ELEMENTS OF THE RANDOM WALK

By U. GAVAY

CONTENTS INTRODUCTION TO TECHNIQUES THE SIMPLEST WALK SOME VERY ELEMENTARY CALCULATIONS ON THE SIMPLEST WALK BACK TO THE PROBABILITY DISTRIBUTION RECURSION RELATION FOR THE ONE-DIMENSIONAL WALK 1 5 BACKING INTO THE GENERATING FUNCTION FOR A RANDOM WALK SUPPLEMENT: METHOD OF STEEPEST DESCENTS GENERATING FUNCTIONS I GENERAL INTRODUCTION TO GENERATING FUNCTIONS SUPPLEMENT 1: GAUSSIAN INTEGRALS SUPPLEMENT 2: FOURIER EXPANSIONS ON A LATTICE SUPPLEMENT 3: ASYMPTOTIC COEF?CIENTS OF POWER SERIES GENERATING FUNCTIONS II: RECURRENCE, SITES VISITED, AND THE ROLE OF DIMENSIONALITY RECURRENCE A NEW GENERATING FUNCTION DERIVATION OF THE NEW GENERATING FUNCTION DIMENSIONALITY AND THE PROBABILITY OF RECURRENCE RECURRENCE IN TWO DIMENSIONS RECURRENCE WHEN THE DIMENSIONALITY, D, LIES BETWEEN 2 AND THE PROBABILITY OF NON-RECURRENCE IN WALKS ON DIFFERENT CUBIC LATTICES IN THREE DIMENSIONS THE NUMBER OF SITES VISITED BY A RANDOM WALK BOUNDARY CONDITIONS, STEADY STATE, AND THE ELECTROSTATIC ANALOGY THE EFFECTS OF SPATIAL CONSTRAINTS ON RANDOM WALK STATISTICS RANDOM WALK IN THE STEADY STATE SUPPLEMENT: BOUNDARY CONDITIONS AT AN ABSORBING BOUNDARY VARIATIONS ON THE RANDOM WALK THE BIASED RANDOM WALK THE PERSISTENT RANDOM WALK THE CONTINUOUS TIME RANDOM WALK THE SHAPE OF A RANDOM WALK THE NOTION AND QUANTI?CATION OF SHAPE WALKS IN D ?3 DIMENSIONS FINAL COMMENTARY SUPPLEMENT 1: PRINCIPAL RADII OF GYRATION AND ROTATIONAL MOTION SUPPLEMENT 2: CALCULATIONS FOR THE MEAN ASPHERICITY SUPPLEMENT 3: DERIVATION OF (6 21) FOR THE RADIUS OF GYRATION TENSOR, ? T , AND THE EIGENVALUES OF THE OPERATOR PATH INTEGRALS AND SELF-AVOIDANCE THE UNRESTRICTED RANDOM WALK AS A PATH INTEGRAL SELF-AVOIDING WALKS PROPERTIES OF THE RANDOM WALK: INTRODUCTION TO SCALING UNIVERSALITY SCALING OF WALKS AND CRITICAL PHENOMENA SCALING AND THE RANDOM WALK CRITICAL POINTS, SCALING, AND BROKEN SYMMETRIES GINZBURG–LANDAU–WILSON EFFECTIVE HAMILTONIAN SCALING AND THE MEAN END-TO-END DISTANCE; ?R2? CONNECTION BETWEEN THE O(N) MODEL AND THE SELF-AVOIDING WALK SUPPLEMENT: EVALUATION OF GAUSSIAN INTEGRALS WALKS AND THE O(N) MODEL: MEAN ?ELD THEORY AND SPIN WAVES 10 1 MEAN ?ELD THEORY AND SPIN WAVES CONTRIBUTIONS 10 2 THE MEAN ?ELD THEORY OF THE O(N) MODEL 10 3 FLUCTUATIONS: LOW ORDER SPIN WAVE THEORY 10 4 THE CORRELATION HOLE SCALING, FRACTALS, AND RENORMALIZATION 11 1 SCALE INVARIANCE IN MATHEMATICS AND NATURE 11 2 MORE ON THE RENORMALIZATION GROUP: THE REAL SPACE METHOD 11 3 RECURSION RELATIONS: ?XED POINTS AND CRITICAL EXPONENTS MORE ON THE RENORMALIZATION GROUP 12 1 THE MOMENTUM-SHELL METHOD 12 2 THE EFFECTIVE HAMILTONIAN WHEN THERE IS FOURTH ORDER INTERACTION BETWEEN THE SPIN DEGREES OF FREEDOM THE O(N) MODEL: DIAGRAMMATICS O(N) RECURSION RELATIONS THE DIAGRAMMATIC METHOD DIAGRAMMATIC ANALYSIS: THE TWO-POINT CORRELATION FUNCTION SUPPLEMENT: LINKED CLUSTER EXPANSION REFERENCES