Effect of higher-order interactions on noisy majority-rule dynamics with random group sizes

We study noisy majority-rule dynamics on annealed hypergraphs to clarify how variability in group interaction sizes reshapes collective ordering. At each update, a group is sampled from a prescribed size distribution and either follows the strict within-group majority or, with probability $q$, updates independently under an external bias $p$. At the symmetric point $p=1/2$, we obtain an explicit analytical expression for the critical independence threshold $q_c$, which separates macroscopic ordering from a fluctuating mixed state and can be interpreted as the largest fraction of independent behavior that can be sustained without destroying order. Because $q_c$ is governed by group-size statistics through an effective majority leverage, broad and heavy-tailed size distributions enhance robustness by enabling rare large-group events to realign a substantial fraction of the population. We further derive analytical predictions, benchmarked against Monte Carlo simulations, for the leading finite-size behavior of relaxation: for narrow distributions the characteristic relaxation time typically grows logarithmically with system size, whereas sufficiently heavy-tailed power laws produce strong crossovers and make the large-system dynamics sensitive to how $q$ approaches the transition. In the pure majority-rule limit, we find a crossover from conventional logarithmic consensus times to rapid ordering driven by occasional macroscopic groups, and the exit probability near coexistence collapses onto a universal error-function form controlled by a single structural parameter.