High‐current, fast‐rise‐time electron beams in air often are subject to powerful hollowing instabilities. Previous work indicates that the instability is triggered by avalanche ionization of the air, which occurs for a ratio of inductive electric field strength to air density exceeding 130 kV/cm‐atm. Here, computer simulations are used to recast this threshold criterion in terms of the beam current rise rate, which is more readily accessible experimentally. The threshold dependence on beam radius and peak current is much weaker. The current rise rate tends to increase during propagation caused by erosion of the beam front, and this behavior is explored with additional simulations. Evidence for hollowing stabilization by electron energy spread also is presented. Numerical algorithms employed in these calculations are included for completeness.