Excited random walks in cookie environments with Markovian cookie stacks

Transcription

1 Excited random walks in cookie environments with Markovian cookie stacks Jonathon Peterson Department of Mathematics Purdue University Joint work with Elena Kosygina December 5, 2015 Jonathon Peterson 12/5/ / 19

2 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p 3 Jonathon Peterson 12/5/ / 19

3 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p 3 1 p 1 p 1 Jonathon Peterson 12/5/ / 19

4 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p 3 1 p 1 p 1 Jonathon Peterson 12/5/ / 19

5 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p 3 1 p 2 p 2 Jonathon Peterson 12/5/ / 19

6 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p 3 1 p 1 p 1 Jonathon Peterson 12/5/ / 19

7 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p 3 1 p 3 p 3 Jonathon Peterson 12/5/ / 19

8 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p 3 Jonathon Peterson 12/5/ / 19

9 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p 3 Jonathon Peterson 12/5/ / 19

10 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p Jonathon Peterson 12/5/ / 19

11 Excited (Cookie) Random Walks Cookie Environment M cookies at each site Cookie strengths p 1, p 2,, p M (0, 1) Eating a cookie induces a drift for the next step 1 p 1 p 1 1 p 2 p 2 1 p 3 p Jonathon Peterson 12/5/ / 19

12 Excited (Cookie) Random Walks Random iid cookie environments ω x (j) strength of j-th cookie at site x Cookie environment ω = {ω x } is iid Cookies within a stack may be dependent Jonathon Peterson 12/5/ / 19

13 Recurrence/Transience and LLN Average drift per site M δ = E (2ω 0 (j) 1) j=1 (deterministic cookie environments: δ = M j=1 (2p j 1)) Theorem ( Zerner 05, Zerner & Kosygina 08) The cookie RW is recurrent if and only if δ [ 1, 1] Jonathon Peterson 12/5/ / 19

14 Recurrence/Transience and LLN Average drift per site M δ = E (2ω 0 (j) 1) j=1 (deterministic cookie environments: δ = M j=1 (2p j 1)) Theorem ( Zerner 05, Zerner & Kosygina 08) The cookie RW is recurrent if and only if δ [ 1, 1] Theorem (Basdevant & Singh 07, Zerner & Kosygina 08) lim n X n /n = v 0, and v 0 > 0 δ > 2 No explicit formula is known for v 0 Jonathon Peterson 12/5/ / 19

15 Limiting Distributions for Theorem (Basdevant & Singh 08, Kosygina & Zerner 08, Kosygina & Mountford 11) Excited random walks have the following limiting distributions Regime Re-scaling Limiting Distribution δ (1, 2) X n ( δ 2 -stable) δ/2 δ (2, 4) δ > 4 n δ/2 X n nv 0 n 2/δ X n nv 0 A n Results are also known for other values of δ Note: similar scaling limits for RWRE Totally asymmetric δ 2 -stable Gaussian Jonathon Peterson 12/5/ / 19

16 Periodic cookie stacks Periodic cookie sequence p 1, p 2,, p M, p 1, p 2, Average value p = 1 M M j=1 p j Jonathon Peterson 12/5/ / 19

17 Periodic cookie stacks Periodic cookie sequence p 1, p 2,, p M, p 1, p 2, Average value p = 1 M M j=1 p j Questions: Recurrence/transience, limiting speed, limiting distributions? Jonathon Peterson 12/5/ / 19

18 Periodic cookie stacks - recurrence/transience Critical case: p = 1 M M j=1 p j = 1 2 Jonathon Peterson 12/5/ / 19

19 Periodic cookie stacks - recurrence/transience Critical case: p = 1 M M j=1 p j = 1 2 M j j=1 i=1 (1 p j)(2p i 1) M j j=1 i=1 p j(1 2p i ) θ = 2 M j=1 p j(1 p j ) and θ = 2 M j=1 p j(1 p j ) Theorem (Kozma, Shinkar, Orenshtein) For excited random walks with periodic cookie stacks and p = 1 2, θ > 1 implies X n + θ > 1 implies X n max{θ, θ} 1 implies X n is recurrent Note: θ + θ < 1 Jonathon Peterson 12/5/ / 19

