  We show that a collection of independent Ising spins evolving stochastically
can display surprisingly large fluctuations towards ordered behaviour, as
quantified by certain types of time-integrated plaquette observables, despite
the underlying dynamics being non-interacting. In the large deviation (LD)
regime of long times and large system size, this can give rise to a phase
transition in trajectory space. As a non-interacting system we consider a
collection of spins undergoing single spin-flip dynamics at
infinite-temperature. For the dynamical observables we study, the associated
tilted generators have an exact and explicit spin-plaquette duality. Such setup
suggests the existence of a transition (in the large size limit) at the
self-dual point of the tilted generator. The nature of the LD transition
depends on the observable. We consider explicitly two situations: (i) for a
pairwise bond observable the LD transition is continuous, and equivalent to
that of the transverse field Ising model; (ii) for a higher order plaquette
observable, in contrast, the LD transition is first order. Case (i) is easy to
prove analytically, while we confirm case (ii) numerically via an efficient
trajectory sampling scheme that exploits the non-interacting nature of the
original dynamics.
