Critical Behaviour in a Planar Dynamical Triangulation Model with a Boundary

We consider a canonical ensemble of dynamical triangulations of a 2-dimensional sphere with a hole where the number $N$ of triangles is fixed. The Gibbs factor is $\exp (-\mu \sum \deg v)$ where $\deg v$ is the degree of the vertex $v$ in the triangulation $T$. Rigorous proof is presented that the free energy has one singularity, and the behaviour of the length $m$ of the boundary undergoes 3 phases: subcritical $m=O(1)$, supercritical (elongated) with $m$ of order $N$ and critical with $m=O(\sqrt{N})$. In the critical point the distribution of $m$ strongly depends on whether the boundary is provided with the coordinate system or not. In the first case $m$ is of order $\sqrt{N}$, in the second case $m$ can have order $N^{\alpha}$ for any $0<\alpha <{1/2}$.