Elezović-Hadzić Suncica, Marcetić Dusanka, Maletić Slobodan
Faculty of Physics, University of Belgrade, P.O. Box 368, Belgrade, Serbia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Jul;76(1 Pt 1):011107. doi: 10.1103/PhysRevE.76.011107. Epub 2007 Jul 12.
We investigate asymptotical behavior of numbers of long Hamiltonian walks (HWs), i.e., self-avoiding random walks that visit every site of a lattice, on various fractal lattices. By applying an exact recursive technique we obtain scaling forms for open HWs on three-simplex lattice, Sierpinski gasket, and their generalizations: Given-Mandelbrot (GM), modified Sierpinski gasket (MSG), and n -simplex fractal families. For GM, MSG and n -simplex lattices with odd values of n , the number of open HWs Z(N), for the lattice with N>>1 sites, varies as omega(N)}N(gamma). We explicitly calculate the exponent gamma for several members of GM and MSG families, as well as for n-simplices with n=3, 5, and 7. For n-simplex fractals with even n we find different scaling form: Z(N) approximately omega(N)mu(N1/d(f), where d(f) is the fractal dimension of the lattice, which also differs from the formula expected for homogeneous lattices. We discuss possible implications of our results on studies of real compact polymers.
我们研究了长哈密顿游走（HWs）的数量的渐近行为，即访问晶格每个位点的自回避随机游走，在各种分形晶格上的情况。通过应用一种精确的递归技术，我们得到了三维单纯形晶格、谢尔宾斯基垫片及其推广形式：给定 - 曼德勃罗（GM）、修正谢尔宾斯基垫片（MSG）和n - 单纯形分形族上开放HWs的标度形式。对于GM、MSG和n为奇数的n - 单纯形晶格，对于具有N>>1个位点的晶格，开放HWs的数量Z(N)的变化规律为ω(N)N(γ)。我们明确计算了GM和MSG族的几个成员以及n = 3、5和7的n - 单纯形的指数γ。对于n为偶数的n - 单纯形分形，我们发现了不同的标度形式：Z(N)≈ω(N)μ(N1/d(f))，其中d(f)是晶格的分形维数，这也与均匀晶格预期的公式不同。我们讨论了我们的结果对实际致密聚合物研究的可能影响。
