Polymer Network of fixed topology: renormalization, exact critical exponentγin two dimensions, andd=4−ε

Abstract

I consider a connected self-avoiding polymer network made of identical long chains, with fixed topology. Using renormalization theory and conformal invariance, I conjecture in 2D, and give in $d=4-\epsilon$, to order $O\left(\epsilon \right)$, the exact value of its critical exponent $\gamma$ as a function of the topological invariants. In 2D, the exact result fits with recent numerical data for three- and four-leg stars by Lipson et al.