The spectral dimension of the branched polymers phase of two-dimensional quantum gravity

TL;DR

This work analyzes the spectral dimension of the branched-polymer phase in two-dimensional quantum gravity coupled to conformal matter. By connecting canonical and grand canonical ensemble formalisms through return-probability generating functions and a careful analysis of pole structures and scaling exponents, it derives the leading diffusion behaviour of branched polymers. The main result is a universal spectral dimension $d_S=\frac{4}{3}$ for the pure BP phase, with a gap exponent $\Delta=\tfrac{3}{2}$ and $\gamma_{str}=\tfrac{1}{2}$, and it discusses subleading corrections and robustness of the result across BP variants. The findings align with numerical simulations at large central charge and clarify the role of BP geometry in the large-$c$ limit of 2D quantum gravity, while delineating conditions under which non-generic BP ensembles may yield different exponents.

Abstract

The metric of two-dimensional quantum gravity interacting with conformal matter is believed to collapse to a branched polymer metric when the central charge c>1. We show analytically that the spectral dimension of such a branched polymer phase is four thirds. This is in good agreement with numerical simulations for large c.

Figures (7)

• Figure 1: The elementary polymer ${{\mathrm B}}_0$.
• Figure 2: The combination of polymers ${{\mathrm B}}_1$ and ${{\mathrm B}}_2$ to form ${{\mathrm B}}_3$.
• Figure 3: The unique decomposition of polymer ${{\mathrm B}}$ to its constituents ${{\mathrm B}}'$ and ${{\mathrm B}}"$.
• Figure 4: Example of a walk contributing to the first return probability on polymer ${{\mathrm A}}={{\mathrm B}}\cup{{\mathrm C}}$.
• Figure 5: Percolation cluster on a Cayley tree.