Left panel: Unconditional first-crossing distribution for walks starting from S = 0, δ = 0. The histogram shows the results of random walks; the solid black and dashed orange curves show the ansatz (3) with q = 0.80 and q = 0.707, respectively; and the dotted grey curve shows the spherical collapse result. Other panels: Conditional first-crossing distribution at z > z₀ for walks starting from S₀ > 0 and δ_ell(z₀, S₀) > 0 as indicated in the plot. The histogram again shows the results of random walks, the solid black curves show the ansatz (4), the dashed orange curves show the ansatz (3) with δ_sp(z)→δ_sp(z)−δ_sp(z₀) and S → S − S₀, and the dotted grey curve shows the spherical collapse result.