viii CONTENTS
6.5. Notes and comments 204
Chapter 7. Independent random walks: large deviations 207
7.1. Feynman-Kac minorations 207
7.2. Moderate deviations in sub-critical dimensions 222
7.3. Laws of the iterated logarithm 226
7.4. What do we expect in critical dimensions? 230
7.5. Large deviations in super-critical dimensions 231
7.6. Notes and comments 247
Chapter 8. Single random walk: large deviations 253
8.1. Self-intersection in one dimension 253
8.2. Self-intersection in d = 2 257
8.3. LDP of Gaussian tail in d = 3 264
8.4. LDP of non-Gaussian tail in d = 3 270
8.5. LDP for renormalized range in d = 2, 3 278
8.6. Laws of the iterated logarithm 287
8.7. What do we expect in d 4? 289
8.8. Notes and comments 291
Appendix 297
A. Green’s function 297
B. Fourier transformation 299
C. Constant κ(d, p) and related variations 303
D. Regularity of stochastic processes 309
E. Self-adjoint operators 313
Bibliography 321
List of General Notations 329
Index 331
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