010 7 / 27
Introduction Totally Asymmetric κ-stable Distribution Characteristic Function of L κ,b : ( exp { b t κ 1 i )} t t tan(πκ/2) If Y 0 and P(Y 1 > x) Cx κ, as x, then Y is in the domain of attraction of L κ,b. T n = n (T i T i 1 ) i=1 Cornell University Jonathon Peterson 9/13/2010 7 / 27
Quenched Limit Laws Quenched Limiting Distribution: Gaussian Regime Theorem (Goldsheid 06, P. 06) If κ > 2 then lim n P ω ( Tn E ω T n σ n ) x = Φ(x), 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. ) = Φ(x), P a.s. Cornell University Jonathon Peterson 9/13/2010 8 / 27
Quenched Limit Laws Quenched Limiting Distribution: Gaussian Regime Theorem (Goldsheid 06, P. 06) If κ > 2 then lim n P ω ( Tn E ω T n σ n ) x = Φ(x), 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. Cannot remove correction term Z n (ω). Scaling is different from averaged CLT. ) = Φ(x), P a.s. Cornell University Jonathon Peterson 9/13/2010 8 / 27
Quenched Limit Laws Quenched Limiting Distribution: Gaussian Regime Quenched CLT: T n = n i=1 (T i T i 1 ) Use Lindberg-Feller Condition Cornell University Jonathon Peterson 9/13/2010 9 / 27
Quenched Limit Laws Quenched Limiting Distribution: Gaussian Regime Quenched CLT: T n = n i=1 (T i T i 1 ) Use Lindberg-Feller Condition Quenched CLT Averaged CLT: T n n/v P n = T n E ω T n n E ωt n n/v P n Terms on right are asymptotically independent. (E ω T n n/v P )/ n is approximately mean zero Gaussian. Cornell University Jonathon Peterson 9/13/2010 9 / 27
Quenched Limit Laws Quenched Limiting Distribution: Gaussian Regime Quenched CLT: T n = n i=1 (T i T i 1 ) Use Lindberg-Feller Condition Quenched CLT Averaged CLT: T n n/v P n = T n E ω T n n E ωt n n/v P n Terms on right are asymptotically independent. (E ω T n n/v P )/ n is approximately mean zero Gaussian. Question What happens when κ < 2? Do we get quenched stable laws? Cornell University Jonathon Peterson 9/13/2010 9 / 27
Quenched Limits: κ < 2 Quenched Limit Laws Theorem (P. 07) If κ < 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 = Φ(x) k Varω T nk (b) lim k P ω ( T mk E ω T mk Varω T mk x ) = { 0 if x < 1 1 e x 1 if x 1 Cornell University Jonathon Peterson 9/13/2010 10 / 27
Quenched Limits: κ < 2 Quenched Limit Laws Theorem (P. 07) If κ < 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 = Φ(x) k Varω T nk FAQ (b) lim k P ω ( T mk E ω T mk Varω T mk Which limit is more likely? x ) = { 0 if x < 1 1 e x 1 if x 1 Cornell University Jonathon Peterson 9/13/2010 10 / 27
Quenched Limits: κ < 2 Quenched Limit Laws Theorem (P. 07) If κ < 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 = Φ(x) k Varω T nk FAQ (b) lim k P ω ( T mk E ω T mk Varω T mk Which limit is more likely? x What other subsequential limits are possible? ) = { 0 if x < 1 1 e x 1 if x 1 Cornell University Jonathon Peterson 9/13/2010 10 / 27
Quenched Limits: κ < 2 Quenched Limit Laws Theorem (P. 07) If κ < 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 = Φ(x) k Varω T nk FAQ (b) lim k P ω ( T mk E ω T mk Varω T mk Which limit is more likely? x What other subsequential limits are possible? Where do the κ-stable distributions come from. ) = { 0 if x < 1 1 e x 1 if x 1 Cornell University Jonathon Peterson 9/13/2010 10 / 27
