  We theoretically study the scale-invariant relaxation dynamics in segregating
two-component Bose-Einstein condensates with large particle-number imbalance,
and uncover that random walk of droplet 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 out 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.
