Description
I. Introduction II. Probabilities, moments, cumulants A. Probabilities, observables, and moments B. Transformation of random variables C. Cumulants D. Connection between moments and cumulants III. Gaussian distribution and Wick’s theorem A. Gaussian distribution B. Moment and cumulant generating function of a Gaussian C. Wick’s theorem D. Graphical representation: Feynman diagrams E. Appendix: Self-adjoint operators F. Appendix: Normalization of a Gaussian IV. Perturbation expansion A. General case B. Special case of a Gaussian solvable theory C. Example: Example: “phi^3 + phi^4” theory D. External sources E. Cancellation of vacuum diagrams F. Equivalence of graphical rules for n-point correlation and n-th moment G. Example: “phi^3 + phi^4” theory V. Linked cluster theorem A. General proof of the linked cluster theorem B. Dependence on j – external sources – two complimentary views C. Example: Connected diagrams of the “phi^3 + phi^4” theory VI. Functional preliminaries A. Functional derivative 1. Product rule 2. Chain rule 3. Special case of the chain rule: Fourier transform B. Functional Taylor series VII. Functional formulation of stochastic differential equations A. Onsager-Machlup path integral* B. Martin-Siggia-Rose-De Dominicis-Janssen (MSRDJ) path integral C. Moment generating functional D. Response function in the MSRDJ formalism VIII. Ornstein-Uhlenbeck process: The free Gaussian theory A. Definition B. Propagators in time domain C. Propagators in Fourier domain IX. Perturbation theory for stochastic differential equations A. Vanishing moments of response fields B. Vanishing response loops C. Feynman rules for SDEs in time domain and frequency domain D. Diagrams with more than a single external leg E. Appendix: Unitary Fourier transform X. Dynamic mean-field theory for random networks A. Definition of the model and generating functional B. Property of self-averaging C. Average over the quenched disorder D. Stationary statistics: Self-consistent autocorrelation of as motion of a particle in a potential E. Transition to chaos F. Assessing chaos by a pair of identical systems G. Schrdinger equation for the maximum Lyapunov exponent H. Condition for transition to chaos XI. Vertex generating function A. Motivating example for the expansion around a non-vanishing mean value B. Legendre transform and definition of the vertex generating function Gamma C. Perturbation expansion of Gamma D. Generalized one-line irreducibility E. Example F. Vertex functions in the Gaussian case G. Example: Vertex functions of the “phi^3 + phi^4”-theory H. Appendix: Explicit cancellation until second order I. Appendix: Convexity of W J. Appendix: Legendre transform of a Gaussian XII. Application: TAP approximation Inverse problem XIII. Expansion of cumulants into tree diagrams of vertex functions A. Self-energy or mass operator Sigma XIV. Loopwise expansion of the effective action – Tree level A. Counting the number of loops B. Loopwise expansion of the effective action – Higher numbers of loops C. Example: phi^3 + phi^4-theory D. Appendix: Equivalence of loopwise expansion and infinite resummation E. Appendix: Interpretation of Gamma as effective action F. Loopwise expansion of self-consistency equation XV. Loopwise expansion in the MSRDJ formalism A. Intuitive approach B. Loopwise corrections to the effective equation of motion C. Corrections to the self-energy and self-consistency D. Self-energy correction to the full propagator E. Self-consistent one-loop F. Appendix: Solution by Fokker-Planck equation XVI. Nomenclature Acknowledgments References
