Randomly branched polymer chains (or trees) are a classical
subject of polymer physics with connections to the theory of magnetic
systems, percolation and critical phenomena. More recently, the model
has been reconsidered for RNA, supercoiled DNA and the crumpling of
topologically constrained polymers. While solvable in the ideal case,
little is known exactly about randomly branched polymers with volume
interactions. In the first part of the presentation, I will review the
Flory theory for interacting trees for good solvent, θ-solutions and
melts and compare its predictions to a wide range of available
analytical and numerical results [1]. Even though the predictions are
surprisingly good, the approach is inherently limited. In the second
part of the presentation, I use a combination of scaling arguments and
computer simulations to analyse distribution functions for a wide
variety of quantities characterizing the tree connectivities and
conformations. We observe [2] a generalized Kramers relation for the
branch weight distributions and find that distributions of contour and
spatial distances are of Redner-des Cloizeaux type, q(x) = C|x|^θ
exp[−(K|x|)^t]. 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.
[1] Flory theory of randomly branched polymers, Ralf Everaers,
Alexander Y. Grosberg, Michael Rubinstein, and Angelo Rosa, Soft Matter,
13, 1223 (2017).
[2] Beyond Flory theory: Distribution functions for interacting
lattice trees, A. Rosa and R. Everaers, Phys. Rev. E, 95, 012117 (2017).