Strict Monotonicity Properties in One-dimensional Excited Random Walks

We consider excited random walk's with M "cookies" where the ith 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 him.(n ->infinity) X-n/n = v(p(1), p(2), ... , 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.