20 Periodic cookie stacks - LLN Limiting speed exists (Kosygina-Zerer 13) X n v 0 = lim n n Jonathon Peterson 12/5/ / 19

21 Periodic cookie stacks - LLN Limiting speed exists (Kosygina-Zerer 13) X n v 0 = lim n n Theorem (Kosygina & P 15) For excited random walks with periodic cookie stacks and p = 1 2, If θ > 2 then v 0 > 0 If θ > 2 then v 0 < 0 If max{θ, θ} 2 then v 0 = 0 Jonathon Peterson 12/5/ / 19

22 Periodic cookie stacks - limiting distributions Limiting distributions when transient to the right (θ > 1) Theorem (Kosygina & P 15) For excited random walks with periodic cookie stacks and p = 1 2, the following limiting distributions hold Regime Re-scaling Limiting Distribution θ (1, 2) X n ( θ 2 -stable) θ/2 θ (2, 4) θ > 4 n θ/2 X n nv 0 n 2/θ X n nv 0 A n Results are also known for θ = 1 and θ = 2 Totally asymmetric θ 2 -stable Gaussian Jonathon Peterson 12/5/ / 19

23 Markovian cookie stacks Markov chain {R j } j 1 on finite state space {1, 2,, N} 1 α α α 1 α Jonathon Peterson 12/5/ / 19

24 Markovian cookie stacks Markov chain {R j } j 1 on finite state space {1, 2,, N} Function p : {1, 2,, N} (0, 1) p 1 p 1 α α 1 α 1 p p α Jonathon Peterson 12/5/ / 19

25 Markovian cookie stacks Markov chain {R j } j 1 on finite state space {1, 2,, N} Function p : {1, 2,, N} (0, 1) Cookie stack at x = 0: ω 0 (j) = p(r j ) p 1 p 1 α α 1 α 1 p p α Jonathon Peterson 12/5/ / 19

26 Markovian cookie stacks Markov chain {R j } j 1 on finite state space {1, 2,, N} Function p : {1, 2,, N} (0, 1) Cookie stack at x = 0: ω 0 (j) = p(r j ) Cookie stacks iid spatially p 1 p 1 α α 1 α 1 p p α Jonathon Peterson 12/5/ / 19

27 Markovian cookie stacks - results Markov chain input transition matrix K initial distribution η function p : {1, 2,, N} (0, 1) Jonathon Peterson 12/5/ / 19

28 Markovian cookie stacks - results Markov chain input transition matrix K initial distribution η function p : {1, 2,, N} (0, 1) Critical case: p = N j=1 π jp(j) = 1 2 (π is stationary for Markov chain) Jonathon Peterson 12/5/ / 19

29 Markovian cookie stacks - results Markov chain input transition matrix K initial distribution η function p : {1, 2,, N} (0, 1) Critical case: p = N j=1 π jp(j) = 1 2 (π is stationary for Markov chain) Theorem (Kosygina & P 15) Previous results for periodic cookie stacks extend to Markovian cookie stacks In particular, in the critical case Recurrence max{θ, θ} 1 Positive speed θ > 2 CLT θ > 4 or θ > 4 Explicit formulas for θ = θ(k, η, p) and θ = θ(k, η, p) Jonathon Peterson 12/5/ / 19

30 Explicit formulas for θ and θ Input parameters: K, η, p µ - stationary distribution for Markov chain D p - diagonal matrix with diagonal entries p = (p(1),, p(m)) Jonathon Peterson 12/5/ / 19

31 Explicit formulas for θ and θ Input parameters: K, η, p µ - stationary distribution for Markov chain D p - diagonal matrix with diagonal entries p = (p(1),, p(m)) r = (I K + 2(1 p)µ(i D p )K ) 1 p 1 ν = 2 + 4µD p K r θ = θ(k, η, 1 p) θ = 2η r ν Jonathon Peterson 12/5/ / 19

