On the equivalence between the Wilson flow and stout-link smearing

We present the numerical equivalence between the Wilson flow and stout-link smearing, both of which are known to be a relatively new technique for smoothing the gauge fields on the lattice. Although the conceptional correspondence between two methods was first pointed out by L\"uscher in his original paper [J. High Energy Phys.~08 (2010) 071], we provide a direct analytical proof of the equivalence between the two methods at finite lattice spacing $a$ in the zero limit of the stout-smearing parameter $\rho$. The leading order corrections start at ${\cal O}(\rho)$, which would induce ${\cal O}(a^2)$ corrections. It is, therefore, not obvious that they remain equivalent even with finite parameters ($a\neq 0$ and $\rho\neq0$) within some numerical precision. In this paper, we demonstrate the equivalence of both methods by directly comparing the expectation value of the action density, which is measured in actual numerical simulations.