Universality classes for self-avoiding walks in a strongly disordered system

We study the behavior of self-avoiding walks (SAWs) on square and cubic lattices in the presence of strong disorder. We simulate the disorder by assigning random energy $\epsilon$ taken from a probability distribution $P\left(\epsilon \right)$ to each site (or bond) of the lattice. We study the strong disorder limit for an extremely broad range of energies with $P\left(\epsilon \right)\propto 1/\epsilon .$ For each configuration of disorder, we find by exact enumeration the optimal SAW of fixed length N and fixed origin that minimizes the sum of the energies of the visited sites (or bonds). We find the fractal dimension of the optimal path to be $d_{\mathrm{opt}}=1.52\mathrm{}\pm 0.10$ in two dimensions (2D) and $d_{\mathrm{opt}}=1.82\mathrm{}\pm 0.08$ in 3D. Our results imply that SAWs in strong disorder with fixed N are much more compact than SAWs in disordered media with a uniform distribution of energies, optimal paths in strong disorder with fixed end-to-end distance $R,$ and SAWs on a percolation cluster. Our results are also consistent with the possibility that SAWs in strong disorder belong to the same universality class as the maximal SAW on a percolation cluster at criticality, for which we calculate the fractal dimension $d_{\max }=1.64\mathrm{}\pm 0.02$ for 2D and $d_{\max }=1.87\mathrm{}\pm 0.05$ for 3D, values very close to the fractal dimensions of the percolation backbone in 2D and 3D.