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We theoretically study the scale-invariant relaxation dynamics in segregatingtwo-component Bose-Einstein condensates with large particle-number imbalance,and uncover that random walk of droplet for the minor component plays afundamental role in the relaxation process. Our numerical simulations based onthe binary Gross-Pitaevskii model reveal the emergence of the dynamical scalingduring the relaxation, which is a hallmark of scale-invariant dynamics, in acorrelation function for the minor condensate. Tracking exponentscharacterizing the dynamical scaling in time, we find out a crossoverphenomenon that features the change in power exponents of the correlationlength. To understand the fundamental mechanism inherent in the scale-invariantrelaxation dynamics, we construct a random walk model for droplets. Employingthe model, we analytically derive the $1/3$ and $1/2$ power laws and predictthe crossover of the scaling. These exponents are in reasonable agreement withthe values obtained in the numerical calculations. We also discuss a possibleexperimental setup for observing the scale-invariant dynamics.

Scale-invariant relaxation dynamics in two-component Bose-Einstein condensates with large particle-number imbalance