Abstract
We introduce a method for studying monotonicity of the speed of excited random walks in high dimensions, based on a formula for the speed obtained via cut-times and Girsanov's transform. While the method gives rise to similar results as have been or can be obtained via the expansion method of van der Hofstad and Holmes, it may be more palatable to a general probabilistic audience. We also revisit the law of large numbers for stationary cookie environments. In particular, we introduce a new notion of $e_1-$exchangeable cookie environment and prove the law of large numbers for this case.
Highlights
1.1 Excited random walk with random cookies (ERWRC)Excited random walks (ERW) were introduced in [2] by I
We introduce a method for studying monotonicity of the speed of excited random walks in high dimensions, based on a formula for the speed obtained via cut-times and Girsanov’s transform
We revisit the law of large numbers for stationary cookie environments
Summary
Excited random walks (ERW) were introduced in [2] by I. In [9], the LLN of Theorems 4.6 and 4.8 is proved for all dimension d 1 using renewal structure To use this technique, the conditions of uniform ellipticity of the cookie environment and the transience of the random walk in some direction l ∈ Rd are needed. The laws of the random environments are allowed to take two values, say ν1 and ν2 with probabilities β and 1−β (where β is a constant in [0, 1]) They proved the monotonicity of the speed with respect to β. We prove the monotonicity of the speed with respect to the law of the random cookie environment Q for the special case of m identical random cookies This model is a partial model of m−ERWRC when the cookie environment is not random and identical, i.e. the cookies are the same for every site:. The rest could be obtained by minor modification of the proof of Theorem 2.3 of [7]
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