  We propose an efficient algorithm for matching two correlated
Erd\H{o}s--R\'enyi graphs with $n$ vertices whose edges are correlated through
a latent vertex correspondence. When the edge density $q= n^{- \alpha+o(1)}$
for a constant $\alpha \in [0,1)$, we show that our algorithm has polynomial
running time and succeeds to recover the latent matching as long as the edge
correlation is non-vanishing. This is closely related to our previous work on a
polynomial-time algorithm that matches two Gaussian Wigner matrices with
non-vanishing correlation, and provides the first polynomial-time random graph
matching algorithm (regardless of the regime of $q$) when the edge correlation
is below the square root of the Otter's constant (which is $\approx 0.338$).
