  Dynamical quantum phase transitions (DQPTs) feature singular temporal
behavior in transient quantum states during nonequilibrium real-time evolution.
In this work we show that DQPTs in random Ising chains exhibit critical
behavior with nontrivial exponents that are not integer valued and not of
mean-field type. By means of an exact renormalization group transformation we
estimate the exponents with high accuracy eliminating largely any finite-size
effects. We further discuss how the considered dynamical phenomena can be made
accessible in current Rydberg atom platforms. In this context we explore
signatures of the DQPTs in the statistics of spin configuration measurements
available in such architectures. Specifically, we study the statistics of
clusters of consecutively aligned spins and observe a marked influence of the
DQPT on the corresponding distribution.
