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The work establishes the exact performance limits of stochastic coded caching when users share a bounded number of cache states, and when the association between users and caches, is random. Under the premise that more balanced user-to-cache associations perform better than unbalanced ones, our work provides a statistical analysis of the average performance of such networks, identifying in closed form, the exact optimal average delivery time. To insightfully capture this delay, we derive easy to compute closed-form analytical bounds that prove tight in the limit of a large number $Λ$ of cache states. In the scenario where delivery involves $K$ users, we conclude that the multiplicative performance deterioration due to randomness -- as compared to the well-known deterministic uniform case -- can be unbounded and can scale as $Θ\left( \frac{\log Λ}{\log \log Λ} \right)$ at $K=Θ\left(Λ\right)$, and that this scaling vanishes when $K=Ω\left(Λ\log Λ\right)$. To alleviate this adverse effect of cache-load imbalance, we consider various load balancing methods, and show that employing proximity-bounded load balancing with an ability to choose from $h$ neighboring caches, the aforementioned scaling reduces to $Θ\left(\frac{\log(Λ/ h)}{ \log \log(Λ/ h)} \right)$, while when the proximity constraint is removed, the scaling is of a much slower order $Θ\left( \log \log Λ\right)$. The above analysis is extensively validated numerically.