The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles

Transcription

1 The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles E. J. JANSE van RENSBURG Associate Professor of Mathematics York University, Toronto OXFORD UNIVERSITY PRESS

2 Contents Introduction Lattice models of polymers and vesicles Walks and polygons Growth constants Generating functions The metric exponent Scaling The correlation length Scaling relations Branched polymers 12 2 Tricriticality Interacting models of polygons Classical tricriticality Finite size scaling Homogeneity of the generating function Uniform asymptotics and the finite size scaling function Asymmetric tricriticality and e-asymptotics Extended tricriticality Uniform asymptotics for the generating function The finite size scaling function 36 Density functions and free energies The density function Density functions Integrated density functions Density functions and free energies Properties of the density function Jump discontinuities at the end-points of [e m,e M ] Finite right- and left-derivatives at e m and м A jump discontinuity in dv#{e)/de logv#(e)=ice+8m[e', "] 57

3 viii Contents 3.4 Examples Chromatic polynomial of path graphs Combinations Directed or staircase walks in the square lattice Partitions Queens on a chessboard Adsorbing walks 63 4 Exact models Introduction Partition polygons (Ferrers diagrams) The area-perimeter generating function of partition polygons Asymptotic analysis of the partition generating function Stack polygons The area-perimeter generating function of stack polygons Asymptotic analysis of the stack generating function Spiral walks Staircase polygons Perimeter generating function by counting paths Perimeter generating function by algebraic languages Area-perimeter generating function by a Temperley method Polya's method for staircase polygons Area-perimeter generating function from a functional equation Other models of convex polygons The adsorption of staircase walks on the main diagonal Staircase walks above the main diagonal Staircase walks adsorbing on the main diagonal Staircase walks adsorbing on to a penetrable diagonal Adsorbing staircase walks with an area activity The constant term formulation and adsorbing staircase walks A staircase walks model of copolymer adsorption Staircase polygons above the main diagonal Partially directed walks with a contact activity Directed animals and directed percolation Interacting models of walks and polygons Walks and polygons Unfolded walks Loops and polygons The pattern theorem Density functions and prime patterns The pattern theorem and interacting models of walks The free energy of//-walks Interacting models of polygons and walks The pattern theorem for interacting models 148

4 Contents ix 5.3 Polygons with curvature Curvature in polygons Curvature and knotted polygons Curvature and writhe Curvature and contacts Polygons interacting with a surface: adsorption Positive polygons Location of the adsorption transition Excursions and the adsorption transition A density of excursions Collapsing and adsorbing polygons Copolymer adsorption Torsion in polygons Dense walks and composite polygons Walks which cross a square Complex composite polygons The unfolding of polygons Simple composite polygons The free energy of simple composite polygons Density functions of simple composite polygons Interacting models of simple composite polygons Animals and trees Lattice animals and trees The growth constant of lattice animals A submultiplicative relation for t n Pattern theorems and interacting models of lattice animals Collapsing animals The cycle model The cycle-contact model Adsorbing trees The free energy of adsorbing and collapsing trees The phase diagram of adsorbing and collapsing trees The location of the adsorption transition Excursions and roots in the adsorption transition Excursions and roots in adsorbing trees Adsorption of percolation clusters Branched copolymer adsorption Embeddings of graphs with specified topologies A pattern theorem for embedded graphs in the cubic lattice Knotted embeddings of graphs Walks and polygons in wedges and uniform animals 278

5 x Contents Lattice vesicles and surfaces Introduction Square lattice vesicles The Fisher-Guttmann-Whittington vesicle The perimeter of a vesicle Punctured disks in two dimensions Submultiplicativity of disks The asymptotic behaviour of punctured disks Pattern theorems for punctured disks in two dimensions Adsorbing disks in three dimensions Unfolded disks Bounds on D n The free energy of positive disks The location of the adsorption transition Crumpling surfaces The density function Bounds on V s (z) A crumpling transition in surfaces with connected skeletons Inflating and crumpling c-surfaces 327 Appendix A: Subadditive functions 333 Appendix B: Convex functions 339 Appendix C: Asymptotics for ^-factorials 346 Appendix D: Bond or edge percolation 350 Bibliography 359 Index 377