Main Results Weak Quenched Limits: κ < 2 For any x R, is a random variable. P ω ( Tn E ω T n n 1/κ ) x Doesn t converge almost surely to a deterministic constant. Maybe converges in some weaker sense (in distribution). Cornell University Jonathon Peterson 9/13/2010 11 / 27
Random distributions Main Results M 1 = Probability distributions on R. - Equip with topology of convergence in distribution. Quenched distributions are M 1 -valued random variables. Quenched CLT (κ > 2): Then, µ n,ω (A) = P ω ( Tn E ω T n σ n lim µ n,ω = γ 0 M 1, n ) A. P a.s. where γ 0 M 1 is the standard normal distribution. Cornell University Jonathon Peterson 9/13/2010 12 / 27
Main Results Weak Quenched Limiting Distributions: κ < 2 µ n,ω (A) = P ω ( Tn E ω T n n 1/κ ) A Theorem (P. & Samorodnitsky 10) If κ < 2, then there exists a λ > 0 such that µ n,ω = n µ λ,κ, where µ λ,κ is a random probability distribution defined by ( ) µ λ,κ (A) = P a i (τ i 1) A a i, i 1 i=1 and N = i 1 δ a i is a non-homogeneous Poisson point process with intensity measure λκx κ 1 dx. Cornell University Jonathon Peterson 9/13/2010 13 / 27
Potential and Traps Trapping Effects i 1 k=0 log ρ k, i > 0 V (i) := 0, i = 0 1 k=i log ρ k, i < 0 Trap: Atypical section where the potential is increasing. Cornell University Jonathon Peterson 9/13/2010 14 / 27
Trapping Effects Cornell University Jonathon Peterson 9/13/2010 15 / 27
Blocks of the environment Trapping Effects Ladder locations {ν n } defined by ν 0 = 0, ν n := inf{i > ν n 1 : V (i) < V (ν n 1 )} ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 ν 7 Cornell University Jonathon Peterson 9/13/2010 16 / 27
Blocks of the environment Trapping Effects Ladder locations {ν n } defined by ν 0 = 0, ν n := inf{i > ν n 1 : V (i) < V (ν n 1 )} ν 1 ν 2 ν 3 ν 4 ν 5 ν 6 ν 7 Q( ) = P ( {V (i) > 0, i < 0}) Under Q, the environment is stationary under shifts of the ν i. Cornell University Jonathon Peterson 9/13/2010 16 / 27
Trapping Effects Reduction to T νn ν n = n i=1 (ν i ν i 1 ). LLN implies ν n n ν = E Pν 1 Cornell University Jonathon Peterson 9/13/2010 17 / 27
Trapping Effects Reduction to T νn ν n = n i=1 (ν i ν i 1 ). LLN implies ν n n ν = E Pν 1 Enough to study ( ) Tνn E ω T νn φ n,ω ( ) = P ω n 1/κ Want to show φ n,ω = µ λ,κ for some λ. Cornell University Jonathon Peterson 9/13/2010 17 / 27
Escaping Traps Trapping Effects Probability of escaping a trap of Height H. P ω (T b < T 1 ) = 1 1 + b j=1 ev (j) V (b) = H 1 0 b Cornell University Jonathon Peterson 9/13/2010 18 / 27
Escaping Traps Trapping Effects Probability of escaping a trap of Height H. P ω (T b < T 1 ) = 1 1 + b j=1 ev (j) e V (b) = e H V (b) = H 1 0 b Cornell University Jonathon Peterson 9/13/2010 18 / 27
Escaping Traps Trapping Effects Probability of escaping a trap of Height H. P ω (T b < T 1 ) = 1 1 + b j=1 ev (j) e V (b) = e H Time to escape trap of height H Exp(e H ). V (b) = H 1 0 b Cornell University Jonathon Peterson 9/13/2010 18 / 27
Comparison with exponentials Want: T ν Law (Eω T ν )τ. Trapping Effects Cornell University Jonathon Peterson 9/13/2010 19 / 27
