Beyond Flory theory: Distribution functions for interacting lattice trees

While Flory theories [J. Isaacson and T. C. Lubensky, J. Physique Lett. 41, 469 (1980); M. Daoud and J. F. Joanny, J. Physique 42, 1359 (1981); A. M. Gutin et al., Macromolecules 26, 1293 (1993)] 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 analyze distribution functions for a wide variety of quantities characterizing the tree connectivities and conformations for the four different statistical ensembles, which we have studied numerically in [A. Rosa and R. Everaers, J. Phys. A: Math. Theor. 49, 345001 (2016) and J. Chem. Phys. 145, 164906 (2016)]: (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)((r) over right arrow) of spatial distances, (r) over right arrow, between monomers, and (iv) p(N)((r) over right arrow vertical bar 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 (x) over right arrow = (r) over right arrow/root < r(2)(N)> or x = l/< l(N)>. In particular, we observe a generalized Kramers relation for the branch weight distributions (i) and find that all the other distributions (ii-iv) are of Redner-des Cloizeaux type, q((x) over right arrow) = C vertical bar x vertical bar(theta) exp (-(K vertical bar x vertical bar)(t)). We propose a coherent framework, including generalized Fisher-Pincus relations, relating most of the RdC exponents to each other and to the contact and Flory exponents for interacting trees.