Random endomorphisms of spherical reflection groups

The goal of this paper is to understand the set $\mathrm{E}\mathrm{n}\mathrm{d}(W)$ of endomorphisms of an irreducible spherical reflection group $W$. We do this in two ways: numerically, by deriving an explicit formula for $\mathrm{\mid }\mathrm{E}\mathrm{n}\mathrm{d}(W)\mathrm{\mid }$; and probabilistically, by exploring the question $\mathit{\text{what}}\mathit{\text{does}}\mathit{\text{a}}\mathit{\text{random}}\mathit{\text{endomorphism}}\mathit{\text{of}}W\mathit{\text{look}}\mathit{\text{like?}}$ For example, we show that as $n\to \mathrm{\infty }$ the probability that a random endomorphism of $W_n$ is an automorphism tends to $\frac{1}{2}$ if $W_n=C_{2n}$ or $D_n$, to $\frac{1}{4}$ if $W_n=C_{2n+1}$, and to $1$ if $W_n=A_n\mathrm{.}$