A Flow in the Forest

TL;DR

This paper develops an exactly solvable matrix model for 2D quantum gravity coupled to massive fermions, revealing universal scaling functions and exploring the flow between different gravity theories.

Contribution

It introduces a novel matrix model with a non-polynomial potential for fermions on dynamical graphs, providing new insights into 2D quantum gravity and matter coupling.

Findings

Universal one-point scaling functions parameterized by Lambert function

Explicit solutions in terms of elliptic curves

Demonstration of flow between c=-2 and c=0 gravity models

Abstract

Using the matrix-forest theorem and the Parisi-Sourlas trick we formulate 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 lattice version of 2d quantum gravity coupled to massive spinless fermions. Our model equivalently describes the ensemble of spanning forests on the same graphs. 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 gravity with c=-2 (large trees regime) and c=0 (small trees regime). We also compute the disc partition functions with Dirichlet and Neumann boundary conditions in the same critical limit.