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Hughes B.D. — Random walks and random enviroments (Vol. 1. Random walks)
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Название: Random walks and random enviroments (Vol. 1. Random walks)Автор: Hughes B.D. Аннотация: Volume 1 can be read without reference to Volume 2. In Volume 1, I expect of the reader only a modest facility in classical analysis, including the theory of functions of a complex variable up to contour integration. Those elements of probability theory which are needed are introduced in Chapter 1 of Volume 1, so that although an acquaintance with elementary
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1995Количество страниц: 631Добавлена в каталог: 13.10.2005Операции: Положить на полку |
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Persistent random walk, mean number of sites visited 338 Persistent random walk, synonymous with correlated random walk 179 Persistent random walk, synonymous with recurrent random walk 122 Pivot algorithm 465 Pochhammer's symbol 572 Poisson distribution, associated with exponential waiting time density 248 Poisson distribution, defined 37 Poisson's summation formula, Borgen's form 216 Poisson's summation formula, classical form 214 Polya type walk 119 Polya walk, Bethe lattice 113 Polya walk, general definition 119 Polya walk, on a ring 161 Polya walk, one-dimensional 112 Polya's Theorem for continuous-time walk 253 Polya's Theorem, corollary to 144 Polya's Theorem, Spitzer's form 160 Polya's Theorem, stated and proved 142—144 Probability, conditional 31 Probability, defining rules 31 Probability, density function 32 Probability, distribution function 32 Probability, measure 46 Probability, space 46 Projection operator 294 Pseudolattice 20 Pseudolattice, Bethe lattice as example of 24 Pseudolattice, self-similar 25 Pseudolattice, tree-like 25 Radius of gyration of long-chain polymer 416—417 Radius of gyration of random walk 96 Random flight after many steps 91—93 102—104 Random flight, closed-form solutions 87—91 Random flight, history 55 Random flight, integral solutions 86—87 Random flight, model of polymer chain 94—98 Random flight, numerical solutions 93—94 Random flight, one-dimensional 62 Random flight, steepest descent analysis 102 Random fractal 14 Random process 1 Random system 3 Random variable, continuous-valued 32 Random variable, integer-valued 32 Random variable, introduced 31 Random walk between barriers, defect technique solution 171 Random walk between barriers, difference equation solution 170 Random walk between barriers, method of images solution 172 Random walk in continuous time see "Continuous-time random walk" Random walk of Pearson type see "Pearson's walk" Random walk on a lattice 111 Random walk on Bethe lattice, correlated 179 Random walk on Bethe lattice, Polya type 113 Random walk on diamond lattice, correlated 179 Random walk on tree-like pseudolattice 179 Random walk with internal states 177 Random walk, cardinality of see "Number of distinct sites visited" Random walk, controlled by a Markov chain 177 Random walk, correlated 179 Random walk, diffusive 268 Random walk, elementary arguments 56—57 Random walk, Gillis' centrally biased 174 Random walk, history 53—56 111 Random walk, integral transform solution 57—59 Random walk, isotropic 59—62 Random walk, maximum displacement 331 Random walk, one-dimensional 62—66 100—102 Random walk, persistent 179 Random walk, randomized (misnomer) 242 Random walk, range of see "Number of distinct sites visited" Random walk, span 331 Random walk, three-dimensional isotropic see "Random flight" Random walk, two-dimensional isotropic see "Pearson's walk" Randomized random walk 242 Range of random walk see "Number of distinct sites visited" Ratio method, history 469 Ratio method, improved 470 Ratio method, simple 470 Rayleigh, Lord 53 Real-space renormalization, analogues in random processes 221 Real-space renormalization, continuous-time random walk 309 Real-space renormalization, self-avoiding walk 514 Realization of a random environment 299 Realization