Trapping Effects Comparison with exponentials Want: T ν Law (Eω T ν )τ. Decompose T ν into trials Where T ν = S + G i=1 F i S (T ν T ν < T + 0 ) F i (T + 0 T + 0 < T ν) G Geo(p ω ), where p ω = P ω (T ν < T + 0 ) Cornell University Jonathon Peterson 9/13/2010 19 / 27
Comparison with exponentials Want: T ν Law (Eω T ν )τ. Decompose T ν into trials Trapping Effects Where T ν = S + G i=1 F i S (T ν T ν < T + 0 ) F i (T + 0 T + 0 < T ν) G Geo(p ω ), where p ω = P ω (T ν < T + 0 ) Couple G with an exponential τ G = τ log(1/(1 p ω )). Cornell University Jonathon Peterson 9/13/2010 19 / 27
Proofs Heuristics of Quenched Limit Laws n n T νn = (T νi T νi 1 ) Law β i τ i i=1 i=1 where τ i are i.i.d. Exp(1) random variables and β i = β i (ω) = E ω (T νi T νi 1 ) Cornell University Jonathon Peterson 9/13/2010 20 / 27
Proofs Heuristics of Quenched Limit Laws n n T νn = (T νi T νi 1 ) Law β i τ i i=1 i=1 where τ i are i.i.d. Exp(1) random variables and β i = β i (ω) = E ω (T νi T νi 1 ) Reduced to study of ( 1 σ n,ω ( ) = P ω n 1/κ ) n β i (τ i 1). i=1 where τ i are i.i.d. exponential(1). Cornell University Jonathon Peterson 9/13/2010 20 / 27
Proofs Properties of β i 1 {β i } is stationary under Q. 2 The β i have heavy tails. 3 {β i } has good mixing properties. Cornell University Jonathon Peterson 9/13/2010 21 / 27
Proofs Properties of β i 1 {β i } is stationary under Q. 2 The β i have heavy tails. 3 {β i } has good mixing properties. Theorem (P. & Zeitouni 08) There exists a constant C > 0 such that Q(β 1 > x) = Q(E ω T ν > x) Cx κ. Moreover, if κ < 1 then there exists a constant b > 0 such that ( ) lim Q 1 n n n 1/κ β i x = L κ,b (x). i=1 Cornell University Jonathon Peterson 9/13/2010 21 / 27
Proofs Properties of β i 1 {β i } is stationary under Q. 2 The β i have heavy tails. 3 {β i } has good mixing properties. Theorem (P. & Zeitouni 08) There exists a constant C > 0 such that Q(β 1 > x) = Q(E ω T ν > x) Cx κ. Moreover, if κ < 1 then there exists a constant b > 0 such that ( ) lim Q 1 n n n 1/κ β i x = L κ,b (x). i=1 Kesten showed that P(E ω T 1 > x) C x κ. Cornell University Jonathon Peterson 9/13/2010 21 / 27
Point Process Convergence Proofs M p = Point processes i 1 δ x i on (0, ). N n = N n (ω) = n i=1 δ βi /n 1/κ Cornell University Jonathon Peterson 9/13/2010 22 / 27
Proofs Point Process Convergence M p = Point processes i 1 δ x i on (0, ). N n = N n (ω) = n i=1 δ βi /n 1/κ Proposition (P. & Samorodnitsky 10) Let κ < 2. Then N n = N λ,κ, where N λ,κ is a non-homogeneous Poisson point process with intensity measure λκx κ 1 dx, for some λ > 0. Cornell University Jonathon Peterson 9/13/2010 22 / 27
Proofs Point Process Convergence M p = Point processes i 1 δ x i on (0, ). N n = N n (ω) = n i=1 δ βi /n 1/κ Proposition (P. & Samorodnitsky 10) Let κ < 2. Then N n = N λ,κ, where N λ,κ is a non-homogeneous Poisson point process with intensity measure λκx κ 1 dx, for some λ > 0. Remark: N λ,κ = i 1 δ a i can be constructed by letting a i = Γ 1/κ i, where Γ i are the arrivals of a homogeneous Poisson(λ) point process. Cornell University Jonathon Peterson 9/13/2010 22 / 27
