Authors: Sumedha; Dhar, Deepak

Source: Journal of Statistical Physics, Volume 125, Number 1, October 2006 , pp. 55-76(22)

Publisher: Springer

Abstract:

We study rooted self avoiding polygons and self avoiding walks on deterministic fractal lattices of finite ramification index. Different sites on such lattices are not equivalent, and the number of rooted open walks W _n (S), and rooted self-avoiding polygons P _n (S) of n steps depend on the root S. We use exact recursion equations on the fractal to determine the generating functions for P _n (S), and W _n(S) for an arbitrary point S on the lattice. These are used to compute the averages <EquationSource Format="TEX"><![CDATA[$$langle P_{n}(S) rangle$$]]></EquationSource> , <EquationSource Format="TEX"><![CDATA[$$langle W_{n}(S) rangle$$]]></EquationSource> , <EquationSource Format="TEX"><![CDATA[$$langle log P_{n}(S) rangle$$]]></EquationSource> and <EquationSource Format="TEX"><![CDATA[$$langle log W_{n}(S) rangle$$]]></EquationSource> over different positions of S. We find that the connectivity constant μ, and the radius of gyration exponent <EquationSource Format="TEX"><![CDATA[$$u$$]]></EquationSource> are the same for the annealed and quenched averages. However, <EquationSource Format="TEX"><![CDATA[$$langle log P_{n}(S) rangle simeq n log mu + (alpha_q - 2)log n$$]]></EquationSource> , and <EquationSource Format="TEX"><![CDATA[$$langle log W_{n}(S) rangle simeq n log mu + (gamma_q-1) log{n}$$]]></EquationSource> , where the exponents <EquationSource Format="TEX"><![CDATA[$$alpha_q$$]]></EquationSource> and <EquationSource Format="TEX"><![CDATA[$$gamma_q$$]]></EquationSource> , take values different from the annealed case. These are expressed as the Lyapunov exponents of random product of finite-dimensional matrices. For the 3-simplex lattice, our numerical estimation gives <EquationSource Format="TEX"><![CDATA[$$alpha_q simeq 0.72837 pm 0.00001;$$]]></EquationSource> and <EquationSource Format="TEX"><![CDATA[$$gamma_q simeq 1.37501 pm 0.00003$$]]></EquationSource> , to be compared with the known annealed values <EquationSource Format="TEX"><![CDATA[$$alpha_a = 0.73421$$]]></EquationSource> and <EquationSource Format="TEX"><![CDATA[$$gamma_q = 1.37522$$]]></EquationSource> .

Keywords: self-avoiding walks; random media; fractals

Document Type: Research article

DOI: http://dx.doi.org/10.1007/s10955-006-9098-7

Publication date: 2006-10-01

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