This paper studies the performance of a randomly Reconfigurable intelligent surface (RIS)-assisted multi-target localization system, in which the configurations of the RIS are randomly set to avoid high-complexity optimization. We first focus on the scenario where the number of RIS elements is significantly large, and then obtain the scaling law of Cramér-Rao bound (CRB) under certain conditions, which shows that CRB decreases in the third or fourth order as the RIS dimension increases. Second, we extend our analysis to large systems where both the number of targets and sensors is substantial. Under this setting, we explore two common RIS models: the constant module model and the discrete amplitude model, and illustrate how the random RIS configuration impacts the value of CRB. Numerical results demonstrate that asymptotic formulas provide a good approximation to the exact CRB in the proposed randomly configured RIS systems.