Excited Random Walks with Non-nearest Neighbor Steps

Davis, Burgess; Peterson, Jonathon

doi:10.1007/s10959-016-0697-1

Cited by 4 publications

(5 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, M τ 2k } the rotors to the left of X τ 2k point right and the rotors to the right of X τ 2k point left. These rotors coincide with the native environment (12) with the origin shifted to X τ 2k . Therefore, starting at time τ 2k until the time τ 2k+1 when it exits the interval I k , the p-rotor walk is a correlated random walk with persistence 1 − p (Definition 2.4).…”

Section: Proposition 26mentioning

confidence: 76%

“…If the local chain

{(X_{t_{k}^{x} + 1})}_{k \geq 1}

is only assumed to be hidden Markov, then the answer is yes, as shown by Pinsky and Travers . For excited random walks with non‐nearest neighbor steps, the question of transience/recurrence was investigated in . It may be useful to understand if one can adapt their results in our setting of locally Markov walks.…”

Section: Higher Dimensions and Longer Jumpsmentioning

confidence: 99%

“…We give a quick proof of this fact for completeness. (0, 1), the native p-rotor walk (U n ) with initial configuration as in (12) when rescaled by √ n, converges weakly on C[0, 1] to a Brownian motion:…”

Section: Definition 24mentioning

confidence: 99%

See 2 more Smart Citations

Interpolating between random walk and rotor walk

Huss

Levine

Sava-Huss

2017

Random Struct Algorithms

View full text Add to dashboard Cite

We introduce a family of stochastic processes on the integers, depending on a parameter p∈[0,1] and interpolating between the deterministic rotor walk (p=0) and the simple random walk (p=1/2). This p‐rotor walk is not a Markov chain but it has a local Markov property: for each x∈ℤ the sequence of successive exits from x is a Markov chain. The main result of this paper identifies the scaling limit of the p‐rotor walk with two‐sided i.i.d. initial rotors. The limiting process takes the form 1−ppX(t), where X is a doubly perturbed Brownian motion, that is, it satisfies the implicit equation X ( t ) = B ( t ) + a sup ⁡ s ≤ t X ( s ) + b inf ⁡ s ≤ t X ( s ) for all t∈[0,∞). Here B(t) is a standard Brownian motion and a,b<1 are constants depending on the marginals of the initial rotors on ℕ and −ℕ respectively. Chaumont and Doney have shown that Equation 1 has a pathwise unique solution X(t), and that the solution is almost surely continuous and adapted to the natural filtration of the Brownian motion. Moreover, lim⁡sup⁡X(t)=+∞ and lim⁡inf⁡X(t)=−∞. This last result, together with the main result of this paper, implies that the p‐rotor walk is recurrent for any two‐sided i.i.d. initial rotors and any 0

show abstract

Section: Proposition 26mentioning

confidence: 76%

“…If the local chain

{(X_{t_{k}^{x} + 1})}_{k \geq 1}

Section: Higher Dimensions and Longer Jumpsmentioning

confidence: 99%

See 1 more Smart Citation

Interpolating between random walk and rotor walk

Huss

Levine

Sava-Huss

2017

Random Struct Algorithms

View full text Add to dashboard Cite

show abstract

“…We call the process X described above (L, 1) non-nearest neighbors excited random walk ((L, 1)-ERW, for brevity). It is worth mentioning that (L, 1)-ERW is a special case of excited random walks with non-nearest neighbour steps considered by Davis and Peterson in [5] in which the particle can also jump to non-nearest neighbours on the right and Λ can be an unbounded subset of Z. In particular, Theorem 1.6 in [5] implies that the process studied in this paper is…”

Section: Introductionmentioning

confidence: 93%

“…Additionally, Davis and Peterson conjectured that the limiting speed of the random walk exists if δ > 1 and it is positive when δ > 2. (see Conjecture 1.8 in [5]).…”

Section: Introductionmentioning

confidence: 99%

Speed of excited random walks with long backward steps

Nguyen

2021

Preprint

View full text Add to dashboard Cite

We study a model of multi-excited random walk with non-nearest neighbour steps on Z, in which the walk can jump from a vertex x to either x + 1 or x − i with i ∈ {1, 2, . . . , L}, L ≥ 1.We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton-Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh [Probab. Theory Related Fields (2008), 141(3-4)], we extend their result (w.r.t the case L = 1) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift δ > 2. This confirms a special case of a conjecture proposed by Davis and Peterson.

show abstract

Speed of Excited Random Walks with Long Backward Steps

Nguyen

2022

J Stat Phys

View full text Add to dashboard Cite

We study a model of multi-excited random walk with non-nearest neighbour steps on $$\mathbb {Z}$$ Z , in which the walk can jump from a vertex x to either $$x+1$$ x + 1 or $$x-i$$ x - i with $$i\in \{1,2,\dots ,L\}$$ i ∈ { 1 , 2 , ⋯ , L } , $$L\ge 1$$ L ≥ 1 . We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton–Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh (Probab Theory Relat Fields 141:3–4, 2008), we extend their result (w.r.t. the case $$L=1$$ L = 1 ) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift $$\delta >2$$ δ > 2 . This confirms a special case of a conjecture proposed by Davis and Peterson.

show abstract

Excited Random Walks with Non-nearest Neighbor Steps

Cited by 4 publications

References 12 publications

Interpolating between random walk and rotor walk

Interpolating between random walk and rotor walk

Speed of excited random walks with long backward steps

Speed of Excited Random Walks with Long Backward Steps

Contact Info

Product

Resources

About