Large Deviations and Slowdown Asymptotics of Excited Random Walks - PowerPoint PPT Presentation

LDP for Excited Random Walks Large Deviations and Slowdown Asymptotics of Excited Random Walks Jonathon Peterson Department of Mathematics Purdue University September 4, 2012 Jonathon Peterson 9/4/2012 1 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks ( M , p ) Cookie Random Walk Initially M cookies at each site. Jonathon Peterson 9/4/2012 2 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks ( M , p ) Cookie Random Walk Initially M cookies at each site. ◮ Cookie available: Eat cookie. Move right with probability p ∈ ( 0 , 1 ) 1 − p p Jonathon Peterson 9/4/2012 2 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks ( M , p ) Cookie Random Walk Initially M cookies at each site. ◮ Cookie available: Eat cookie. Move right with probability p ∈ ( 0 , 1 ) Jonathon Peterson 9/4/2012 2 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks ( M , p ) Cookie Random Walk Initially M cookies at each site. ◮ Cookie available: Eat cookie. Move right with probability p ∈ ( 0 , 1 ) 1 − p p Jonathon Peterson 9/4/2012 2 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks ( M , p ) Cookie Random Walk Initially M cookies at each site. ◮ Cookie available: Eat cookie. Move right with probability p ∈ ( 0 , 1 ) ◮ No cookies: Move right/left with probability 1 2 . Jonathon Peterson 9/4/2012 2 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks ( M , p ) Cookie Random Walk Initially M cookies at each site. ◮ Cookie available: Eat cookie. Move right with probability p ∈ ( 0 , 1 ) ◮ No cookies: Move right/left with probability 1 2 . 1 1 2 2 Jonathon Peterson 9/4/2012 2 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks Unequal cookies ◮ M cookies at each site. ◮ Cookie strengths p 1 , p 2 , . . . , p M ∈ ( 0 , 1 ) . Jonathon Peterson 9/4/2012 3 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks Unequal cookies ◮ M cookies at each site. ◮ Cookie strengths p 1 , p 2 , . . . , p M ∈ ( 0 , 1 ) . 1 − p 1 p 1 Jonathon Peterson 9/4/2012 3 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks Unequal cookies ◮ M cookies at each site. ◮ Cookie strengths p 1 , p 2 , . . . , p M ∈ ( 0 , 1 ) . Jonathon Peterson 9/4/2012 3 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks Unequal cookies ◮ M cookies at each site. ◮ Cookie strengths p 1 , p 2 , . . . , p M ∈ ( 0 , 1 ) . 1 − p 1 p 1 Jonathon Peterson 9/4/2012 3 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks Unequal cookies ◮ M cookies at each site. ◮ Cookie strengths p 1 , p 2 , . . . , p M ∈ ( 0 , 1 ) . Jonathon Peterson 9/4/2012 3 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks Unequal cookies ◮ M cookies at each site. ◮ Cookie strengths p 1 , p 2 , . . . , p M ∈ ( 0 , 1 ) . 1 − p 2 p 2 Jonathon Peterson 9/4/2012 3 / 13

LDP for Excited Random Walks Excited Random Walks Excited (Cookie) Random Walks Random i.i.d. cookie environments ◮ M cookies per site. ◮ ω x ( j ) – strength of j -th cookie at site x . ◮ Cookie environment ω = { ω x } is i.i.d. Cookies within a stack may be dependent. Jonathon Peterson 9/4/2012 4 / 13

LDP for Excited Random Walks Excited Random Walks Recurrence/Transience and LLN Average drift per site   M � δ = E ( 2 ω 0 ( j ) − 1 )   j = 1 Theorem ( Zerner ’05, Zerner & Kosygina ’08) The cookie RW is recurrent if and only if δ ∈ [ − 1 , 1 ] . Jonathon Peterson 9/4/2012 5 / 13

LDP for Excited Random Walks Excited Random Walks Recurrence/Transience and LLN Average drift per site   M � δ = E ( 2 ω 0 ( j ) − 1 )   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 9/4/2012 5 / 13

LDP for Excited Random Walks Excited Random Walks Limiting Distributions for Excited Random Walks Theorem (Basdevant & Singh ’08, Kosygina & Zerner ’08, Dolgopyat ’11) Excited random walks have the following limiting distributions. Regime Re-scaling Limiting Distribution � δ � − δ/ 2 X n δ ∈ ( 1 , 2 ) 2 -stable n δ/ 2 X n − nv 0 Totally asymmetric δ δ ∈ ( 2 , 4 ) 2 -stable n 2 /δ X n − nv 0 δ > 4 Gaussian A √ n Results are also known for other values of δ . Note: δ > 1 results similar to transient RWRE. Jonathon Peterson 9/4/2012 6 / 13

LDP for Excited Random Walks Excited Random Walks Limiting Distributions for Excited Random Walks Theorem (Basdevant & Singh ’08, Kosygina & Zerner ’08, Dolgopyat ’11) Hitting times T n = min { k ≥ 0 : X k = n } of excited random walks have the following limiting distributions. Regime Re-scaling Limiting Distribution T n Totally asymmetric δ δ ∈ ( 1 , 2 ) 2 -stable n 2 /δ T n − n / v 0 Totally asymmetric δ δ ∈ ( 2 , 4 ) 2 -stable n 2 /δ T n − n / v 0 δ > 4 Gaussian A √ n Results are also known for other values of δ . Note: δ > 1 results similar to transient RWRE. Jonathon Peterson 9/4/2012 7 / 13

