  This thesis discusses the worldsheet geometries in the (p,q) minimal model
coupled to 2D euclidean Quantum Gravity with particular focus on non-compact
geometries. We first calculate the FZZT-FZZT cylinder amplitudes for all pairs
of Cardy matter states thereby generalizing the result obtained in
arXiv:hep-th/0406030 and discuss the decompositions of the cylinder amplitudes
in terms of solutions to the homogeneous Wheeler-DeWitt equation. We then show,
that the principal ZZ boundary conditions in Liouville theory can be viewed as
effective boundary conditions obtained by integrating out the matter degrees of
freedom on the worldsheet. We now consider the (2,2m-1) minimal model coupled
to 2D euclidean Quantum Gravity obtained in the scaling limit of dynamical
triangulations, in which the mth multi-critical hyper-surface is approached and
the conformal background is turned on. We show, that only one concrete
realization of matter boundary condition, the (1,1) Cardy boundary condition,
is obtained in this scaling limit. Finally, we study the cylinder amplitude
with fixed distance and provide some evidence of a transition from a FZZT-brane
to a ZZ-brane on the exit loop in the limit, where the distance approaches
infinity, and for particular values of the exit-boundary cosmological constant.

High Energy Physics - Theory