Splitting of Heterogeneous Boundaries in a System of the Tricritical Ising Model Coupled to 2-Dim Gravity

We study disk amplitudes whose boundaries have heterogeneous matter states in a system of $(4,5)$ conformal matter coupled to 2-dim gravity. They are analysed by using the 3-matrix chain model in the large $N$ limit. Each of the boundaries is composed of two or three parts with distinct matter states. From the obtained amplitudes, it turns out that each heterogeneous boundary loop splits into several loops and we can observe properties in the splitting phenomena that are common to each of them. We also discuss the relation to boundary operators.