Circular law for the sum of random permutation matrices

Let Pₙ¹,, Pₙᵈ be n× n permutation matrices drawn independently and uniformly at random, and set Sₙᵈ:=∑₌₁ᵈ Pₙ^. We show that if log¹²n/(log log n)⁴ ≤ d=O(n), then the empirical spectral distribution of Sₙᵈ/√d converges weakly to the circular law in probability as n → ∞.