TY - JOUR

T1 - Speed of Excited Random Walks with Long Backward Steps

AU - Nguyen, Tuan Minh

N1 - Funding Information:
The author would like to thank Andrea Collevecchio, Kais Hamza and the anonymous referees for their thorough reading and their constructive suggestions which improved the quality of the manuscript.
Publisher Copyright:
© 2022, The Author(s).

PY - 2022/7

Y1 - 2022/7

N2 - 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 Relat Fields 141:3–4, 2008), 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.

AB - 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 Relat Fields 141:3–4, 2008), 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.

KW - Excited random walks

KW - Multi-type branching processes with emigration

KW - Non-nearest neighbour random walks

UR - http://www.scopus.com/inward/record.url?scp=85129722993&partnerID=8YFLogxK

U2 - 10.1007/s10955-022-02926-2

DO - 10.1007/s10955-022-02926-2

M3 - Article

AN - SCOPUS:85129722993

SN - 0022-4715

VL - 188

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

IS - 1

M1 - 3

ER -