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Grimmett G. — Percolation
Grimmett G. — Percolation

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Название: Percolation

Автор: Grimmett G.

Аннотация:

Percolation theory is the study of an idealized random medium in two or more dimensions. It is a cornerstone of the theory of spatial stochastic processes with applications in such fields as statistical physics, epidemiology, and the spread of populations. Percolation plays a pivotal role in studying more complex systems exhibiting phase transition. The mathematical theory is mature, but continues to give rise to problems of special beauty and difficulty. The emphasis of this book is upon core mathematical material and the presentation of the shortest and most accessible proofs. The book is intended for graduate students and researchers in probability and mathematical physics. Almost no specialist knowledge is assumed beyond undergraduate analysis and probability. This new volume differs substantially from the first edition through the inclusion of much new material, including: the rigorous theory of dynamic and static renormalization; a sketch of the lace expansion and mean field theory; the uniqueness of the infinite cluster; strict inequalities between critical probabilities; several essays on related fields and applications; numerous other results of significant. There is a summary of the hypotheses of conformal invariance. A principal feature of the process is the phase transition. The subcritical and supercritical phases are studied in detail. There is a guide for mathematicians to the physical theory of scaling and critical exponents, together with selected material describing the current state of the rigorous theory. To derive a rigorous theory of the phase transition remains an outstanding and beautiful problem of mathematics.


Язык: en

Рубрика: Математика/Математическая Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: second edition

