Exact-enumeration approach to multifractal structure for diffusion-limited aggregation

Using an exact-enumeration approach, we study the multifractal spectrum of diffusion-limited aggregation. The enormous number of possible configurations is reduced by many orders of magnitude using symmetry considerations. The most interesting result is that we find evidence which strongly suggests the existence of a phase transition in the multifractal spectrum: specifically, the ‘‘free energy,’’ ‘‘energy,’’ and ‘‘specific heat’’ develop singularities near a critical ‘‘temperature’’ ${\beta }_c$. Moreover, the energy shows large fluctuations near ${\beta }_c$. We also find that for β<${\beta }_c$, the free energy is dominated by the maximum energy $E_{\max (\mathrm{L}}$), which increases with system size L: $E_{\max \left(\mathrm{L}\right)\mathrm{\sim }{L}^2}$/lnL. The implications of this phase transition are that the free energy is not defined for β<${\beta }_c$, and that the large energy part of the ‘‘entropy’’ function is a straight line of slope ${\beta }_c$. We provide a phenomenological explanation for the origin of this phase transition.