Beyond Flory theory: Distribution functions for interacting lattice trees

While Flory theories provide an extremely useful framework for understanding the behavior of interacting, randomly branching polymers, the approach is inherently limited. Here we use a combination of scaling arguments and computer simulations to go beyond a Gaussian description. We analyse distributions functions for a wide variety of quantities characterising the tree connectivities and conformations for the four different statistical ensembles, which we have studied numerically in [Rosa and Everaers, J. Phys. A (2016, published) and J. Chem. Phys. (2016, to appear)]: (a) ideal randomly branching polymers, (b) $2d$ and $3d$ melts of interacting randomly branching polymers, (c) $3d$ self-avoiding trees with annealed connectivity and (d) $3d$ self-avoiding trees with quenched ideal connectivity. In particular, we investigate the distributions (i) $p_N(n)$ of the weight, $n$, of branches cut from trees of mass $N$ by severing randomly chosen bonds; (ii) $p_N(l)$ of the contour distances, $l$, between monomers; (iii) $p_N(\vec r)$ of spatial distances, $\vec r$, between monomers, and (iv) $p_N(\vec r|l)$ of the end-to-end distance of paths of length $l$. Data for different tree sizes superimpose, when expressed as functions of suitably rescaled observables $\vec x = \vec r/\langle r^2(N) \rangle$ or $x =l/\langle l(N) \rangle$. In particular, we observe a generalised Kramers relation for the branch weight distributions (i) and find that all the other distributions (ii-iv) are of Redner-des Cloizeaux type, $q(\vec x) = C \, |x|^\theta\ \exp \left( -(K |x|)^t \right)$. We propose a coherent framework, including generalised Fisher-Pincus relations, relating most of the RdC exponents to each other and to the contact and Flory exponents for interacting trees.