A relativistic electron beam propagating through dense gas may evolve into a variety of current density profiles Jb(r) depending on the beam temperature T and plasma current profile Jp(r). Four broad classes of equilibria are discernible in particle simulations: (1) Bennett-like or compact, (2) core and halo, (3) on-axis hollowed, and (4) off-axis hollowed. The qualitative features of each class are reproduced in a simple analytical model that assumes an isothermal beam and a Bennett plasma current profile and iterates about an initial ansatz of a Bennett beam current profile. Bennett-like equilibria are produced when the return current fraction f is small or when T/TB=1−f, where TB=eIb/2c is the Bennett temperature. For T/TB >1−f and moderate or high f, the equilibrium consists of a compact core surrounded by a halo, which may contain most of the beam current. In extreme cases (T/TB≫1−f and f close to unity) an off-axis density minimum occurs. (This is also referred to as off-axis hollowing.) For T/TB<1−f and moderate or high f, the beam core is broader than the plasma current profile, and on-axis hollowing results. Simulations with the ultrarelativistic sarlac simulation code [Bull. Am. Phys. Soc. 31, 1429 (1986); Phys. Fluids 29, 3056 (1986)] exhibit all of these types of equilibria. sarlac also treats transverse beam distortion arising from the resistive hose instability. Hose instability growth appears to be strongly enhanced when halos are generated.