Fix a sequence of $d$-regular graphs $(G_d)_{ d\in \mathbb{ N}}$ and denote by $G_{ d,p}$ the graph obtained from $G_d$ after edge-percolation with probability $p=c/d$, for a constant $c>0$. We prove a quantitative local convergence of $(G_{ d,p})_{ d\in \mathbb{ N}}$. In combination with results of Bordenave, Lelarge and Salez, it implies that the rescaled matching number of $G_{ d,p}$ is asymptotically equivalent to that of the binomial random graph $G(n,c/n)$.
Categories: math.CO math.PR