LDP for Excited Random Walks Excited Random Walks Large Deviations for Excited Random Walks Theorem (P . ’12) X n / n has a large deviation principle with rate function I X ( x ) . That is, for any open G ⊂ [ − 1 , 1 ] 1 lim inf n log P ( X n / n ∈ G ) ≥ − inf x ∈ G I X ( x ) n →∞ and for any closed F ⊂ [ − 1 , 1 ] 1 n log P ( X n / n ∈ F ) ≤ − inf x ∈ F I X ( x ) lim sup n →∞ Informally, P ( X n ≈ xn ) ≈ e − nI X ( x ) . Jonathon Peterson 9/4/2012 8 / 13

LDP for Excited Random Walks Excited Random Walks Large Deviations for Hitting Times of Excited Random Walks T x = inf { n ≥ 0 : X n = x } , x ∈ Z . Theorem (P . ’12) T n / n has a large deviation principle with rate function I T ( t ) . Jonathon Peterson 9/4/2012 9 / 13

LDP for Excited Random Walks Excited Random Walks Large Deviations for Hitting Times of Excited Random Walks T x = inf { n ≥ 0 : X n = x } , x ∈ Z . Theorem (P . ’12) T n / n has a large deviation principle with rate function I T ( t ) . T − n / n has a large deviation principle with rate function I T ( t ) . Jonathon Peterson 9/4/2012 9 / 13

LDP for Excited Random Walks Excited Random Walks Large Deviations for Hitting Times of Excited Random Walks T x = inf { n ≥ 0 : X n = x } , x ∈ Z . Theorem (P . ’12) T n / n has a large deviation principle with rate function I T ( t ) . T − n / n has a large deviation principle with rate function I T ( t ) . Implies LDP for X n / n . P ( X n > xn ) ≈ P ( T xn < n ) .  xI T ( 1 / x ) x ∈ ( 0 , 1 ]   I X ( x ) = 0 x = 0  | x | I T ( 1 / | x | ) x ∈ [ − 1 , 0 )  Jonathon Peterson 9/4/2012 9 / 13

LDP for Excited Random Walks Excited Random Walks Properties of the rate function I X ( x ) δ ∈ [ − 2 , 2] δ > 2 − 1 x 2 v 0 x 1 x 2 − 1 x 2 x 2 1 1 I X ( x ) is a convex function. 1 Jonathon Peterson 9/4/2012 10 / 13

LDP for Excited Random Walks Excited Random Walks Properties of the rate function I X ( x ) δ ∈ [ − 2 , 2] δ > 2 − 1 x 2 v 0 x 1 x 2 − 1 x 2 x 2 1 1 I X ( x ) is a convex function. 1 Zero Set 2 ◮ δ ∈ [ − 2 , 2 ] : I X ( x ) = 0 ⇐ ⇒ x = 0. ◮ δ > 2: I X ( x ) = 0 ⇐ ⇒ x ∈ [ 0 , v 0 ] . Jonathon Peterson 9/4/2012 10 / 13

LDP for Excited Random Walks Excited Random Walks Properties of the rate function I X ( x ) δ ∈ [ − 2 , 2] δ > 2 − 1 x 2 v 0 x 1 x 2 − 1 x 2 x 2 1 1 I X ( x ) is a convex function. 1 Zero Set 2 ◮ δ ∈ [ − 2 , 2 ] : I X ( x ) = 0 ⇐ ⇒ x = 0. ◮ δ > 2: I X ( x ) = 0 ⇐ ⇒ x ∈ [ 0 , v 0 ] . Derivatives 3 ◮ I ′ X ( 0 ) = lim x → 0 I X ( x ) / x = 0. Jonathon Peterson 9/4/2012 10 / 13

LDP for Excited Random Walks Excited Random Walks Properties of the rate function I T ( t ) δ > 2 δ ≤ 2 t 2 t 1 1 /v 0 t 2 1 1 I T ( t ) is a convex function. 1 Jonathon Peterson 9/4/2012 11 / 13

LDP for Excited Random Walks Excited Random Walks Properties of the rate function I T ( t ) δ > 2 δ ≤ 2 t 2 t 1 1 /v 0 t 2 1 1 I T ( t ) is a convex function. 1 Zero Set 2 ◮ δ ∈ [ − 2 , 2 ] : I T ( t ) > 0 but lim t →∞ I T ( t ) = 0. ◮ δ > 2: I T ( t ) = 0 ⇐ ⇒ t ≥ 1 / v 0 . Jonathon Peterson 9/4/2012 11 / 13

LDP for Excited Random Walks Excited Random Walks Slowdown probability asymptotics I X ( x ) = 0 ⇐ ⇒ x ∈ [ 0 , v 0 ] . P ( X n < xn ) decays sub-exponentially for x ∈ [ 0 , v 0 ] . Jonathon Peterson 9/4/2012 12 / 13