Universality classes for self-avoiding walks in a strongly disordered system Lidia A. Braunstein, 1,2 Sergey V. Buldyrev, 1 Shlomo Havlin, 1,3 and H. Eugene Stanley 1 1 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02115 2 Departamento de Fı ´sica, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, 7600 Mar del Plata, Argentina 3 Minerva Center and Department of Physics, Bar-Ilan University, 52900 Ramat-Gan, Israel Received 16 January 2002; published 21 May 2002 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  taken from a probability distribution P (  ) to each site or bond of the lattice. We study the strong disorder limit for an extremely broad range of energies with P (  )  1/ . 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 ˜ opt =1.520.10 in two dimensions 2D and d ˜ opt =1.82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0.02 for 2D and d max =1.870.05 for 3D, values very close to the fractal dimensions of the percolation backbone in 2D and 3D. DOI: 10.1103/PhysRevE.65.056128 PACS numbers: 64.60.Ak, 05.45.Df I. INTRODUCTION The problem of self-avoiding walks SAWs in different types of disorder is related to problems such as polymers in porous media and spin glasses. For SAWs in the absence of disorder, the average root mean square of the end-to-end dis- tance R scales with the length N as R N  . Hence SAWs are fractals with a fractal dimension d SAW =1/ . The values of d SAW in two dimensions 2D and 3D are well known see Table I. The effects of disorder on d SAW has been the subject of many studies 1–7. Recently, there has been much inter- est in the problem of finding the optimal path in a disordered energy landscape. The optimal path can be defined as fol- lows: consider a d-dimensional lattice, where each site or bond is assigned by a random energy  taken from a given distribution. The optimal path is the path for which the sum of the energies along the path is minimal. There are two kinds of the optimal path problems. In the first kind fixed-R problem, the starting and the ending sites of the path are fixed, but the length of the path N is not fixed. In the second kind fixed-N problem, the starting site origin and the length of the path N are fixed, but the ending point is not fixed. These problems are relevant in many fields such as spin glasses 1, protein folding 2, and the traveling sales- man problem 8. Cieplak et al. 4 and Porto et al. 5 studied numerically the behavior of the average path length N for the fixed-R minimum-energy SAW. If the distribution of energies  is uniform or Gaussian, N is proportional to R and hence d opt =1. The situation is different in the strong disorder limit. In this case, the total energy E is dominated by the maximum value of  along the path. This case can be realized if the probability density P (  )  1/ for an extremely broad range of energies. It was found 4,5 that N  R d opt , where d opt 1.22 in 2D and d opt 1.42 in 3D. These values are similar to the fractal dimensions of the typical path of a passive tracer in the problem of the ideal flow through the percola- tion cluster, a problem relevant for oil recovery 9. This fact is consistent with the possibility that the strong disorder limit is related to the percolation problem. Smailer et al. 3 studied the problem of minimum-energy fixed-N SAWs in which the energies are taken from uniform and Gaussian distributions. This kind of disorder is called weak disorder and is different from the strong disorder case studied here. They studied the asymptotic behavior of the mean square end-to-end distance R versus N and found that the fractal dimension in this case is smaller than in the case of the pure SAW see Table I. The fixed-N problem for strong disorder has not yet been studied. Numerical studies of fixed-N SAWs on percolation clus- TABLE I. Fractal exponents characterizing the end-to-end dis- tance of SAWs as a function of the length as well as the fractal dimension of the backbone, d B . 2D 3D d SAW 4/3 a 1.699 b d ˜ opt weak, fixed N ) 1.25 c 1.4 c d ˜ opt strong, fixed N ) 1.520.10 f 1.820.08 f d SAW percolation at p c ) 1.29 d 1.55 h d max 1.640.02 f 1.870.05 f d opt strong, fixed R ) 1.21 g 1.44 g d B backbone 1.6432 e 1.87 e a Ref. 20. b Ref. 21. c Ref. 3. d Ref. 7. e Ref. 23. f Ref. 24. g Ref. 6. h Ref. 22. PHYSICAL REVIEW E, VOLUME 65, 056128 1063-651X/2002/655/0561286/$20.00 ©2002 The American Physical Society 65 056128-1