Wreath Macdonald polynomials as eigenstates

We show that the wreath Macdonald polynomials for $\mathbb{Z}\mathrm{/}\mathrm{\ell }\mathbb{Z}\wr {\mathrm{\Sigma }}_n$, when naturally viewed as elements in the vertex representation of the quantum toroidal algebra $U_{\mathfrak{q},\mathfrak{d}}({\ddot{\mathfrak{s}\mathfrak{l}}}_{\mathrm{\ell }})$, diagonalize its horizontal Heisenberg subalgebra. Our proof makes heavy use of shuffle algebra methods, and we also obtain a new proof of existence of wreath Macdonald polynomials.