Title Page
Abstract
Contents
Chapter 1. Introduction 16
1.1. Early results 17
1.2. Freidlin-Wentzell theory and pathwise approach to metastability 18
1.3. Potential-theoretic approach: Eyring-Kramers formula 19
1.4. Martingale approach: Markov chain model reduction 20
1.5. Organization of the thesis 21
Part Ⅰ. Ferromagnetic Ising/Potts and related models 22
Chapter 2. Classic result: Ising model with positive external field 24
Chapter 3. Ising/Potts models with zero external fields 29
3.1. Main results 32
3.2. Potential theory and H¹-approximation 40
3.3. Other non-reversible models 54
3.4. Energy landscape analysis: Metropolis-Hastings dynamics 59
3.5. Test functions: Metropolis-Hastings dynamics 78
3.6. Energy landscape analysis: cyclic dynamics 96
3.7. Test functions: cyclic dynamics 107
3.8. Decomposition lemma 117
3.9. Proof of results in Section 3.4 118
3.10. Dirichlet form on gateway configurations 136
3.11. Proof of results in Section 3.6 137
Chapter 4. Three-dimensional model 149
4.1. Main results 150
4.2. Outline of the proof 158
4.3. Canonical configurations and paths 163
4.4. Gateway configurations 178
4.5. Energy barrier between ground states 185
4.6. Typical configurations and optimal paths 207
4.7. Construction of test function 220
4.8. Remarks on open boundary condition 234
Chapter 5. Large-volume limit 238
5.1. Main result 243
5.2. Sharp threshold for the Gibbs distribution 249
5.3. Preliminaries for the energy landscape 259
5.4. Energy barrier 262
5.5. Saddle structure 290
5.6. Capacity estimates 300
5.7. Upper bound for capacities 302
5.8. Lower bound for capacities 317
Chapter 6. Blume-Capel model 331
6.1. Main results 331
6.2. Outline of proofs 339
6.3. Canonical configurations and energy barrier 341
6.4. Typical and gateway configurations 348
6.5. Edge typical configurations 353
6.6. Construction of fundamental test functions and flows 359
6.7. Capacity estimates 370
6.8. Periodic boundary conditions 375
6.9. Auxiliary process 378
Chapter 7. General interaction constants 385
7.1. Main results 387
7.2. Projection operator 399
7.3. Comparison with the original Ising path 404
7.4. Proofs 406
7.5. Appendix 426
Part Ⅱ. Inclusion process 434
Chapter 8. Reversible inclusion process 435
Chapter 9. Second time scale of metastability 443
9.1. Main results 443
9.2. Outline of proof of Theorems 9.1.2 and 9.1.4 451
9.3. Hitting times on tubes 453
9.4. Upper bound for capacities: simple case 457
9.5. Lower bound for capacities: simple case 470
9.6. Upper bound for capacities: general case 481
9.7. Lower bound for capacities: general case 496
9.8. Proof of condition (9.16) 499
9.9. Proof of the main theorem 501
Chapter 10. Non-reversible inclusion process 504
10.1. Condensation of non-reversible inclusion process 504
10.2. Main results 507
10.3. Movements of condensate: general results 528
10.4. Metastable behavior of inclusion process under condition (UI) 540
10.5. Metastable behavior of inclusion process with condensation 541
10.6. Condensation under condition (UP) 546
10.7. Inclusion process in thermodynamic limit regime 570
10.8. Appendix 581
Bibliography 584
국문초록 594