A Law of Iterated Logarithm on Lamplighter Diagonal Products

We prove a Law of Iterated Logarithm for random walks on a family of diagonal products constructed by Brieussel and Zheng (2021). This provides a wide variety of new examples of Law of Iterated Logarithm behaviours for random walks on groups. In particular, it follows that for any $\frac{1}{2}\leq β\leq 1$ there is a group $G$ and random walk $W_n$ on $G$ with $\mathbb{E}|W_n|\simeq n^β$ such that $0<\limsup \frac{|W_n|}{n^β(\log\log n)^{1-β}}<\infty$ and $0<\liminf \frac{|W_n|(\log\log n)^{1-β}}{n^β}<\infty.$

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