Numerical estimates of square lattice star vertex exponents

Abstract

Numerical values of lattice star vertex exponents are estimated using parallel implementations of the GARM and Wang-Landau algorithms in the square and cubic lattices. In the square lattice the results are consistent with exact values of the exponents, but in the cubic lattice there are deviations from the predictions of the -expansion. In addition, the entropic exponents of acyclic branched lattice networks are calculated, and found to be consistent with the predicted values in terms of star vertex exponents.

Entropic exponents of grafted lattices stars

“…This approach was developed in reference [23] where it was used effectively for estimating γ 1 and γ 11 using PERM simulations in the cubic half-lattice. See also references [15][16][17].…”

“…Thus, we select as our error bar on γ 1 this value of , giving our best estimate γ 1 = 0.95325 ± 0.00020. (16) The estimate for γ 11 was similarly obtained using the data of grafted self-avoiding walks with their endpoints in the hard wall forming a surface loop (these are 11-stars in our notation). The results of the analysis are shown in the bottom panel of figure 3.…”

“…+ , m f is the number of nodes of degree f in the bulk lattice, c(G) is the number of independent circuits in the network, and (G) is the number of independent surface loops [21,22]. Testing of equation ( 5) are limited to self-avoiding walks in a half-lattice [23], and for branched networks, to star polymers in bulk [15][16][17] and a few cases of branched acyclic networks in bulk [16,17].…”

“…where γ f is the f -star entropic exponent. High quality numerical results and estimates of the entropic exponents of star polymers were made in references [12][13][14][15][16][17].…”

Lattice star and acyclic branched polymer vertex exponents in 3d

“…Lattice self-avoiding walk models of star polymers have been studied since the 1970s [25] and remain of considerable interest in the statistical mechanics of polymeric systems. Numerical simulation of lattice stars stretches back decades [1,19,23,41], and their properties and scaling exponents have been calculated numerically [1,5,13,17,18,23,28,29,41,43] and by using field theoretic approaches [9,10,22,26,27,33]. A general survey can be found in references [11,21].…”