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
Contents Introduction 1 1.1 Lattice models of polymers and vesicles 1 1.2 Walks and polygons 4 1.2.1 Growth constants 4 1.2.2 Generating functions 6 1.2.3 The metric exponent 8 1.3 Scaling 9 1.3.1 The correlation length 9 1.3.2 Scaling relations 11 1.3.3 Branched polymers 12 2 Tricriticality 15 2.1 Interacting models of polygons 15 2.2 Classical tricriticality 18 2.3 Finite size scaling 23 2.4 Homogeneity of the generating function 26 2.5 Uniform asymptotics and the finite size scaling function 28 2.5.1 Asymmetric tricriticality and e-asymptotics 28 2.5.2 Extended tricriticality 30 2.5.3 Uniform asymptotics for the generating function 33 2.5.4 The finite size scaling function 36 Density functions and free energies 39 3.1 The density function 39 3.1.1 Density functions 39 3.1.2 Integrated density functions 44 3.2 Density functions and free energies 50 3.3 Properties of the density function 53 3.3.1 Jump discontinuities at the end-points of [e m,e M ] 54 3.3.2 Finite right- and left-derivatives at e m and м 54 3.3.3 A jump discontinuity in dv#{e)/de 56 3.3.4 logv#(e)=ice+8m[e', "] 57
viii Contents 3.4 Examples 57 3.4.1 Chromatic polynomial of path graphs 57 3.4.2 Combinations 57 3.4.3 Directed or staircase walks in the square lattice 59 3.4.4 Partitions 60 3.4.5 Queens on a chessboard 62 3.4.6 Adsorbing walks 63 4 Exact models 65 4.1 Introduction 65 4.2 Partition polygons (Ferrers diagrams) 67 4.2.1 The area-perimeter generating function of partition polygons 67 4.2.2 Asymptotic analysis of the partition generating function 70 4.3 Stack polygons 74 4.3.1 The area-perimeter generating function of stack polygons 74 4.3.2 Asymptotic analysis of the stack generating function 76 4.3.3 Spiral walks 78 4.4 Staircase polygons 80 4.4.1 Perimeter generating function by counting paths 81 4.4.2 Perimeter generating function by algebraic languages 82 4.4.3 Area-perimeter generating function by a Temperley method 86 4.4.4 Polya's method for staircase polygons 89 4.4.5 Area-perimeter generating function from a functional equation 93 4.4.6 Other models of convex polygons 96 4.5 The adsorption of staircase walks on the main diagonal 98 4.5.1 Staircase walks above the main diagonal 98 4.5.2 Staircase walks adsorbing on the main diagonal 101 4.5.3 Staircase walks adsorbing on to a penetrable diagonal 103 4.5.4 Adsorbing staircase walks with an area activity 104 4.5.5 The constant term formulation and adsorbing staircase walks 105 4.5.6 A staircase walks model of copolymer adsorption 108 4.5.7 Staircase polygons above the main diagonal 111 4.6 Partially directed walks with a contact activity 117 4.7 Directed animals and directed percolation 122 5 Interacting models of walks and polygons 126 5.1 Walks and polygons 126 5.1.1 Unfolded walks 127 5.1.2 Loops and polygons 133 5.1.3 The pattern theorem 135 5.1.4 Density functions and prime patterns 140 5.2 The pattern theorem and interacting models of walks 143 5.2.1 The free energy of//-walks 144 5.2.2 Interacting models of polygons and walks 147 5.2.3 The pattern theorem for interacting models 148
Contents ix 5.3 Polygons with curvature 157 5.3.1 Curvature in polygons 157 5.3.2 Curvature and knotted polygons 160 5.3.3 Curvature and writhe 161 5.3.4 Curvature and contacts 165 5.4 Polygons interacting with a surface: adsorption 167 5.4.1 Positive polygons 168 5.4.2 Location of the adsorption transition 174 5.4.3 Excursions and the adsorption transition 177 5.4.4 A density of excursions 181 5.4.5 Collapsing and adsorbing polygons 183 5.4.6 Copolymer adsorption 189 5.5 Torsion in polygons 194 5.6 Dense walks and composite polygons 200 5.6.1 Walks which cross a square 201 5.6.2 Complex composite polygons 207 5.6.3 The unfolding of polygons 209 5.6.4 Simple composite polygons 215 5.6.5 The free energy of simple composite polygons 218 5.6.6 Density functions of simple composite polygons 221 5.6.7 Interacting models of simple composite polygons 224 6 Animals and trees 227 6.1 Lattice animals and trees 227 6.1.1 The growth constant of lattice animals 227 6.1.2 A submultiplicative relation for t n 230 6.2 Pattern theorems and interacting models of lattice animals 232 6.3 Collapsing animals 237 6.3.1 The cycle model 238 6.3.2 The cycle-contact model 242 6.4 Adsorbing trees 250 6.4.1 The free energy of adsorbing and collapsing trees 251 6.4.2 The phase diagram of adsorbing and collapsing trees 255 6.4.3 The location of the adsorption transition 257 6.4.4 Excursions and roots in the adsorption transition 260 6.4.5 Excursions and roots in adsorbing trees 262 6.4.6 Adsorption of percolation clusters 264 6.4.7 Branched copolymer adsorption 267 6.5 Embeddings of graphs with specified topologies 271 6.5.1 A pattern theorem for embedded graphs in the cubic lattice 272 6.5.2 Knotted embeddings of graphs 276 6.5.3 Walks and polygons in wedges and uniform animals 278
x Contents Lattice vesicles and surfaces 285 7.1 Introduction 285 7.2 Square lattice vesicles 285 7.2.1 The Fisher-Guttmann-Whittington vesicle 286 7.2.2 The perimeter of a vesicle 291 7.3 Punctured disks in two dimensions 293 7.3.1 Submultiplicativity of disks 294 7.3.2 The asymptotic behaviour of punctured disks 298 7.3.3 Pattern theorems for punctured disks in two dimensions 301 7.4 Adsorbing disks in three dimensions 303 7.4.1 Unfolded disks 303 7.4.2 Bounds on D n 306 7.4.3 The free energy of positive disks 308 7.4.4 The location of the adsorption transition 311 7.5 Crumpling surfaces 313 7.5.1 The density function 315 7.5.2 Bounds on V s (z) 319 7.5.3 A crumpling transition in surfaces with connected skeletons 326 7.5.4 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