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 -