On eigenvalues of random partial wreath product

Partial wreath 𝑛-th power of symmetric inverse semigroup ℐ𝑑 is a semigroup defined recursively by 𝒫𝑛 = (𝒫𝑛-1) ≀𝑝 ℐ𝑑 = {(𝑓,𝑎)|𝑎 ∈ ℐ𝑑,𝑓 : dom(𝑎) → 𝒫𝑛-1},𝑛 ≥ 2, with composition (𝑓,𝑎) · (𝑔, 𝑏)=(𝑓𝑔𝑎, 𝑎𝑏), and 𝒫1 = ℐ𝑑. To a randomly chosen element 𝑥 ∈ 𝒫𝑛, we assign the matrix 𝐴𝑥 = ( 1{𝑥(𝑣𝑛 𝑖 )=𝑣𝑛 𝑗 } )𝑑𝑛 𝑖,𝑗=1 . In other words, (𝑖, 𝑗)-th entry of 𝐴𝑥 is equal to 1 if transformation 𝑥 maps 𝑖 to 𝑗, and 0 otherwise. Let 𝜒𝑥(𝜆) be the characteristic …