Wick’s Theorem beyond the Gaussian

An extension of Wick’s theorem

We propose an extension of a result by Repetowicz et al. (Repetowicz 2004) about Wick’s theorem and its applications: we first show that Wick’s theorem can be extended to the uniform distribution on the sphere and then to the whole class of elliptical distributions. Then, as a special case, we detail this result for distributions that are scale mixtures of Gaussians. Finally, we show that these...

Non-Gaussian generalizations of Wick’s theorems, related to the Schwinger-Dyson equation.

Cumulant operators and moments of the Itô and Skorohod integrals

where the sum runs over the partitions B1, . . . , Ba of {1, . . . , n} with cardinal |Bi| by the Faà di Bruno formula, cf. [5], [6] and references therein for background on combinatorial probability. When X is centered Gaussian, e.g. X is the Wiener integral of a deterministic function with respect to a standard Brownian motion (Bt)t∈R+ , we have κ X n = 0, n 6= 2, and (1.1) reads as Wick’s th...

The Enumeration of Planar Graphs via Wick’s Theorem

A seminal technique of theoretical physics calledWick’s theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enume...

Math 247a: Introduction to Random Matrix Theory

1. Wigner Matrices 2 2. Wigner’s Semicircle Law 4 3. Markov’s Inequality and Convergence of Expectation 6 4. First Proof of Wigner’s Semicircle Law 7 4.1. Convergence of matrix moments in Expectation 7 4.2. Convergence of Matrix Moments in Probability 12 4.3. Weak Convergence 15 5. Removing Moment Assumptions 17 6. Largest Eigenvalue 21 7. The Stiletjes Transform 31 8. The Stieltjes Transform a...