Strict Monotonicity Properties in One-dimensional Excited Random Walks

#### J. Peterson

2013, v.19, Issue 4, 721-734

ABSTRACT

We consider excited random walks with $M$ "cookies" where the $i$th cookie at each site has strength $p_i$. There are certain natural monotonicity results that are known for the excited random walk under some partial orderings of the cookie environments. For instance the limiting speed $$\lim_{n \to \infty} X_n/n = v(p_1,p_2,\ldots, p_M)$$ is increasing in each $p_j$. We improve these monotonicity results to be strictly monotone under a partial ordering of cookie environments introduced by Holmes and Salisbury. While the self-interacting nature of the excited random walk makes a direct coupling proof difficult, we show that there is a very natural coupling of the associated branching process from which the monotonicity results follow.

Keywords: excited random walk