| Joseph Rudnick earned his PhD in 1970. He has held faculty positions at Tufts University and the University of California, Santa Cruz. He has held a visiting position at Harvard University. He is currently a Professor in the Department of Physics and Astronomy at the University of California, Los Angeles. |
| Preface 1 Introduction to techniques 1.1 The simplest walk 1.2 Some very elementary calculations on the simplest walk 1.3 Back to the probability distribution 1.4 Recursion relation for the one-dimensional walk 1.5 Backing into the generating function for a random walk 1.6 Supplement: method of steepest descents 2 Generating functions I 2.1 General introduction to generating functions 2.2 Supplement 1: Gaussian integrals 2.3 Supplement 2: Fourier expansions on a lattice 2.4 Supplement 3: asymptotic coefficients of power series 3 Generating functions II: recurrence, sites visited, and the role of dimensionality 3.1 Recurrence 3.2 A new generating function 3.3 Derivation of the new generating function 3.4 Dimensionality and the probability of recurrence 3.5 Recurrence in two dimensions 3.6 Recurrence when the dimensionality, d, lies between 2 and 4 3.7 The probability of non-recurrence in walks on different cubic lattices in three dimensions 3.8 The number of sites visited by a random walk 4 Boundary conditions, steady state, and the electrostatic analogy 4.1 The effects of spatial constraints on random walk statistics 4.2 Random walk in the steady state 4.3 Supplement: boundary conditions at an absorbing boundary 5 Variations on the random walk 5.1 The biased random walk 5.2 The persistent random walk 5.3 The continuous time random walk 6 The shape of a random walk 6.1 The notion and quantification of shape 6.2 Walks in d >> 3 dimensions 6.3 Final commentary 6.4 Supplement 1: principal radii of gyration and rotational motion 6.5 Supplement 2: calculations for the mean asphericity 6.6 Supplement 3: derivation of (6.21) for the radius of gyration tensor, T, and the eigenvalues of the operator 7 Path integrals and self-avoidance 7.1 The unrestricted random walk as a path integral 7.2 Self-avoiding walks 8 Properties of the random walk: introduction to scaling 8.1 Universality 9 Scaling of walks and critical phenomena 9.1 Scaling and the random walk 9.2 Critical points, scaling, and broken symmetries 9.3 Ginzburg-Landau-Wilson effective Hamiltonian 9.4 Scaling and the mean end-to-end distance; (R2) 9.5 Connection between the O(n) model and the self-avoiding walk 9.6 Supplement: evaluation of Gaussian integrals 10 Walks and the O(n) model: mean field theory and spin waves 10.1 Mean field theory and spin waves contributions 10.2 The mean field theory of the O(n) model 10.3 Fluctuations: low order spin wave theory 10.4 The correlation hole 11 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: fixed points and critical exponents 12 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 …… References Index |
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