A flow in the forest

I will tell about the interpolation between the critical behavior of two models of quantum gravity with matter with central charges $c=-2$ and $c=0$. We will start with the discretized Polyakov action for massive spinless fermions on the world surface. Using the Parisi-Sourlas trick, we can reduce the problem to the calculation of the partition function of a matrix model with a polynomial potential. This model interpolates between the regime describing random graphs and the regime of random trees, and in general describes the statistics of random forests on random graphs. The solution of the matrix model is formulated in terms of its spectral curve. We will focus on the almost critical scaling limit when both graphs and trees in forests are macroscopically large. In this limit, we obtain universal single-point scaling functions (condensates) parametrized via the Lambert function.