Appears in collection : Combinatorics and Arithmetic for Physics : Special Days
Using the matrix-forest theorem and the Parisi-Sourlas trick we formu-
late and solve a one-matrix model with non-polynomial potential which provides
perturbation theory for massive spinless fermions on dynamical planar graphs.
This is a version of 2d quantum gravity discretized via RRG coupled to massive
spinless fermions. Our model equivalently describes the ensemble of spanning
forests on the same graph. The solution is formulated in terms of an elliptic curve.
We then focus on a near-critical scaling limit when both the graphs and the trees in
the forests are macroscopically large. In this limit we obtain universal one-point
scaling functions (condensates), parameterized in terms of the Lambert function.
Our results provide a rare example where one can explore the flow between two
gravity models – in this case, the theories of conformal matter coupled to 2d grav-
ity with c=-2 (large trees regime) and c=0 (small trees regime). We shall also
present the results of numerical simulations concerning phase transitions in RRG
ensemble and their relation with Anderson localization.