Regular dessins with primitive automorphism groups

We classify the dessins $\mathcal{D}$ for which the automorphism group $G$ acts primitively and faithfully on the points over one of the three critical values (without loss of generality the black vertices in the usual bipartite map representation). We show that they are all generalised Paley dessins, in which the black vertices are the elements of a finite field ${\mathbb{F}}_q$, and $G$ is a subgroup of the affine group ${\mathrm{A}\mathrm{G}\mathrm{L}}_1(q)$. Using earlier results obtained with Streit and Wolfart, we determine the orbits of the absolute Galois group on these dessins, we show that they are all defined over certain cyclotomic fields, and we obtain defining equations in some special cases. Relaxing the condition of a faithful action allows only cyclic regular coverings of these dessins.