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We study disk amplitudes whose boundaries have heterogeneous matter states ina system of $(4,5)$ conformal matter coupled to 2-dim gravity. They areanalysed by using the 3-matrix chain model in the large $N$ limit. Each of theboundaries is composed of two or three parts with distinct matter states. Fromthe obtained amplitudes, it turns out that each heterogeneous boundary loopsplits into several loops and we can observe properties in the splittingphenomena that are common to each of them. We also discuss the relation toboundary operators.Comment: 10 pages, Latex, 3 figure

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