Proofs Define H : M p M 1 by ζ = δ xi = H(ζ)(A) = P x i (τ i 1) A x i, i 1. i 1 i 1 Remark: Only defined if x 2 i <. Cornell University Jonathon Peterson 9/13/2010 23 / 27
Proofs Define H : M p M 1 by ζ = δ xi = H(ζ)(A) = P x i (τ i 1) A x i, i 1. i 1 i 1 Remark: Only defined if x 2 i <. Recall σ n,ω = H(N n ) µ λ,κ D = H(Nλ,κ ) Since N n = N λ,κ, would like to conclude that µ n,ω = H(N n ) = H(N λ,κ ). Cornell University Jonathon Peterson 9/13/2010 23 / 27
Proofs Define H : M p M 1 by ζ = δ xi = H(ζ)(A) = P x i (τ i 1) A x i, i 1. i 1 i 1 Remark: Only defined if x 2 i <. Recall σ n,ω = H(N n ) µ λ,κ D = H(Nλ,κ ) Since N n = N λ,κ, would like to conclude that H is NOT continuous. µ n,ω = H(N n ) = H(N λ,κ ). Cornell University Jonathon Peterson 9/13/2010 23 / 27
Proofs For ε > 0 let H ε : M p M 1 be defined by ζ = δ xi = H ε (ζ)(a) = P x i 1 {xi >ε}(τ i 1) A x i, i 1. i 1 i 1 Claim: H ε is continuous on {ζ M p : ζ({ε}) = 0} M p. Cornell University Jonathon Peterson 9/13/2010 24 / 27
Proofs For ε > 0 let H ε : M p M 1 be defined by ζ = δ xi = H ε (ζ)(a) = P x i 1 {xi >ε}(τ i 1) A x i, i 1. i 1 i 1 Claim: H ε is continuous on {ζ M p : ζ({ε}) = 0} M p. Let ζ n ζ with ζ({ε}) = 0. ζ [ε, ) = M i=1 δ x i. For n large ζ n [ε, ) = M i=1 δ x (n) i, and x (n) i x i for all i M. Cornell University Jonathon Peterson 9/13/2010 24 / 27
Proofs Since P(N({ε}) = 0) = 1, H ε (N n ) = H ε (N). It remains to show lim ε 0 lim ρ(h ε(n), H(N)) = 0, P a.s. (1) ε 0 lim sup Q (ρ(h ε (N n ), H(N n )) δ) = 0. (2) n Cornell University Jonathon Peterson 9/13/2010 25 / 27
Proofs Since P(N({ε}) = 0) = 1, H ε (N n ) = H ε (N). It remains to show lim ε 0 lim ρ(h ε(n), H(N)) = 0, P a.s. (1) ε 0 lim sup Q (ρ(h ε (N n ), H(N n )) δ) = 0. (2) n where ρ is the Prohorov metric on M 1. { } ρ(µ, π) = inf δ : µ(a) π(a (δ) ) + δ, for all Borel A Cornell University Jonathon Peterson 9/13/2010 25 / 27
H ε (N) H(N) Proofs H ε (N)(A) = P a i 1 {ai >ε}(τ i 1) A i 1 P a i (τ i 1) A (δ) + P a i 1 {ai ε}(τ i 1) i 1 i 1 δ H(N)(A (δ) ) + 1 δ 2 ai 2 1 {a i ε} i 1 Cornell University Jonathon Peterson 9/13/2010 26 / 27
H ε (N) H(N) Proofs H ε (N)(A) = P a i 1 {ai >ε}(τ i 1) A i 1 P a i (τ i 1) A (δ) + P a i 1 {ai ε}(τ i 1) i 1 i 1 δ H(N)(A (δ) ) + 1 δ 2 ai 2 1 {a i ε} i 1 Thus ρ(h ε (N), H(N)) ai 2 1 {a i ε} i 1 1/3 Cornell University Jonathon Peterson 9/13/2010 26 / 27
Proofs Similarly Q (ρ(h ε (N n ), H(N n )) > δ) Q 1 n 2/κ βi 2 1 {βi /n 1/κ ε} > δ3 i 1 n1 2/κ [ ] δ 3 E Q β1 2 1 {β 1 /n 1/κ ε} Cornell University Jonathon Peterson 9/13/2010 27 / 27
Proofs Similarly Q (ρ(h ε (N n ), H(N n )) > δ) Q 1 n 2/κ βi 2 1 {βi /n 1/κ ε} > δ3 i 1 n1 2/κ [ ] δ 3 E Q β1 2 1 {β 1 /n 1/κ ε} Control expectation with tail decay: ] E Q [β 1 2 1 {β 1 /n 1/κ ε} C ε 1 κ/2 n 2/κ 1. Cornell University Jonathon Peterson 9/13/2010 27 / 27
Proofs Similarly Q (ρ(h ε (N n ), H(N n )) > δ) Q 1 n 2/κ βi 2 1 {βi /n 1/κ ε} > δ3 i 1 n1 2/κ [ ] δ 3 E Q β1 2 1 {β 1 /n 1/κ ε} Control expectation with tail decay: ] E Q [β 1 2 1 {β 1 /n 1/κ ε} C ε 1 κ/2 n 2/κ 1. Thus, Q (ρ(h ε (N n ), H(N n )) > δ) C δ 3 ε1 κ/2. Cornell University Jonathon Peterson 9/13/2010 27 / 27