32 Markov cookie stacks - examples Periodic cookie stacks Jonathon Peterson 12/5/ / 19

33 Markov cookie stacks - examples Periodic cookie stacks Deterministic M-cookie stacks In this case θ = δ and θ = δ Jonathon Peterson 12/5/ / 19

34 Markov cookie stacks - examples Two-type, critical p 1 p 1 α α 1 α 1 p p α Jonathon Peterson 12/5/ / 19

35 Markov cookie stacks - examples Two-type, critical p 1 p 1 α α 1 α 1 p p α < θ < 1 1 < θ < 2 2 < θ < 4 θ > 4 θ = (2p 1)((2α 1)p α) 4(2α 1)(p 1)p + α 1 α p Jonathon Peterson 12/5/ / 19

36 Markov cookie stacks - examples Geometric cookie stacks 1 p p 1 α α θ = 2p 1 α Jonathon Peterson 12/5/ / 19

37 Markov cookie stacks - examples Geometric cookie stacks 1 p p 1 α α θ = 2p 1 α [ Note that also δ = E j=1 (2ω 0(j) 1)] = 2p 1 α Jonathon Peterson 12/5/ / 19

38 Backward branching-like process Branching-like processes Jonathon Peterson 12/5/ / 19

39 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i Jonathon Peterson 12/5/ / 19

40 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i Jonathon Peterson 12/5/ / 19

41 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i -2-1 D 5 5 = Jonathon Peterson 12/5/ / 19

42 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i -2-1 D 5 5 = Jonathon Peterson 12/5/ / 19

43 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i -2-1 D 5 4 = 1 D 5 5 = Jonathon Peterson 12/5/ / 19

44 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i -2-1 D 5 3 = 4 D 5 4 = 1 D 5 5 = Jonathon Peterson 12/5/ / 19

45 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i -2-1 D 5 2 = 2 D 5 3 = 4 D 5 4 = 1 D 5 5 = Jonathon Peterson 12/5/ / 19

46 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i -2-1 D 5 1 = 1 D 5 2 = 2 D 5 3 = 4 D 5 4 = 1 D 5 5 = Jonathon Peterson 12/5/ / 19

47 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i -2-1 D 5 0 = 2 D 5 1 = 1 D 5 2 = 2 D 5 3 = 4 D 5 4 = 1 D 5 5 = Jonathon Peterson 12/5/ / 19

48 Backward branching-like process Branching-like processes D n i = # steps from i before T n T n = n + i<n D n i -2 D 5 1 = 0-1 D 5 0 = 2 D 5 1 = 1 D 5 2 = 2 D 5 3 = 4 D 5 4 = 1 D 5 5 = Jonathon Peterson 12/5/ / 19

49 Branching-like processes Branching-like processes - LLN Backward branching process Markov chain {V i } i 0 (V 0, V 1,, V n ) Law = (D n n, D n n 1,, Dn 0 ) Jonathon Peterson 12/5/ / 19

50 Branching-like processes Branching-like processes - LLN Backward branching process Markov chain {V i } i 0 (V 0, V 1,, V n ) Law = (D n n, D n n 1,, Dn 0 ) T n n Jonathon Peterson 12/5/ / 19

51 Branching-like processes Branching-like processes - LLN Backward branching process Markov chain {V i } i 0 (V 0, V 1,, V n ) Law = (D n n, D n n 1,, Dn 0 ) T n n 1 n ( n 1 n + 2 i=0 D n i ) Jonathon Peterson 12/5/ / 19

52 Branching-like processes Branching-like processes - LLN Backward branching process Markov chain {V i } i 0 (V 0, V 1,, V n ) Law = (D n n, D n n 1,, Dn 0 ) T n n 1 n ( n 1 n + 2 i=0 D n i ) Law = n n V i i=1 Jonathon Peterson 12/5/ / 19

