Beyond Flory theory: Distribution functions for interacting lattice trees

While Flory theories [J. Isaacson and T. C. Lubensky, J. Physique Lett. 41, 469 (1980)JPSLBO0302-072X10.1051/jphyslet:019800041019046900; M. Daoud and J. F. Joanny, J. Physique 42, 1359 (1981)JOPQAG0302-073810.1051/jphys:0198100420100135900; A. M. Gutin et al., Macromolecules 26, 1293 (1993)MAMOBX0024-929710.1021/ma00058a016] 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)1751-811310.1088/1751-8113/49/34/345001 and J. Chem. Phys. 145, 164906 (2016)JCPSA60021-960610.1063/1.4965827]: (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 ⃗]) of spatial distances, r[over ⃗], between monomers, and (iv) p_{N}(r[over ⃗]|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 ⃗]=r[over ⃗]/sqrt[〈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 ⃗])=C|x|^{θ}exp(-(K|x|)^{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.