On the diffraction spectrum of two-dimensional random substitutions

We extend the diffraction calculations made by Spindeler [(2017), PhD thesis, Bielefeld University] and Gohlke [(2017), Master's thesis, Bielefeld University] to a family of two-dimensional constant-shape random substitutions and a nonconstant-shaped example. Specifically, we consider a family of binary constant-shape random block substitutions of size k × k defined by the rule \vartheta:b\ \ \mapsto\ \ a^{k\times k}, a\ \mapsto {\vartheta^{(i)}(a) with probability p_{1}\mid 1\leq i\leq N-1}, where k\in{\bb N}, p_i > 0 for all 0 ≤ i ≤ N - 1 and \sum_{i = 0}^{N-1}p_{i} = 1 with a^k×k and each ϑ_i(a) is a square block of size k × k. We illustrate that their corresponding systems have a mixed spectral type consisting of a pure-point part and a continuous part that is only fully absolutely continuous. We also explore extending the computation to random Fibonacci direct product variation, presenting a heuristic diffraction framework for the class of nonconstant-shape but geometrically compatible inflation rules.