53 Branching-like processes Branching-like processes - LLN Backward branching process Markov chain {V i } i 0 (V 0, V 1,, V n ) Law = (D n n, D n n 1,, Dn 0 ) T n n 1 n ( n 1 n + 2 LLN for excited random walk 1 X n n v Tn n 1 v i=0 D n i ) Law = n n V i i=1 Jonathon Peterson 12/5/ / 19

54 Branching-like processes - LLN Branching-like processes Backward branching process Markov chain {V i } i 0 (V 0, V 1,, V n ) Law = (D n n, D n n 1,, Dn 0 ) T n n 1 n ( n 1 n + 2 LLN for excited random walk 1 X n n v Tn n 1 v 2 T n n 1 + 2E [V 0 ] i=0 D n i ) Law = n n V i i=1 Jonathon Peterson 12/5/ / 19

55 Branching-like processes - LLN Branching-like processes Backward branching process Markov chain {V i } i 0 (V 0, V 1,, V n ) Law = (D n n, D n n 1,, Dn 0 ) T n n 1 n ( n 1 n + 2 LLN for excited random walk 1 X n n v Tn n 1 v 2 T n n 1 + 2E [V 0 ] i=0 Ballistic: v > 0 E [V 0 ] < D n i ) Law = n n V i i=1 Jonathon Peterson 12/5/ / 19

56 Branching-like processes - LLN Branching-like processes Ballistic: v > 0 E [V 0 ] < Jonathon Peterson 12/5/ / 19

57 Branching-like processes - LLN Branching-like processes Ballistic: v > 0 E [V 0 ] < Regeneration time: σ = inf{i > 0 : V i = 0} Jonathon Peterson 12/5/ / 19

58 Branching-like processes - LLN Branching-like processes Ballistic: v > 0 E [V 0 ] < Regeneration time: σ = inf{i > 0 : V i = 0} E [V 0 ] = E [ σ i=1 V i] E[σ] Jonathon Peterson 12/5/ / 19

59 Branching-like processes - LLN Branching-like processes Ballistic: v > 0 E [V 0 ] < Regeneration time: σ = inf{i > 0 : V i = 0} E [V 0 ] = E [ σ i=1 V i] E[σ] Theorem (Kosygina & P 15) There exist constants C 1, C 2 > 0 such that ( σ ) P(σ > n) C 1 n θ, and P V i > n C 2 n θ/2 i=1 Jonathon Peterson 12/5/ / 19

60 Branching-like processes - LLN Branching-like processes Ballistic: v > 0 E [V 0 ] < Regeneration time: σ = inf{i > 0 : V i = 0} E [V 0 ] = E [ σ i=1 V i] E[σ] Theorem (Kosygina & P 15) There exist constants C 1, C 2 > 0 such that ( σ ) P(σ > n) C 1 n θ, and P V i > n C 2 n θ/2 Key idea: V i is like a squared Bessel process of dimension 2(1 θ) i=1 Jonathon Peterson 12/5/ / 19

61 Branching-like processes Comparison with squared Bessel process Mean: α = α(k, p, η) E[V 1 V 0 V 0 = n] α Ce cn Jonathon Peterson 12/5/ / 19

62 Branching-like processes Comparison with squared Bessel process Mean: α = α(k, p, η) E[V 1 V 0 V 0 = n] α Ce cn Variance: Var(V 1 V 0 = n) ν n C n ν = ν(k, p) Jonathon Peterson 12/5/ / 19

63 Branching-like processes Comparison with squared Bessel process Mean: α = α(k, p, η) E[V 1 V 0 V 0 = n] α Ce cn Variance: Var(V 1 V 0 = n) ν n C n ν = ν(k, p) { V tn n } t 0 is similar to dy t = α dt + ν Y t dw t (4/ν)Y t is a Squared Bessel process of dimendion 4α ν =: 2(1 θ) Jonathon Peterson 12/5/ / 19