  Motivated by recent progress on the scaling behavior of entanglement entropy,
we study the scaling behavior of the number of clusters crossing the boundary
between two subsystems for several classical statistical models in two
dimension. This number exhibits a subleading logarithmic dependence of the
linear dimension of the boundary when the model is at critical, in analogy to
the entanglement entropy of a quantum system. It is shown that the logarithmic
scaling of the cluster number originates from the conformal invariance of the
critical system. We check this numerically for Ising and Ashkin-Teller models
by using Monte Carlo simulations, and show that whether a universal coefficient
of the logarithmic term can be observed numerically may strongly depend on the
geometry and boundary condictions of the system.
