Cumulative incidence rate of LAHs, dn/dz(LAH,  M₁₅₀₀ ≤ M_1500, 0), as a function of M_1500, 0 at different redshifts. The different colors (blue, green, and red) indicate different redshifts (z ≃ 3.65, 4.9, and 5.9 in panels a–c, respectively). The solid black vertical lines show the M₁₅₀₀ of the brightest and faintest sources in our sample. The green vertical line indicates the upper limit of M₁₅₀₀ for the UV-bright sample. (a) At z ≃ 3.65. The blue solid line and shading show the dn/dz(LAH,  M₁₅₀₀ ≤ M_1500, 0) and the uncertainty, respectively. The blue-dashed thin line, medium-thick line, and thick line represent the dn/dz for DLAs, DLAs+sub-DLAs, and DLAs+sub-DLAs+LLSs at the same redshift, respectively. (b) At z ≃ 4.9. (c) At z ≃ 5.9.