Small exponent Schatten class Toeplitz operators on convex domains of finite type

In this paper, we study the characterizations of Schatten $p$-class Toeplitz operators on smoothly bounded convex domains of finite type in ${\text{C}}^n$. On one hand, by using discrete Kobayashi lattice, we characterize the Schatten $p$-class Toeplitz operators by employing a probabilistic selection method for $0<p<1$. At the same time, we show that the characterizations obtained by Kobayashi lattice are equivalent to the Berezin transform for $p>\frac{n}{n+1}$. On the other hand, as an application, the criterion gives a geometric characterization for Schatten $p$-class weighted composition operators for $0<p<2$ and also gives an analytic characterization for $p>\frac{2n}{n+1}$. Our main results give an answer to the question that find a geometric or analytic characterization of the Schatten $p$-class weighted composition operators whenever $0<p<2$ raised by Xiao-Yang-Yuan \cite{Xiao2026}.