  In this work, we study a noiseless broadcast link serving $K$ users whose
requests arise from a library of $N$ files. Every user is equipped with a cache
of size $M$ files each. It has been shown that by splitting all the files into
packets and placing individual packets in a random independent manner across
all the caches, it requires at most $N/M$ file transmissions for any set of
demands from the library. The achievable delivery scheme involves linearly
combining packets of different files following a greedy clique cover solution
to the underlying index coding problem. This remarkable multiplicative gain of
random placement and coded delivery has been established in the asymptotic
regime when the number of packets per file $F$ scales to infinity.
  In this work, we initiate the finite-length analysis of random caching
schemes when the number of packets $F$ is a function of the system parameters
$M,N,K$. Specifically, we show that existing random placement and clique cover
delivery schemes that achieve optimality in the asymptotic regime can have at
most a multiplicative gain of $2$ if the number of packets is sub-exponential.
Further, for any clique cover based coded delivery and a large class of random
caching schemes, that includes the existing ones, we show that the number of
packets required to get a multiplicative gain of $\frac{4}{3}g$ is at least
$O((N/M)^g)$. We exhibit a random placement and an efficient clique cover based
coded delivery scheme that approximately achieves this lower bound. We also
provide tight concentration results that show that the average (over the random
caching involved) number of transmissions concentrates very well requiring only
polynomial number of packets in the rest of the parameters.