of a random walk 400 Realization of a self-trapping walk 413 Recurrence Theorem, stated and proved 124—125 Recurrence Theorem, used 352 Recurrent random walk 122 Reduced partition function 392 Reflecting boundary in continuum diffusion 228 Reflection formula 570 Relative dispersion 271 Renewal theory 249 Renormalization group transformation for continuous-time random walk 309 Renormalization group transformation for self-avoiding walk 514 Renormalization group transformation, analogue in lattice walk 221 Reptation 418 Resistance dimension 233 Retained site 309 Reversible trap 166 RG transformation 309 Riemann hypothesis 573 Riemann random walk, continuum limit of 196—197 Riemann random walk, evaluation of structure function 596 Riemann random walk, introduced 154 Riemann random walk, structure function asymptotics 154—156 Riemann relation 214 573 Riemann walk see "Riemann random walk" Riemann zeta function, properties 573—574 Riemann zeta function, random walk based on 154 196 Riemann zeta function, used as a probability distribution 37 Riemann's nondifferentiable function 220 Ring closure probability, numerical, estimation of 480 Rosenstock approximation 373 Rosenstock model on a random lattice 390 Rosenstock model on fractal lattice 390 Rosenstock model, behaviour at long time 385—389 Rosenstock model, cumulant analysis 373 Rosenstock model, defined 330 Rosenstock model, exact results in one dimension 377—384 Rosenstock model, exact results on mean lifetime 374—377 Rosenstock model, history 372—373 Rosenstock model, survival probability 373 Row-convex polygons (self-avoiding) 513 Rushbrooke's scaling law 527 Saddle point method 98 Sample space 30 Scaling dimension 10 Scaling field 221 Scaling law, Rushbrooke's 527 Scaling law, Widom's 526 Scaling theory for exponents in magnetic models 525 Scaling theory for self-avoiding walk 453—457 Schwarz inequality, stated and proved 46 Schwarz inequality, used 346 Second characteristic 374 Self-affine set 16 Self-avoiding bridges 429 436 445 Self-avoiding Levy flights 499 Self-avoiding n-circuit 440 Self-avoiding polygon, connective constant 440 Self-avoiding polygon, defined 439 Self-avoiding polygon, exponent h 444 Self-avoiding polygon, Hammersley's Theorem on connective constant 440 Self-avoiding trail, analysed 494—497 Self-avoiding trail, introduced 417 Self-avoiding walk on Penrose tiling 484 Self-avoiding walk, 'true' 419 Self-avoiding walk, backbone of 431 Self-avoiding walk, complete 507 Self-avoiding walk, connective constant 423 Self-avoiding walk, defined 414 Self-avoiding walk, directed 491 Self-avoiding walk, Euler 507 Self-avoiding walk, exponent 426 Self-avoiding walk, exponent 415 Self-avoiding walk, Hamiltonian 507 Self-avoiding walk, Kasteleyn's problem 507 Self-avoiding walk, loop-erased 505 Self-avoiding walk, model non-Markovian process 3 Self-avoiding walk, spiral 508 Self-avoiding walk, unfolding 437 Self-similar lattice 25 Self-similar set 11 Self-similarity 11 Self-trapping walk 413 Sequence, subadditive 423 Sequence, submultiplicative 424 Series expansions, data for self-avoiding walks and polygons 458—464 Series expansions, history for self-avoiding walk 458 Sgn function 88 Shift theorem for Laplace transform 291 Sierpinski carpet 11 Sierpinski gasket 11 Sierpinski lattice, continuous-time random walk on 314 Sierpinski lattice, defined 25 Sierpinski lattice, random walk on 335 359 Sierpinski lattice, self-avoiding walk on 515 Simple cubic lattice, defined 21 Simple cubic lattice, Green function evaluated 610 Simple cubic lattice, return probability 153 Simple cubic lattice, structure function for 139 Simple hypercubic lattice see "Hypercubic lattice" Singular continuous function 34 Site Green function 302 Site of a lattice 19 Slowly-varying function in discrete Tauberian Theorem 118 Slowly-varying function in Tauberian Theorems for Laplace transform 248 Smirnov density 203 206 Smoluchowski's equation 209 Solvent, 416 Solvent, 'good' 416 Solvent, interaction with 98 Sonine's integral of the product of three Bessel functions 581 Span of one-dimensional Polya