Год издания: 1999

Количество страниц: 444

Добавлена в каталог: 24.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
'S-shape' theorem      47 246
A B percolation      351
adjacency      10
Aharony, A.      29 249 252 279
Ahlswede, R.      51
Aizenman, M.      ix 29—30 50 52 58 66 75—76 86 88 102 115 130 133 144—145 196 216 229—231 252—253 263 266 270—271 278—279 347 353—356 359—361 364 396
Alexander, K.S.      21 145 229 231 347 371
Andjel, E.      196
Animals      79 86 142 224
Annulus      315
Annulus, open circuits in a.      316
Ansatz      240
Ansatz for cluster size      239
Ansatz for connectivity function      242
Ant in labyrinth      251
Antal, P.      196
Antipercolation      351
Appel, M.J.      351
Asmussen, S.      278
Association      310 312
asymptotic relations      12 239
Athreya, K.B.      278
Augmented percolation      66 75
Backbone      252
Barlow, R.N.      46 51
Barsky, D.J.      ix—x 88 102 115 147 163 170 174 196 229—230 266 278—279
Batty, C.J.K.      51
Baxter, R.J.      396
Ben-Avraham, D.      253
Benjamini, I.      29 231 253 350—351 390
Berg, J. van den      8 30 37—39 51—52 86 144 204 230 325 348 393
Berlyand, L.      253
Bernoulli measure      10
Bethe lattice      255
Bethe, H.      255
Bezuidenhout, C.E.      76 86 196 229 231 253 369 385 392
Biggs, N.L.      349
Binary tree      254
BK inequality      37
Block argument      145 147 148 222 245
Bochner's theorem      271
Bollman, H.W.      51
Bolthausen, E.      311
Bonay, R.      x
Bond percolation      3 9 71
Bond-box      155
Bondy, J.A.      285 345
Boolean model      371
Borgs, C.      252 279
Bories, S.      367
Boundary condition      395
Bow-tie lattice      53 75
box      12
Branching      173
Branching process      15 253 256 260 278 354
Branvall, G.      86 231 312 347
Brick      163
Broadbent, S.R.      vii 1 15 29—30
Brownian motion      389
Bunimovitch, L.A.      383—384
Burton, R.M.      ix 198 229 355
Campanino, M.      51
Cantor set      385
Capacitated network      378
Capacity      378
Cardy's formula      253 346 396
Cardy, J.      ix 253 346 396
Cayley graph      29 350
Central limit theorem      196 231 253 309 311
Cerf, R.      21 145 231 347
Chandler, R.      367
Chayes, J.T.      21 29 50—52 75 115 144—145 196 206 210 214 230—231 252—253 278—279 297 347—348 353 360—361 367 381 386—387 396
Chayes, L.      21 29 50—52 75 115 144—145 196 206 210 214 230—231 253 278 297 347—348 353 360—361 367 381 386—387 396
Chow, Y.S.      95
Circuit      11
Circuit in annulus      316
Circuit, interior      288
Closed circuit      11
Closed edge      10
Cluster analysis      374
Cluster diameter      176 206
Cluster of spheres      372
cluster size      12
Cluster size, distribution      139
Cluster size, exponential decay      132
Cluster size, exponential decay for continuum percolation      373
Cluster size, moments      234 278
Cluster size, power law in two dimensions      324
Cluster size, sub-exponential decay      216 296
Cluster size, width      206
Cluster, open      11
Cluster, sequential construction      28 154 171 211
Cohen, E.G.D.      383—384
Communication system      374
Compatible 3-partition      200
Complement of event      10
Concentration inequality      52
Configuration      10
Configuration, sphere of c.      49
Conformal invariance      253 346 396
Coniglio, A.      253
Connective constant      15 30
Connectivity function      126 248
Connectivity function, continuity      204
Connectivity function, exponential decay      126 214
Connectivity function, truncated      197 213 295
Contact process      368 390
Continuum percolation      86 371
correlation length      127 214 235 237 304
Coverage process      374
Covering graph, lattice      24 54
Cox, J.T.      8 86 289 310—314 347 369—370
Critical curve      59 331 332
Critical dimension      238 253 278
Critical exponents      22 84 89 102 232 236 252 256
Critical exponents, exact values in high dimensions      270 280
Critical exponents, exact values in two dimensions      252 279
Critical exponents, inequalities      262 278 325
Critical exponents, values for tree      256 279
Critical percolation process      14 22 232 254
Critical phenomenon      13 87
Critical probability      13 22 24 35 53 88 304
Critical probability for oriented percolation      368
Critical probability in high dimensions      75 280
Critical probability of half-space      162
Critical probability of slabs      146 148 361
Critical probability, exact values      53 285 287 332 345
Critical probability, inequalities      25 30 57 72
Critical probability, series expansion      75 279
Critical surface      59 331 332
Crossing      382
Crossing c. cluster      176
Cubic lattice      10 349
Curie point      7
Cutset      378
Dal Maso, G.      381
Daykin, D.E.      51
Debierre, J.M.      351
Decreasing event      32
Decreasing random variable      32
Dekking, F.M.      386—390
Delyon, F.      86 216 231
DeMasi, A.      231 253
Deuschel, J.-D.      196 231
