  Around the snow line, icy pebbles and silicate dust may locally pile-up and
form icy and rocky planetesimals via streaming instability and/or gravitational
instability. We perform 1D diffusion-advection simulations that include the
back-reaction to radial drift and diffusion of icy pebbles and silicate dust,
ice sublimation, release of silicate dust, and their recycling through
recondensation and sticking onto pebbles outside the snow line. We use a
realistic description of the scale height of silicate dust obtained from Ida et
al. and that of pebbles including the effects of a Kelvin-Helmholtz
instability. We study the dependence of solid pile-up on distinct effective
viscous parameters for turbulent diffusions in the radial and vertical
directions ($\alpha_{\rm Dr}$ and $\alpha_{\rm Dz}$) and for the gas accretion
to the star ($\alpha_{\rm acc}$) as well as that on the pebble-to-gas mass flux
($F_{\rm p/g}$). We derive the sublimation width of drifting icy pebbles which
is a critical parameter to characterize the pile-up of silicate dust and
pebbles around the snow line. We identify a parameter space (in the $F_{\rm
p/g}-\alpha_{\rm acc}-\alpha_{\rm Dz}(=\alpha_{\rm Dr})$ space) where pebbles
no longer drift inward to reach the snow line due to the back-reaction that
slows down radial velocity of pebbles. We show that the pile-up of solids
around the snow line occurs in a broader range of parameters for $\alpha_{\rm
acc}=10^{-3}$ than for $\alpha_{\rm acc}=10^{-2}$. Above a critical $F_{\rm
p/g}$ value, the runaway pile-up of silicate dust inside the snow line is
favored for $\alpha_{\rm Dr}/\alpha_{\rm acc} \ll 1$, while that of pebbles
outside the snow line is favored for $\alpha_{\rm Dr}/\alpha_{\rm acc} \sim 1$.
Our results imply that a distinct evolutionary path could produce a diversity
of outcomes in terms of planetesimal formation around the snow line.
