Abstract
AbstractWe 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.
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference26 articles.
1. Athreya, K.B., Ney, P.E.: Branching processes, Die Grundlehren der mathematischen Wissenschaften, vol. 196. Springer, New York (1972)
2. Basdevant, A.-L., Singh, A.: On the speed of a cookie random walk. Probab. Theory Relat. Fields 141(3–4), 625–645 (2008)
3. Benjamini, I., Wilson, D.B.: Excited random walk. Electron. Commun. Probab. 8, 86–92 (2003)
4. Collevecchio, A., Hamza, K., Nguyen, T.-M.: Long range one-cookie random walk with positive speed. Stoch. Process. Appl. 142, 462–478 (2021)
5. Davis, B., Peterson, J.: Excited random walks with non-nearest neighbor steps. J. Theoret. Probab. 30(4), 1255–1284 (2017)