We develop and implement a parallel flatPERM algorithm (Grassberger 1997 Phys. Rev. E 56 3682-3693, Prellberg and Krawczyk 2004 Phys. Rev. Lett. 92 120602) with mutually interacting parallel flatPERM sequences and use it to sample self-avoiding walks in two and three dimensions. Our data show that the parallel implementation accelerates the convergence of the flatPERM algorithm. Moreover, increasing the number of interacting flatPERM sequences (rather than running longer simulations) improves the rate of convergence. This suggests that a more efficient implementation of flatPERM will be a massively parallel implementation, rather than long simulations of one, or a few parallel sequences. We also use the algorithm to estimate the growth constant of the self-avoiding walk in two and in three dimensions using simulations over 12 parallel sequences. Our best results are μ d = 2.638 1585(1), if d = 2; 4.684 039(1), if d = 3.

Numerical values of lattice star entropic exponents $\gamma_f$, and star vertex exponents $\sigma_f$, are estimated using parallel implementations of the PERM and Wang-Landau algorithms. Our results show that the numerical estimates of the vertex exponents deviate from predictions of the $\eps$-expansion and confirms and improves on estimates in the literature. We also estimate the entropic exponents $\gamma_\mathcal{G}$ of a few acyclic branched lattice networks with comb and brush connectivities. In particular, we confirm within numerical accuracy the scaling relation $$ \gamma_{\mathcal{G}}-1 = \sum_{f\geq 1} m_f \, \sigma_f $$ for a comb and two brushes (where $m_f$ is the number of nodes of degree $f$ in the network) using our independent estimates of $\sigma_f$.

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.

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