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E-raamat: Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles

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(Professor of Mathematics, York University, Toronto)
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Statistical Mechanics of Interacting Walks, Polygons, Animals and VesiclesSuurem pilt
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The self-avoiding walk is a classical model in statistical mechanics, probability theory and mathematical physics. It is also a simple model of polymer entropy which is useful in modelling phase behaviour in polymers.

This monograph provides an authoritative examination of interacting self-avoiding walks, presenting aspects of the thermodynamic limit, phase behaviour, scaling and critical exponents for lattice polygons, lattice animals and surfaces. It also includes a comprehensive account of constructive methods in models of adsorbing, collapsing, and pulled walks, animals and networks, and for models of walks in confined geometries. Additional topics include scaling, knotting in lattice polygons, generating function methods for directed models of walks and polygons, and an introduction to the Edwards model.

This essential second edition includes recent breakthroughs in the field, as well as maintaining the older but still relevant topics. New chapters include an expanded presentation of directed models, an exploration of methods and results for the hexagonal lattice, and a chapter devoted to the Monte Carlo methods.
1 Lattice models of linear and ring polymers 1(37)
1.1 The self-avoiding walk
2(3)
1.2 Lattice polygons
5(4)
1.3 Self-avoiding walks with fixed endpoints
9(1)
1.4 Scaling
10(9)
1.5 Walk and polygon generating functions
19(2)
1.6 Tadpoles, figure eights, dumbbells and thetas
21(4)
1.7 Knotted lattice polygons
25(13)
2 Lattice models of branched polymers 38(38)
2.1 Lattice animals and lattice trees
39(13)
2.2 Stars, combs, brushes and uniform networks
52(5)
2.3 Conformal invariance
57(6)
2.4 The Edwards model
63(13)
3 Interacting lattice clusters 76(35)
3.1 The free energy of lattice clusters
76(3)
3.2 Free energies and generating functions
79(5)
3.3 The microcanonical density function
84(14)
3.4 Integrated density functions
98(5)
3.5 Combinatorial examples
103(8)
4 Scaling, criticality and tricriticality 111(24)
4.1 Tricritical scaling
112(7)
4.2 Finite size scaling
119(4)
4.3 Homogeneity of the generating function
123(2)
4.4 Uniform asymptotics and the finite size scaling function
125(10)
5 Directed lattice paths 135(83)
5.1 Dyck paths
135(17)
5.2 Directed paths above the line y = rx
152(2)
5.3 Dyck path models of adsorbing copolymers
154(6)
5.4 Motzkin paths
160(6)
5.5 Partially directed paths
166(10)
5.6 Staircase polygons
176(12)
5.7 Dyck paths in a layered environment
188(13)
5.8 Paths in wedges and the kernel method
201(14)
5.9 Spiral walks
215(3)
6 Convex lattice vesicles and directed animals 218(36)
6.1 Partitions
218(5)
6.2 Stacks
223(3)
6.3 Staircase vesicles
226(6)
6.4 Convex polygons
232(1)
6.5 Dyck path vesicles
233(2)
6.6 Bargraph and column convex vesicles
235(3)
6.7 Heaps of dimers, and directed animals
238(5)
6.8 Directed percolation
243(11)
7 Self-avoiding walks and polygons 254(43)
7.1 Walks, bridges, polygons and pattern theorems
254(20)
7.2 Patterns in interacting models of walks and polygons
274(5)
7.3 Patterns, curvature and knotting in stiff lattice polygons
279(7)
7.4 Writhe in stiff polygons
286(3)
7.5 Torsion in polygons
289(8)
8 Self-avoiding walks in slabs and wedges 297(29)
8.1 Self-avoiding walks in slabs
298(8)
8.2 Generating functions of walks in slabs
306(6)
8.3 A pattern theorem for walks in Sw
312(3)
8.4 Growth constants and free energies of walks in slabs
315(2)
8.5 Polygons and walks in wedges
317(9)
9 Interaction models of self-avoiding walks 326(69)
9.1 Adsorbing self-avoiding walks and polygons
326(26)
9.2 Adsorbing polygons
352(8)
9.3 Copolymer adsorption
360(6)
9.4 Collapsing self-avoiding walks
366(5)
9.5 Collapsing and adsorbing polygons
371(5)
9.6 Walks crossing a square as a model of the 0-transition
376(6)
9.7 Pulled self-avoiding walks
382(13)
10 Adsorbing walks in the hexagonal lattice 395(20)
10.1 Walks and half-space walks in the hexagonal lattice
395(10)
10.2 Adsorption of walks in a slit in the hexagonal lattice
405(10)
11 Interacting models of animals, trees and networks 415(46)
11.1 The pattern theorem for interacting lattice animals
416(6)
11.2 Self-interacting or collapsing lattice animals
422(14)
11.3 Adsorbing lattice trees
436(10)
11.4 Adsorbing percolation clusters
446(3)
11.5 Embeddings of abstract graphs
449(5)
11.6 Uniform networks
454(7)
12 Interacting models of vesicles and surfaces 461(17)
12.1 Square lattice vesicles
461(7)
12.2 Crumpling self-avoiding surfaces
468(10)
13 Monte Carlo methods for the self-avoiding walk 478(50)
13.1 Dynamic Markov chain Monte Carlo algorithms
479(9)
13.2 The Beretti-Sokal algorithm
488(2)
13.3 The BFACF algorithm
490(3)
13.4 The pivot algorithm
493(7)
13.5 The Rosenbluth method and the PERM algorithm
500(9)
13.6 The GARM algorithm
509(9)
13.7 The GAS algorithm
518(10)
A Subadditivity 528(8)
A.1 The basic subadditive theorem
528(1)
A.2 The Wilker and Whittington generalisation of Fekete's lemma
528(2)
A.3 The generalisation by JM Hammersley
530(4)
A.4 A ratio limit theorem by H Kesten
534(2)
B Convex functions 536(11)
B.1 Convex functions and the midpoint condition
536(2)
B.2 Derivatives of convex functions
538(5)
B.3 Convergence
543(2)
B.4 The Legendre transform
545(2)
C Kesten's pattern theorem 547(11)
C.1 Patterns
547(3)
C.2 Proving Kesten's pattern theorem
550(5)
C.3 Kesten's pattern theorem
555(3)
D Asymptotic approximations 558(23)
D.1 Approximation of the binomial coefficient
559(3)
D.2 Approximation of trinomial coefficients
562(2)
D.3 The Euler-Maclaurin formula
564(2)
D.4 Saddle point approximations of the integral
566(1)
D.5 Asymptotic formulae for the q-factorial and related functions
567(11)
D.6 Asymptotics from the generating function
578(1)
D.7 Convergence of continued fractions
579(2)
E Percolation in Zd 581(14)
E.1 Edge percolation
581(2)
E.2 The decay of the percolation cluster
583(1)
E.3 Exponential decay of the subcritical cluster
584(6)
E.4 Subexponential decay of the supercritical cluster
590(5)
References 595(23)
Index 618
E J Janse van Rensburg is Professor of Mathematics at York University in Toronto, Ontario. He was educated at the University of Stellenbosch and at the University of the Witwatersrand in Johannesburg, South Africa, where he earned a B.Sc. (Hons) in Mathematics and Physics. He earned a Ph.D. in 1988 from Cambridge University. After post-doctoral positions at the University of Toronto, Florida State University and at RMC in Kingston, Ontario, he became an Assistant Professor of Mathematics at York University in 1992, where he was promoted to Associated Professor in 1996 and to Professor in 2000.
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