Numerical estimates of square lattice star vertex exponents

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.