  The Bradley-Terry-Luce (BTL) model is a benchmark model for pairwise
comparisons between individuals. Despite recent progress on the first-order
asymptotics of several popular procedures, the understanding of uncertainty
quantification in the BTL model remains largely incomplete, especially when the
underlying comparison graph is sparse. In this paper, we fill this gap by
focusing on two estimators that have received much recent attention: the
maximum likelihood estimator (MLE) and the spectral estimator. Using a unified
proof strategy, we derive sharp and uniform non-asymptotic expansions for both
estimators in the sparsest possible regime (up to some poly-logarithmic
factors) of the underlying comparison graph. These expansions allow us to
obtain: (i) finite-dimensional central limit theorems for both estimators; (ii)
construction of confidence intervals for individual ranks; (iii) optimal
constant of $\ell_2$ estimation, which is achieved by the MLE but not by the
spectral estimator. Our proof is based on a self-consistent equation of the
second-order remainder vector and a novel leave-two-out analysis.
