Upper panels: summary of the Lagrangian trajectory integration in the active-nqL and passive-nqL disks. Yellow-green color shading denotes the total time spent in each grid cell by the water particles, which are released from the outer boundary. The black dashed lines denote the root-mean-square of the positions of particles in the vertical direction. The stars denote r_pk of different runs (Table 1). The thick grey lines represents the location where ρ_p,ice/ρ_vap = 1 and 10⁻³, respectively, depicting the regions where particles are mostly in ice or vapor phase, respectively (see Appendix A for detailed explanation). The blue and red arrows represent the velocity of ice and vapor, respectively. The length of the arrows is proportional to vH_g /D (D, v are the diffusivity and velocity of gas or pebble), respectively, which represent the ratio of the diffusion and advection timescales, $t_{\text{dif}}/t_{\text{adv}}\sim \left(H_{\text{g}}^2/D\right)/\left(H_{\text{g}}/v\right)$. We limit the length of the arrows so that so that vH_g/D ≤ 3 for visualization purposes. Also, the depth of color of blue arrows represents t_dif,peb/t_adv,peb. Lower panels: crossing frequency of Lagrangian particles at various altitudes along the radial direction. Circles are plotted at the center of each vertical interval, with their surface area proportional to the total number of crossings (N_c). Color indicates the ratio of net crossings to the total particle number (N_net/N_p). Here, negative (blue) values denote net accretion, whereas positive (red) values denote net outflow. The domain is zoomed-in to focus on the snowline region. The checkerboard pattern in the net crossing fraction around the snowline seen in the active-nqL run is a manifestation of vigorous water cycling.