Random graphs, phase transitions, and the Gaussian free field

Scientific Commitee......Page 6
 Sponsors......Page 7
Foreword......Page 9
Preface......Page 13
Contents......Page 16
Scaling Limits of Random Trees  and Random Graphs......Page 17
  1.1 Uniform Random Trees......Page 19
  1.2 Ordered Trees and Their Encodings......Page 22
  1.3 Galton–Watson Trees......Page 24
  1.4 mathbbR-Trees Encoded by Continuous Excursions......Page 29
  1.5 Convergence to the Brownian CRT......Page 30
 2 The Critical Erdős–Rényi Random Graph......Page 33
  2.1 The Phase Transition and Component Sizes in the Critical Window......Page 34
  2.2 Component Structures......Page 37
  3.1 The Configuration Model......Page 41
  3.2 Scaling Limit for the Critical Component Sizes......Page 43
 4 Sources for These Notes and Suggested Further Reading......Page 47
 References......Page 48
Lectures on the Ising and Potts Models on the Hypercubic Lattice......Page 50
  1.1 Lattice Spin Models......Page 52
  1.2 Graphical Representation of Potts Models......Page 57
  1.3 The Mean-Field Model......Page 61
  1.4 The Percolation Phase Transition for the Random-Cluster Model......Page 63
  2.1 Kesten's Theorem......Page 75
  2.2 Two Proofs of Sharpness for Bernoulli Percolation......Page 80
  2.3 Sharpness for Random-Cluster Models......Page 91
  2.4 Computation of the Critical Point for Random-Cluster Models on mathbbZ2......Page 95
 3 Where Are We Standing? And a Nice Conjecture.........Page 98
  4.1 An Elementary Argument in Dimension d=2......Page 99
  4.2 High-Temperature Expansion, Random Current Representation and Percolation Interpretation  of Truncated Correlations......Page 100
  4.3 Continuity of the Phase Transition for Ising Models  on mathbbZd for dge3......Page 107
  4.4 Polynomial Decay at Criticality for dge3......Page 114
 5 Continuity/Discontinuity of the Phase Transition  for the Planar Random-Cluster Model......Page 115
  5.1 Crossing Probabilities in Planar Random-Cluster Models......Page 116
  5.2 Proving Continuity for qle4: The Parafermionic Observables......Page 127
 6 Conformal Invariance of the Ising Model on mathbbZ2......Page 153
  6.1 Conformal Invariance of the Fermionic Observable......Page 156
  6.2 Conformal Invariance of the Exploration Path......Page 165
 7 Where Are We Standing? And More Conjectures........Page 168
 References......Page 171
Extrema of the Two-Dimensional Discrete Gaussian Free Field......Page 177
 1.1  Definitions......Page 182
  1.2  Why d=2 Only?......Page 186
  1.3  Green Function Asymptotic......Page 188
  1.4  Continuum Gaussian Free Field......Page 193
  2.1  Level Set Geometry......Page 195
  2.2  Growth of Absolute Maximum......Page 197
  2.3  Intermediate Level Sets......Page 202
  2.4  Link to Liouville Quantum Gravity......Page 205
  3.1  Gibbs-Markov Property of DGFF......Page 209
  3.2  First Moment of Level-Set Size......Page 213
  3.3  Second Moment Estimate......Page 217
  3.4  Second-Moment Asymptotic and Factorization......Page 220
  4.1  Gibbs-Markov Property in the Scaling Limit......Page 223
  4.2  Properties of ZDλ-Measures......Page 226
  4.3  Representation via Gaussian Multiplicative Chaos......Page 228
  4.4  Finishing Touches......Page 232
  4.5  Dealing with Truncations......Page 235
  5.1  Kahane's Inequality......Page 237
  5.2  Kahane's Theory of Gaussian Multiplicative Chaos......Page 239
  5.3  Comparisons for the Maximum......Page 243
  5.4  Stochastic Domination and FKG Inequality......Page 246
  6.1  Inheritance of Gaussian Tails......Page 250
  6.2  Fernique Majorization......Page 253
  6.3  Proof of Fernique's Estimate......Page 255
  6.4  Consequences for Continuity......Page 260
  6.5  Binding Field Regularity......Page 262
  7.1  Dekking–Host Argument for DGFF......Page 263
  7.2  Upper Bound by Branching Random Walk......Page 266
  7.3  Maximum of Gaussian Branching Random Walk......Page 269
  7.4  Bootstrap to Exponential Tails......Page 274
  8.1  Upper Tail of DGFF Maximum......Page 278
  8.2  Concentric Decomposition......Page 282
  8.3  Bounding the Bits and Pieces......Page 285
  8.4  Random Walk Representation......Page 288
  8.5  Tightness of DGFF Maximum: Lower Tail......Page 292
  9.1  Extremal Level Sets......Page 294
  9.2  Distributional Invariance......Page 298
  9.3  Dysonization-Invariant Point Processes......Page 301
  9.4  Characterization of Subsequential Limits......Page 304
  10.1  Connection to the DGFF Maximum......Page 309
  10.2  Gumbel Convergence......Page 314
  10.3  Properties of ZD-Measures......Page 316
  10.4  Uniqueness up to Overall Constant......Page 320
  10.5  Connection to Liouville Quantum Gravity......Page 326
  11.1  Cluster at Absolute Maximum......Page 329
  11.2  Random Walk Based Estimates......Page 330
  11.3  Full Process Convergence......Page 337
  11.4  Some Corollaries......Page 342
  12.1  Spatial Tightness of Extremal Level Sets......Page 346
  12.2  Limit of Atypically Large Maximum......Page 351
  12.3  Precise Upper Tail Asymptotic......Page 355
  12.4  Convergence of the DGFF Maximum......Page 360
  12.5  The Local Limit Theorem......Page 364
  13.1  A Charged Particle in an Electric Field......Page 365
  13.2  Statement of Main Results......Page 367
  13.3  A Crash Course on Electrostatic Theory......Page 370
  13.4  Markov Chain Connections and Network Reduction......Page 374
  14.1  Path-Cut Representations......Page 378
  14.2  Duality and Effective Resistance Across Squares......Page 383
  14.3  RSW Theory for Effective Resistance......Page 389
  14.4  Upper Tail of Effective Resistance......Page 393
  15.1  Hitting and Commute-Time Identities......Page 396
  15.2  Upper Bounds on Expected Exit Time and Heat Kernel......Page 398
  15.3  Bounding Voltage from Below......Page 400
  15.4  Wrapping Up......Page 404
  16.1  DGFF Level Sets......Page 406
  16.2  At and Near the Absolute Maximum......Page 407
  16.3  Universality......Page 409
  16.4  Random Walk in DGFF Landscape......Page 412
  16.5  DGFF Electric Network......Page 414
 References......Page 416