Contents -- Preface to the First Edition -- Preface to the Second Edition -- Preface to the Third Edition -- Introduction -- I. SIMPLE SYMMETRIC RANDOM WALK IN Z1 -- Notations and abbreviations -- 1 Introduction of Part I -- 1.1 Random walk -- 1.2 Dyadic expansion -- 1.3 Rademacher functions -- 1.4 Coin tossing -- 1.5 The language of the probabilist -- 2 Distributions -- 2.1 Exact distributions -- 2.2 Limit distributions -- 3 Recurrence and the Zero-One Law -- 3.1 Recurrence -- 3.2 The zero-one law -- 4 From the Strong Law of Large Numbers to the Law of Iterated Logarithm -- 4.1 Borel Cantelli lemma and Markov inequality -- 4.2 The strong law of large numbers -- 4.3 Between the strong law of large numbers and the law of iterated logarithm -- 4.4 The LIL of Khinchine -- 5 Lévy Classes -- 5.1 Definitions -- 5.2 EFKP LIL -- 5.3 The laws of Chung and Hirsch -- 5.4 When will Sn be very large? -- 5.5 A theorem of Csáki -- 6 Wiener Process and Invariance Principle -- 6.1 Four lemmas -- 6.2 Joining of independent random walks -- 6.3 Definition of the Wiener process -- 6.4 Invariance Principle -- 7 Increments -- 7.1 Long head-runs -- 7.2 The increments of a Wiener process -- 7.3 The increments of SN -- 8 Strassen Type Theorems -- 8.1 The theorem of Strassen -- 8.2 Strassen theorems for increments -- 8.3 The rate of convergence in Strassen's theorems -- 8.4 A theorem of Wichura -- 9 Distribution of the Local Time -- 9.1 Exact distributions -- 9.2 Limit distributions -- 9.3 Definition and distribution of the local time of a Wiener process -- 10 Local Time and Invariance Principle -- 10.1 An invariance principle -- 10.2 A theorem of Lévy -- 11 Strong Theorems of the Local Time -- 11.1 Strong theorems for (x, n) and (n) -- 11.2 Increments of (x, t) -- 11.3 Increments of (x, n) -- 11.4 Strassen type theorems -- 11.5 Stability -- 12 Excursions.

12.1 On the distribution of the zeros of a random walk -- 12.2 Local time and the number of long excursions (Mesure du voisinage) -- 12.3 Local time and the number of high excursions -- 12.4 The local time of high excursions -- 12.5 How many times can a random walk reach its maximum? -- 13 Frequently and Rarely Visited Sites -- 13.1 Favourite sites -- 13.2 Rarely visited sites -- 14 An Embedding Theorem -- 14.1 On the Wiener sheet -- 14.2 The theorem -- 14.3 Applications -- 15 A Few Further Results -- 15.1 On the location of the maximum of a random walk -- 15.2 On the location of the last zero -- 15.3 The Ornstein-Uhlenbeck process and a Darling-Erdos theorem -- 15.4 A discrete version of the Itô formula -- 16 Summary of Part I -- II. SIMPLE SYMMETRIC RANDOM WALK IN Zd -- Notations -- 17 The Recurrence Theorem -- 18 Wiener Process and Invariance Principle -- 19 The Law of Iterated Logarithm -- 20 Local Time -- 20.1 (0, n) in Z2 -- 20.2 (n) in Zd -- 20.3 A few further results -- 21 The Range -- 21.1 The strong law of large numbers -- 21.2 CLT, LIL and Invariance Principle -- 21.3 Wiener sausage -- 22 Heavy Points and Heavy Balls -- 22.1 The number of heavy points -- 22.2 Heavy balls -- 22.3 Heavy balls around heavy points -- 22.4 Wiener process -- 23 Crossing and Self-crossing -- 24 Large Covered Balls -- 24.1 Completely covered discs centered in the origin of Z2 -- 24.2 Completely covered disc in Z2 with arbitrary centre -- 24.3 Almost covered disc centred in the origin of Z2 -- 24.4 Discs covered with positive density in Z2 -- 24.5 Completely covered balls in Zd -- 24.6 Large empty balls -- 24.7 Summary of Chapter 24 -- 25 Long Excursions -- 25.1 Long excursions in Z2 -- 25.2 Long excursions in high dimension -- 26 Speed of Escape -- 27 A Few Further Problems -- 27.1 On the Dirichlet problem -- 27.2 DLA model -- 27.3 Percolation.

III. RANDOM WALK IN RANDOM ENVIRONMENT -- Notations -- 28 Introduction of Part III -- 29 In the First Six Days -- 30 After the Sixth Day -- 30.1 The recurrence theorem of Solomon -- 30.2 Guess how far the particle is going away in an RE -- 30.3 A prediction of the Lord -- 30.4 A prediction of the physicist -- 31 What Can a Physicist Say About the Local Time (0, n)? -- 31.1 Two further lemmas on the environment -- 31.2 On the local time (0, n) -- 32 On the Favourite Value of the RWIRE -- 33 A Few Further Problems -- 33.1 Two theorems of Golosov -- 33.2 Non-nearest-neighbour random walk -- 33.3 RWIRE in Zd -- 33.4 Non-independent environments -- 33.5 Random walk in random scenery -- 33.6 Random environment and random scenery -- 33.7 Reinforced random walk -- IV. RANDOM WALKS IN GRAPHS -- 34 Introduction of Part IV -- 35 Random Walk in Comb -- 35.1 Definitions and legend -- 35.2 Approximation -- 35.3 Extremes of a comb-walk -- 35.4 Local time of a comb-walk -- 35.5 Large square covered by a comb-random-walk -- 36 Random Walk in a Comb and in a Brush with Crossings -- 36.1 Comb with crosslines -- 36.2 A question on a brush with crossings -- 37 Random Walk on a Spider -- 38 Random Walk in Half-Plane-Half-Comb -- References -- Author Index -- Subject Index.