Lattice star and acyclic branched polymer vertex exponents in 3d

TL;DR

This paper numerically estimates lattice star and acyclic branched polymer exponents in 3D, revealing deviations from theoretical predictions and confirming scaling relations for complex networks.

Contribution

It provides improved numerical estimates of vertex and entropic exponents for lattice stars and branched networks, challenging existing theoretical predictions.

Findings

Vertex exponents deviate from $$-expansion predictions.

Confirmed scaling relation for acyclic branched networks.

Improved numerical estimates over previous literature.

Abstract

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 $\epsilon$-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 $\sigma_f$.