Deutscher, G.      253
Diagonal lattice      383
Diameter      176 206
Differential inequality      52 61 71 102 106 115
Digital sundial      388
Diminishment      65 366
Disjoint graphs      11
Disjoint occurrence      37
Distance function      9 255
Double-sided mirror      382
Doubly connected      277
Droplet      231
Dual graph      53 281 283
Dual percolation process      281 284
Dunford, N.      78 228
Durrett, R.T.      ix 8 51 144 196 253 259 278 289 297 347 368—370 386—387 392—393
Dyson, F.J.      352
Edge boundary      79 152
Edge-disjoint graphs      11
Edwards, R.G.      395
Effective resistance      6 380
Efros, A.L.      29
Ehrenfest, P.      382
El Gamal, A.      9
Electrical network      6 251 380
Elementary particle      367
Embedding      364
Enhancement      63 75 366
Enhancement, critical point      64
Enhancement, density      64
Enhancement, essential      64 66
Enhancement, monotonic      64
Enhancement, percolation probability      64
Entanglement      52 57 65 75 362
Entanglement, critical point      363
Entanglement, probability      363
Entanglement, transition      57
Epidemic      8 390
Epidemic, without recovery      393
Esary, J.D.      46
Essam, J.W.      29—30 75—77 86 114 249 252 285 345 348 388
Euler's formula      285
Event      33
Event, complement      10
Event, decreasing      32
Event, disjoint occurrence      36
Event, increasing      32
Event, interior      49
Exterior vertex boundary      152
External boundary      8 223
Falconer, K.J.      386—389
Family, F.      249
Feller, W.      30 266
Fernandez, R.      115
Ferrari, P.A.      231 253
ferromagnetism      7 393
Fiebig, U.      51
Fires in orchards      8
First-order phase transition      306 352 356 396
First-passage, percolation      29 369 380
First-passage, time      370
Fisher, D.S.      52 278
Fisher, M.E.      ix 30 247—249 253 345 400
FKG inequality      18 34 38 51
Flory, P.J.      29
Flow      378
Ford, L.R.      378
Fortuin, C.M.      8 34 51 115 394
Fourier transform      270
Fractal dimension      251 387
Fractal dimension, percolation      383 389
Framework      364
Free energy      23
Friedgut, E.      52
Frieze, A.M.      30
Frisch, H.L.      8
Frohlich, J.      50—52 144 196 230 352 360—361
Fulkerson, D.R.      378
Galaxies      367
Gallavotti, G.      115
Gandolfi, A.      229 345 348
Gap exponent      234 242
Gennes, P.G. de      251 367
Ghost vertex      105 115
Gibbs state      229
Ginibre, J.      34 51
Gluck, H.      365
Godsil, C.      29 349
Golden, K.      381
Goldstein, S.      231 253
Gordon, D.M.      30
Graf, S.      387
Graphical representation      391
Graver, J.      365
Green vertex      105 115
Greene, J.W.      9
Griffiths, R.B.      115
Grigorchuk, R.I.      115
Grimmett, G.R.      8 19 28—30 36 52 58 66 72 75—76 86 95 139 144—149 155 163 170 174 196 206 214 228—231 253 297 306 310—314 347—350 354 357 364—365 369—371 380—386 389—396
Grimmett, H.      x
Guyon, E.      367
Haggstrom, O.      230—231 350 396
Half-space      162
Half-way box      155
Hall, P.      374
Halley, J.W.      351
Hammersley, J.M.      vii x 1 8 15 29—30 51 115—119 144 230 345—347 350 369 381 400
Hankey, A.      231
Hara, T.      ix 30 56 75 144 230 243 269—271 275 278—280 352
Harris's inequality      51
Harris, T.E.      30 34 51 114 196 230 260 278 281 287—288 345 348 390—391
Hassold, G.N.      75 364
Hausdorff dimension      387
Hausdorff measure function      387
Havlin, S.      253
Hawkes, J.      387
Hering, H.      278
Herrndorf, N.      312 347
Hexagonal lattice      30 53 333 345
High density percolation      29
Higuchi, Y.      76
Hille, E.      399
Hoffman, C.      231
Hollander, W.Th. F. den      51
Holroyd, A.E. x,      75 364—366
Homogenization      381
Honeycomb lattice      54
Horizontally bounded      339
Hughes, B.D.      29—30 56 75 249 252 279 382
Hypercubic lattice      9 349
Hyperscaling relations      235 238 243 248 252 279
incidence      10
Incipient infinite cluster      249 253
Increasing event      32
Increasing random variable      32
Indicator function      10
Inequalities for critical points      25 30 57 72
Infinite cluster, boundary/volume ratio      226
Infinite cluster, existence      19
Infinite cluster, geometry      226
Infinite cluster, non-uniqueness      230
Infinite cluster, random walk on i.e.      231 251
Infinite cluster, uniqueness      198 229
Infinite cluster, uniqueness for continuum percolation      373
Infinite cluster, uniqueness, simultaneous      229
Infinite-volume limit      106 397
Infra-red bound      275
Inhomogeneous percolation      3 29 331 350
Integrated super-Brownian excursion      253 280
Interior of circuit      288
Interior of event      49
Intersection-equivalence      389
Invasion percolation      66 75 366
Invasion percolation with trapping      367
Ising model      7 16 76 102 115 144 352 393
Ising, E.      7—8 16 23 76 102 106 115 126 144 188 196 206 233 244 293 310 317 352—355 362 390—396
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