walk 379 Span of random walk 331 Specific heat per site 526 Spectral dimension 30 336 Spherical model, applications of Pearson's walk to 85 spherical polar coordinates 61 532 Spiral self-avoiding walk 508 Spontaneous magnetization of n-vector model 524 Square lattice, defined 20—21 Square lattice, Green function evaluated 600 St Petersburg game, embodied in Weierstrass random walk 212 St Petersburg game, history and description 39 Stable density for d 206—207 Stable density in modern sense 202 Stable density in sense of Levy 200 Stable density, one-sided 203—206 594 Stable density, series expansions via Mellin transforms 593—595 Stable density, symmetric 199 283 Stable fixed point 518 Staircase polygons 512—513 Sticky site 166 Stieltjes integrals 33 Stieltjes transform 243 Stirling's formula 571 Stochastic integration 47 Stokes — Einstein relation for continuum diffusion 230 Stokes' Law 229 Stopping time, not introduced in this work 48 Strong Law of Large Numbers 402 Strong Tauberian Theorem, stated 249 Strong Tauberian Theorem, used 249 270 Strongly transient walk, 344 Strongly transient walk, defined 126 Strongly transient walk, mean number of distinct sites visited 340 Strongly transient walk, mean number of interections 130 Strongly transient walk, necessary and sufficient condition 127 Strongly transient walk, typically needs d 144 Structure function in lattice dynamics 28 Structure function, body-centred cubic lattice 140 Structure function, defined for random walk 137 Structure function, face-centred cubic lattice 140 Structure function, hypercubic lattice 139 Structure function, Riemann walk 154 Structure function, simple cubic lattice 139 Structure function, Weierstraes random walk 213 Structure function, where equal to 1 138—139 Subadditive sequence 423 Subdiffusive behaviour, applications 284 Subdiffusive behaviour, continuous-time random walk 268 Submultiplicative sequence 424 Subordination in Markov processes 267 Superdiffusive behaviour with mean-square displacement still finite 284 Superdiffusive behaviour, continuous-time random walk 268 Superdiffusive behaviour, extreme case of 199 Supremum (sup) 9 Survival probability 373 susceptibility 525 Symmetric Levy density see "Stable density symmetric" Taboo probabilities 168 Taboo site 168 Tail event 399 Tauberian Theorems for the Laplace transform, introduced 248—250 Tauberian Theorems for the Laplace transform, used 252—253 256 262—263 265—267 282 299 Tensor 97 142 285 287 353 355 Tensor, use of boldface symbol 117 Theorem of Bovier, Felder and Frolich 446 Theorem of Guttmann 495 Theorem of Hammersley and Morton, analogue for lattice animals 503 Theorem of Hammersley and Morton, applies to directed walks 491 Theorem of Hammersley and Morton, modified 436 495 Theorem of Hammersley and Morton, shows that 426 Theorem of Hammersley and Morton, stated and proved 424—425 Theorem of Hara, Slade and Sokal 431 Theorem of Jain and Pruitt on 357 Theorem of Jain and Pruitt on 353 Theorem of Jain, Orey, and Pruitt, proved 346 Theorem of Jain, Orey, and Pruitt, stated 344 Theorem of Kasteleyn and den Hollander on coloured lattices 370 Theorem of Kenkre, Montroll, and Shlesinger 295 Theorem of Lawler 130 Theorem of Lehman and Weiss 413 Theorem of Montroll and Weiss on 334 Theorem of Montroll and Weiss on continuous-time random walk 253 Theorem on Drift, proved 400 Theorem on Drift, stated 399 Thermodynamic limit 523 Theta point 98 416 Theta solvent 416 Timid random walker 391 Topological dimension 7 Trail of a random walk, fractal, dimension of 210 Trail, self-avoiding 494 Transient random walk 122 Translational invariance of periodic lattice 21 TREE see "Bethe lattice" Tree-like pseudolattices 25 Triangular lattice as modified square lattice 602 Triangular lattice, Green function evaluated 602 True self-avoiding walk 419 Truncated tetrahedron lattice 27 Turbulent dispersion 289 Turning point 284 Universality in random walk 199 Universality in self-avoiding walk 418 Universality, concept introduced 42 Upper critical dimension 421
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