A polynomial time iterative algorithm for matching Gaussian matrices with non-vanishing correlation

Motivated by the problem of matching vertices in two correlated Erd\H{o}s-R\ényi graphs, we study the problem of matching two correlated Gaussian Wigner matrices. We propose an iterative matching algorithm, which succeeds in polynomial time as long as the correlation between the two Gaussian matrices does not vanish. Our result is the first polynomial time algorithm that solves a graph matching type of problem when the correlation is an arbitrarily small constant.