| Preface | p. ix |
| Introduction | p. 1 |
| Basic definitions | p. l |
| Continuous-time random walk | p. 6 |
| Other lattices | p. 7 |
| Other walks | p. 11 |
| Generator | p. 11 |
| Filtrations and strong Markov property | p. 14 |
| A word about constants | p. 17 |
| Local central limit theorem | p. 21 |
| Introduction | p. 21 |
| Characteristic functions and LCLT | p. 25 |
| Characteristic functions of random variables in Rd | p. 25 |
| Characteristic functions of random variables in Zd | p. 27 |
| LCLT - characteristic function approach | p. 28 |
| Exponential moments | p. 46 |
| Some corollaries of the LCLT | p. 51 |
| LCLT - combinatorial approach | p. 58 |
| Stirling's formula and one-dimensional walks | p. 58 |
| LCLT for Poisson and continuous-time walks | p. 64 |
| Approximation by Brownian motion | p. 72 |
| Introduction | p. 72 |
| Construction of Brownian motion | p. 74 |
| Skorokhod embedding | p. 79 |
| Higher dimensions | p. 82 |
| An alternative formulation | p. 84 |
| The Green's function | p. 87 |
| Recurrence and transience | p. 87 |
| The Green's generating function | p. 88 |
| The Green's function, transient case | p. 95 |
| Asymptotics under weaker assumptions | p. 99 |
| Potential kernel | p. 101 |
| Two dimensions | p. 101 |
| Asymptotics under weaker assumptions | p. 107 |
| One dimension | p. 109 |
| Fundamental solutions | p. 113 |
| The Green's function for a set | p. 114 |
| One-dimensional walks | p. 123 |
| Gambler's ruin estimate | p. 123 |
| General case | p. 127 |
| One-dimensional killed walks | p. 135 |
| Hitting a half-line | p. 138 |
| Potential theory | p. 144 |
| Introduction | p. 144 |
| Dirichlet problem | p. 146 |
| Difference estimates and Harnack inequality | p. 152 |
| Further estimates | p. 160 |
| Capacity, transient case | p. 166 |
| Capacity in two dimensions | p. 176 |
| Neumann problem | p. 186 |
| Beurling estimate | p. 189 |
| Eigenvalue of a set | p. 194 |
| Dyadic coupling | p. 205 |
| Introduction | p. 205 |
| Some estimates | p. 207 |
| Quantile coupling | p. 210 |
| The dyadic coupling | p. 213 |
| Proof of Theorem 7.1.1 | p. 216 |
| Higher dimensions | p. 218 |
| Coupling the exit distributions | p. 219 |
| Additional topics on simple random walk | p. 225 |
| Poisson kernel | p. 225 |
| Half space | p. 226 |
| Cube | p. 229 |
| Strips and quadrants in Z2 | p. 235 |
| Eigenvalues for rectangles | p. 238 |
| Approximating continuous harmonic functions | p. 239 |
| Estimates for the ball | p. 241 |
| Loop measures | p. 247 |
| Introduction | p. 247 |
| Definitions and notations | p. 247 |
| Simple random walk on a graph | p. 251 |
| Generating functions and loop measures | p. 252 |
| Loop soup | p. 257 |
| Loop erasure | p. 259 |
| Boundary excursions | p. 261 |
| Wilson's algorithm and spanning trees | p. 268 |
| Examples | p. 271 |
| Complete graph | p. 271 |
| Hypercube | p. 272 |
| Sierpinski graphs | p. 275 |
| Spanning trees of subsets of Z2 | p. 277 |
| Gaussian free field | p. 289 |
| Intersection probabilities for random walks | p. 297 |
| Long-range estimate | p. 297 |
| Short-range estimate | p. 302 |
| One-sided exponent | p. 305 |
| Loop-erased random walk | p. 307 |
| h-processes | p. 307 |
| Loop-erased random walk | p. 311 |
| LERW in Zd | p. 313 |
| d≥3 | p. 314 |
| d=2 | p. 315 |
| Rate of growth | p. 319 |
| Short-range intersections | p. 323 |
| Appendix | p. 326 |
| Some expansions | p. 326 |
| Riemann sums | p. 326 |
| Logarithm | p. 327 |
| Martingales | p. 331 |
| Optional sampling theorem | p. 332 |
| Maximal inequality | p. 334 |
| Continuous martingales | p. 336 |
| Joint normal distributions | p. 337 |
| Markov chains | p. 339 |
| Chains restricted to subsets | p. 342 |
| Maximal coupling of Markov chains | p. 346 |
| Some Tauberian theory | p. 351 |
| Second moment method | p. 353 |
| Subadditivity | p. 354 |
| Bibliography | p. 360 |
| Index of Symbols | p. 361 |
| Index | p. 363 |
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