Preface......Page 7
Contents......Page 9
1 Introduction......Page 11
2 Immobile Traps......Page 13
2.1 Annealed Asymptotics......Page 14
2.2 Quenched Asymptotics......Page 17
3 Mobile Traps......Page 19
3.1 Decay of Survival Probabilities......Page 20
3.2 Path Measures......Page 25
4 Some Open Questions......Page 28
References......Page 30
1.1 Background and Outline......Page 33
1.2 RWRE......Page 35
1.3 RWCRE......Page 37
1.4 Main Theorems......Page 39
1.5 Discussion, Open Problems and a Conjecture......Page 41
2 No Cooling: Proof of Theorem 1......Page 43
3 Cooling: Proof of Theorem 2......Page 45
4 Slow and Fast Cooling: Proof of Theorem 3......Page 47
B Central Limit Theorem......Page 48
C RWRE: Lp Convergence Under Recurrence......Page 49
References......Page 51
1 Introduction......Page 53
2.1 Preliminary Definitions......Page 57
2.2 Our Motivations and Approach......Page 60
3.1 Graphical Construction......Page 65
3.2 Hydrodynamic Limit and Invariant Measures......Page 67
4 Scalar Conservation Laws and Entropy Solutions......Page 73
4.1 Definition and Properties of Entropy Solutions......Page 74
4.2 The Riemann Problem......Page 77
4.3 From Riemann to Cauchy Problem......Page 80
5 Proof of Hydrodynamics......Page 82
5.1 Riemann Problem......Page 83
5.2 Cauchy Problem......Page 86
6.1 Framework......Page 89
6.2 Examples......Page 92
References......Page 97
1 Introduction......Page 100
1.1 Related Works......Page 101
1.2 Outline of the Proof......Page 102
1.3 Parameters......Page 103
2.1 Recursive Construction of Blocks......Page 104
2.2 Corner to Corner, Corner to Side and Side to Side Mapping Probabilities......Page 106
2.3 Good Blocks......Page 109
3 Recursive Estimates......Page 110
3.2 The Main Recursive Theorem......Page 111
3.3 Proving the Recursive Estimates at Level 1......Page 112
4 Geometric Constructions......Page 116
4.1 Admissible Connections......Page 117
6 Corner to Corner Estimate......Page 125
6.1 Case 1......Page 126
6.2 Case 2......Page 129
6.3 Case 3......Page 132
6.4 Case 4......Page 135
6.5 Case 5......Page 137
7 Side to Corner and Corner to Side Estimates......Page 139
8 Side to Side Estimate......Page 142
9 Good Blocks......Page 145
References......Page 146
1 Introduction......Page 148
2.1 Quadrangulations with a Boundary......Page 152
2.3 Zipper......Page 154
2.4 Annealed Infinite Self-Avoiding Walks on the UIPQ......Page 156
2.5 Annealed and Quenched Connective Constants......Page 158
3.1 Building One Fence......Page 161
3.2 One Fence in the Simple Boundary UIHPQ......Page 163
3.3 The Peeling Process......Page 164
3.4 Overshoot Estimates......Page 166
3.5 Building the Final Fences in Q and Q......Page 167
4.2 Proof of Theorem 2......Page 169
4.3 Lemma 4: Decoupling the Scales......Page 171
References......Page 173
1 Introduction......Page 176
2.1 Derivatives and Evolution......Page 180
3 Proof of Theorem 1......Page 182
4 Proof of Theorem 2......Page 184
5 Around Conjectures 1 and 4b......Page 186
6 About Conjectures 2, 3 and 4a......Page 187
7.1 Proof of Proposition 1......Page 188
7.2 Proof of Proposition 2......Page 191
8 Proof of Theorem 3......Page 193
9 Some Additional Rigorous Results......Page 194
References......Page 196
1.1 Introduction......Page 197
1.2 Results......Page 199
1.3 Comments......Page 201
1.4 Paper Organization......Page 202
2 Preliminaries: Hypergeometric Orthogonal Polynomials......Page 203
3 Self-duality: Proof of Theorem 1 Part (i)......Page 206
3.1 The Symmetric Exclusion Process, SEP(j)......Page 207
3.2 The Symmetric Inclusion Process, SIP(k)......Page 210
3.3 The Independent Random Walker, IRW......Page 213
4.1 The Brownian Momentum Proces, BMP......Page 215
4.2 The Brownian Energy Process, BEP(k)......Page 218
4.3 The KipnisโMarchioroโPresutti Process, KMP(k)......Page 221
References......Page 222
Self-Avoiding Walks and Connective Constants......Page 225
1.1 Self-avoiding Walks......Page 226
1.2 Connective Constants, Exact Values......Page 227
1.3 Three Problems on the Square Lattice......Page 228
1.4 Critical Exponents for SAWs......Page 229
2.2 Bounds for in Terms of Degree......Page 231
3.1 The Fisher Transformation......Page 233
3.2 Bounds for Connective Constants of Cubic Graphs......Page 235
4.1 Outline of Results......Page 236
4.2 Quotient Graphs......Page 237
4.3 Quasi-Transitive Augmentations......Page 239
5.2 Strict Inequalities for Cayley Graphs......Page 240
6.1 Bridges and Graph Height Functions......Page 241
6.3 Weighted Cayley Graphs......Page 242
7.2 Locality Theorem......Page 243
8.1 Elementary Amenable Groups......Page 244
9 Speed, and the Exponent......Page 245
References......Page 248
1 Introduction......Page 252
2.2 Quasi-Stationary Distributions......Page 254
3.1 BBM and FโKPP Equation......Page 256
3.2 N-BBM and DurrettโRemenik Equation......Page 257
3.3 FlemingโViot and QSD......Page 258
3.4 Choose the Fittest and FโKPP Equation......Page 260
4 Traveling Waves and QSD for Lรฉvy Processes......Page 261
References......Page 263
1 Introduction......Page 265
1.1 The Model......Page 266
1.2 Discussion......Page 268
2 Preliminary Results......Page 269
3 The Coupling......Page 270
3.1 The Dynamics......Page 272
3.2 Properties of the Coupling......Page 273
4 Proofs of Theorem 3 and Theorem 4......Page 275
4.1 Bounds on the Decoupling Events......Page 276
4.2 Proofs of Proposition 3 and Proposition 4......Page 281
References......Page 282
1 Introduction and Main Results......Page 284
2.1 General Notation......Page 289
2.2 Ballisticity Conditions......Page 291
3 Proof of Theorem2......Page 292
3.1 Law of Large Numbers for Hitting Times......Page 293
3.2 Introducing Kalikow's Walk......Page 294
3.3 (QLD) Implies (P)K......Page 299
3.4 Finishing the Proof of Theorem2......Page 302
4.1 (LD) Implies (P)K......Page 303
4.2 Exit Measure from Small Slabs......Page 304
4.3 Exit Measures from Small Slabs Within a Seed Box......Page 308
4.4 Renormalization Scheme to Obtain a Seed Estimate......Page 310
4.5 Proof of (27)......Page 314
5.1 The Renormalization Scheme......Page 317
References......Page 335
Publications of Charles M. Newman......Page 336