The Randomized Competitive Ratio of Weighted $k$-server is at least Exponential

The weighted $k$-server problem is a natural generalization of the $k$-server problem in which the cost incurred in moving a server is the distance traveled times the weight of the server. Even after almost three decades since the seminal work of Fiat and Ricklin (1994), the competitive ratio of this problem remains poorly understood, even on the simplest class of metric spaces -- the uniform metric spaces. In particular, in the case of randomized algorithms against the oblivious adversary, neither a better upper bound that the doubly exponential deterministic upper bound, nor a better lower bound than the logarithmic lower bound of unweighted $k$-server, is known. In this article, we make significant progress towards understanding the randomized competitive ratio of weighted $k$-server on uniform metrics. We cut down the triply exponential gap between the upper and lower bound to a singly exponential gap by proving that the competitive ratio is at least exponential in $k$, substantially improving on the previously known lower bound of about $\ln k$.