Phase Transition in the Multifractal Spectrum of Diffusion-Limited Aggregation

Based on a novel "exact enumeration" approach, we find evidence suggesting the existence of a phase transition in the multifractal spectrum of diffusion-limited aggregation. Above a critical point ${\beta }_c$, the moment expansion shows an infinite hierarchy of phases, while below ${\beta }_c$ we find a single phase. At ${\beta }_c$ we find fluctuations of all energy scales and singular behavior of the energy and specific heat. We also find that the maximum energy scales with system size $L$ as $E_{max}\mathrm{}\left(L\right)\mathrm{}\mathbf{\propto }\frac{L^2}{\mathrm{In}L}$. Consequently, for $\beta <{\beta }_c$ the partition function does not scale with $L$, which implies that the conventional moment expansion must break down.