Grimmett G. Probability on graphs: random processes on graphs and lattices (Cambridge, 2010). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаGrimmett G. Probability on graphs: random processes on graphs and lattices. - Cambridge: Cambridge University Press, 2010. - xi, 247 p.: ill. - (Institute of Mathematical Statistics textbooks). - Ref.: p.226-242. - Ind.: p.243-247. - ISBN 978-0-521-14735-4
 

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Оглавление / Contents
 
    Preface .................................................... ix

 1  Random walks on graphs ...................................... 1
    1.1  Random walks and reversible Markov chains .............. 1
    1.2  Electrical networks .................................... 3
    1.3  Flows and energy ....................................... 8
    1.4  Recurrence and resistance ............................. 11
    1.5  Pólya's theorem ....................................... 14
    1.6  Graph theory .......................................... 16
    1.7  Exercises ............................................. 18

 2  Uniform spanning tree ...................................... 21
    2.1  Definition ............................................ 21
    2.2  Wilson's algorithm .................................... 23
    2.3  Weak limits on lattices ............................... 28
    2.4  Uniform forest ........................................ 31
    2.5  Schramm-Löwner evolutions ............................. 32
    2.6  Exercises ............................................. 37

 3  Percolation and self-avoiding walk ......................... 39
    3.1  Percolation and phase transition ...................... 39
    3.2  Self-avoiding walks ................................... 42
    3.3  Coupled percolation ................................... 45
    3.4  Oriented percolation .................................. 45
    3.5  Exercises ............................................. 48

 4  Association and influence .................................. 50
    4.1  Holley inequality ..................................... 50
    4.2  FKG inequality ........................................ 53
    4.3  BK inequality ......................................... 54
    4.4  Hoeffding inequality .................................. 56
    4.5  Influence for product measures ........................ 58
    4.6  Proofs of influence theorems .......................... 63
    4.7  Russo's formula and sharp thresholds .................. 75
    4.8  Exercises ............................................. 78

 5  Further percolation ........................................ 81
    5.1  Subcritical phase ..................................... 81
    5.2  Supercritical phase ................................... 86
    5.3  Uniqueness of the infinite cluster .................... 92
    5.4  Phase transition ...................................... 95
    5.5  Open paths in annuli .................................. 99
    5.6  The critical probability in two dimensions ........... 103
    5.7  Cardy's formula ...................................... 110
    5.8  The critical probability via the sharp-threshold
         theorem .............................................. 121
    5.9  Exercises ............................................ 125

 6  Contact process ........................................... 127
    6.1  Stochastic epidemics ................................. 127
    6.2  Coupling and duality ................................. 128
    6.3  Invariant measures and percolation ................... 131
    6.4  The critical value ................................... 133
    6.5  The contact model on a tree .......................... 135
    6.6  Space-time percolation ............................... 138
    6.7  Exercises ............................................ 141

 7  Gibbs states .............................................. 142
    7.1  Dependency graphs .................................... 142
    7.2  Markov fields and Gibbs states ....................... 144
    7.3  Ising and Potts models ............................... 148
    7.4  Exercises ............................................ 150

 8  Random-cluster model ...................................... 152
    8.1  The random-cluster and Ising/Potts models ............ 152
    8.2  Basic properties ..................................... 155
    8.3  Infmile-volume limits and phase transition ........... 156
    8.4  Open problems ........................................ 160
    8.5  In two dimensions .................................... 163
    8.6  Random even graphs ................................... 168
    8.7  Exercises ............................................ 171

 9  Quantum Ising model ....................................... 175
    9.1  The model ............................................ 175
    9.2  Continuum random-cluster model ....................... 176
    9.3  Quantum Ising via random-cluster ..................... 179
    9.4  Long-range order ..................................... 184
    9.5  Entanglement in one dimension ........................ 185
    9.6  Exercises ............................................ 189

10  Interacting particle systems .............................. 190
    10.1 Introductory remarks ................................. 190
    10.2 Contact model ........................................ 192
    10.3 Voter model .......................................... 193
    10.4 Exclusion model ...................................... 196
    10.5 Stochastic Ising model ............................... 200
    10.6 Exercises ............................................ 203

11  Random graphs ............................................. 205
    11.1 Erdős-Rényi graphs ................................... 205
    11.2 Giant component ...................................... 206
    11.3 Independence and colouring ........................... 211
    11.4 Exercises ............................................ 217

12  Lorentz gas ............................................... 219
    12.1 Lorentz model ........................................ 219
    12.2 The square Lorentz gas ............................... 220
    12.3 In the plane ......................................... 223
    12.4 Exercises ............................................ 224

    References ................................................ 226
    Index ..................................................... 243


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