  For $N$ compatible substitution rules on $M$ prototiles $t_1,\dots,t_M$,
consider tilings and tiling spaces constructed by applying the different
substitution rules at random. These give (globally) random substitution
tilings. In this paper I obtain bounds for the growth on twisted ergodic
integrals for the $\mathbb{R}^d$ action on the tiling space which give lower
bounds on the lower local dimension of spectral measures. For functions with
some extra regularity, uniform bounds on the lower local dimension are
obtained. The results here extends results of Bufetov-Solomyak to tilings of
higher dimensions.
