  Previous studies of oscillator populations with two-simplex interaction
report novel phenomena such as discontinuous desynchronization transitions and
multistability of synchronized states. However, the noise effect has not been
well understood. Here, we find that when oscillators with two-simplex
interaction alone are subjected to external noise, synchrony is eroded and
eventually completely disappears even when the noise is infinitesimally weak.
Nonetheless, synchronized states may persist for extended periods, with the
lifetime increasing approximately exponentially with the strength of the
two-simplex interaction. Assuming weak noise and using Kramers' rate theory, we
derive a closed dynamical equation for the Kuramoto order parameter, by which
the exponential dependence is derived. Further, when sufficiently strong
one-simplex coupling is additionally introduced, noise erosion is prevented and
synchronized states become persistent. The bifurcation analysis of the
desynchronized state reveals that as one-simplex coupling increases, the
synchronized state appears supercritically or subscritically depending on the
strength of two-simplex coupling. Our study uncovers the processes of
synchronization and desynchronization of oscillator assemblies in higher-order
networks and is expected to provide insight into the design and control
principles in such systems.
