Factorization of the N -point amplitudes in two-dimensional c = 1 quantum gravity is understood in terms of short-distance singularities arising from the operator product expansion of vertex operators after the Liouville zero mode integration. Apart from the tachyon states, there are infinitely many topological states contributing to the intermediate states. It is very important to understand t...

In this work, we prove the existence of a spectral function for singular q-Sturm-Liouville operator. Further, we establish a Parseval equality and expansion formula in eigenfunctions by terms of the spectral function.

We construct and analyze a continuum dynamical percolation process which evolves in random environment given by $$\gamma $$ -Liouville measure. The homogeneous counterpart of this describes the scaling limit discrete on rescaled triangular lattice. Our focus here is to study same limiting dynamics, but where speed microscopic updates highly inhomogeneous space driven measure associated with two...