We theoretically study the scale-invariant relaxation dynamics in segregating two-component Bose-Einstein condensates with large particle-number imbalance and uncover that the random walk of droplets for the minor component plays a fundamental role in the relaxation process. Our numerical simulations based on the binary Gross-Pitaevskii model reveal the emergence of the dynamical scaling during the relaxation, which is a hallmark of scale-invariant dynamics, in a correlation function for the minor condensate. Tracking exponents characterizing the dynamical scaling in time, we find a crossover phenomenon that features the change in power exponents of the correlation length. To understand the fundamental mechanism inherent in the scale-invariant relaxation dynamics, we construct a random-walk model for droplets. Employing the model, we analytically derive the 1/3 and 1/2 power laws and predict the crossover of the scaling. These exponents are in reasonable agreement with the values obtained in the numerical calculations. We also discuss a possible experimental setup for observing the scale-invariant dynamics.
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