Numerical estimates of square lattice star vertex exponents

TL;DR

This paper uses advanced computational algorithms to estimate star vertex exponents and network entropic exponents for branched polymers on a square lattice, confirming theoretical predictions.

Contribution

It introduces parallel implementations of GARM and Wang-Landau algorithms to accurately estimate exponents for branched polymers in two dimensions.

Findings

Verified predicted exact values of vertex exponents.

Confirmed the scaling relation for branched networks.

Provided numerical estimates supporting theoretical models.

Abstract

We implement parallel versions of the GARM and Wang-Landau algorithms for stars and for acyclic uniform branched networks in the square lattice. These are models of monodispersed branched polymers, and we estimate the star vertex exponents $\sigma_f$ for $f$-stars, and the entropic exponent $\gamma_\mathcal{G}$ for networks with comb and brush connectivity in two dimensions. Our results verify the predicted (but not rigorously proven) exact values of the vertex exponents and we test the scaling relation [5] $$ \gamma_{\mathcal{G}}-1 = \sum_{f\geq 1} m_f \, \sigma_f $$ for the branched networks in two dimensions.