Copolymer with pinning: variational characterization of the phase diagram

This paper studies a polymer chain in the vicinity of a linear interface separating two immiscible solvents. The polymer consists of random monomer types, while the interface carries random charges. Both the monomer types and the charges are given by i.i.d. sequences of random variables. The configurations of the polymer are directed paths that can make i.i.d. excursions of finite length above and below the interface. The Hamiltonian has two parts: a monomer-solvent interaction ("copolymer") and a monomer-interface interaction ("pinning"). The quenched and the annealed version of the model each undergo a transition from a localized phase (where the polymer stays close to the interface) to a delocalized phase (where the polymer wanders away from the interface). We exploit the approach developed in [5] and [3] to derive variational formulas for the quenched and the annealed free energy per monomer. These variational formulas are analyzed to obtain detailed information on the critical curves separating the two phases and on the typical behavior of the polymer in each of the two phases. Our main results settle a number of open questions.Comment: 46 pages, 9 figure

A brief review of results on the linking probability for 2-component links which span a lattice tube(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)

この論文は国立情報学研究所の電子図書館事業により電子化されました。After a brief introduction to lattice models of ring polymers and their application to studying random knotting and linking, a review is given of recent results due to Atapour, Soteros, Ernst and Whittington on the probability of topological and homological linking for 2-component links for which each component spans a lattice tube

Knots In Graphs In Subsets Of Z³

The probability that an embedding of a graph in Z³ is knotted is investigated. For any given graph (embeddable in Z³) without cut edges, it is shown that this probability approaches 1 at an exponential rate as the number of edges in the embedding goes to infinity. Furthermore, at least for a subset of these graphs, the rate at which the probability approaches 1 does not depend on the particular graph being embedded. Results analogous to these are proved to be true for embeddings of graphs in a subset of Z³ bounded by two parallel planes (a slab). In order to investigate the knotting probability of embeddings of graphs in a rectangular prism (an infinitely long rectangular tube in Z³), a pattern theorem for selfavoiding polygons in a prism is proved. From this it is possible to prove that for any given eulerian graph, the probability that an embedding of the graph in a prism is knotted goes to 1 as the number of edges in the embedding goes to infinity. Then, just as for ..

Lattice Animals: Rigorous Results and Wild Guesses

Introduction We consider the d-dimensional hypercubic lattice with vertices being the integer points in R . Two points are connected by an edge if they are unit distance apart. We write (x 1 , x 2 , . . . , x d ) for the coordinates of a vertex v and e = (v 1 , v 2 ) for the edge joining the vertices v 1 and v 2 whose coordinates must di#er by unity in exactly one coordinate. A bond animal is a connected subgraph of the lattice and a site animal is a connected section graph of the lattice. The distinction is that for each pair of vertices v 1 and v 2 in a site animal, which di#er by unity in exactly one coordinate, the edge e = (v 1 , v 2 ) must be in the site animal. That is, for site animals, edges are induced by the vertices. We shall be interested in the number of bond or site animals, with n vertices, where two animals are identical if one can be translated into the other. We write An for the number of site animals with n vertices and an for the number of bond animals with