In this paper, we investigate the exact asymptotic behavior of the connectivity probability in the Erdos-Renyi graph G(n,p), under different asymptotic assumptions on the edge probability p=p(n). We propose a novel approach based on the analysis of inhomogeneous random walks to derive this probability. We show that the problem of graph connectivity can be reduced to determining the probability that an inhomogeneous random walk with Poisson-distributed increments, conditioned to form a bridge, is actually an excursion