Exact "hydrophobicity" in deterministic circuits: dynamical fluctuations in the Floquet-East model

We study the dynamics of a classical circuit corresponding to a discrete-time deterministic kinetically constrained East model. We show that -- despite being deterministic -- this "Floquet-East" model displays pre-transition behaviour, which is a dynamical equivalent of the hydrophobic effect in water. By means of exact calculations we prove: (i) a change in scaling with size in the probability of inactive space-time regions (akin to the "energy-entropy" crossover of the solvation free energy in water), (ii) a first-order phase transition in the dynamical large deviations, (iii) the existence of the optimal geometry for local phase separation to accommodate space-time solutes, and (iv) a dynamical analog of "hydrophobic collapse". We discuss implications of these exact results for circuit dynamics more generally.