  We study the role of topology in governing deep thermalization, the
relaxation of a local subsystem towards a maximally-entropic, uniform
distribution of post-measurement states, upon observing the complementary
subsystem in a local basis. Concretely, we focus on a class of (1+1)d systems
exhibiting `maximally-chaotic' dynamics, and consider how the rate of the
formation of such a universal wavefunction distribution depends on boundary
conditions of the system. We find that deep thermalization is achieved
exponentially quickly in the presence of either periodic or open boundary
conditions; however, the rate at which this occurs is twice as fast for the
former than for the latter. These results are attained analytically using the
calculus of integration over unitary groups, and supported by extensive
numerical simulations. Our findings highlight the nonlocal nature of deep
thermalization, and clearly illustrates that the physics underlying this
phenomenon goes beyond that of standard quantum thermalization, which only
depends on the net build-up of entanglement between a subsystem and its
complement.
