The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.

Analytic Functions of Several Complex Variables

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Principles Of Random Walk

This review covers recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT). We introduce several RMT methods and analytical techniques, such as the Replica formalism and Free Probability, with an emphasis on the Marchenko-Pastur equation that provides information on the resolvent of multiplicatively corrupted noisy matrices. Special care is devoted to the statistics of the eigenvectors of the empirical correlation matrix, which turn out to be crucial for many applications. We show in particular how these results can be used to build consistent "Rotationally Invariant" estimators (RIE) for large correlation matrices when there is no prior on the structure of the underlying process. The last part of this review is dedicated to some real-world applications within financial markets as a case in point. We establish empirically the efficacy of the RIE framework, which is found to be superior in this case to all previously proposed methods. The case of additively (rather than multiplicatively) corrupted noisy matrices is also dealt with in a special Appendix. Several open problems and interesting technical developments are discussed throughout the paper.

Cleaning large correlation matrices: tools from random matrix theory

Book review: Random walks in the quarter-plane

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

Martin boundary of random walks in convex cones

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.

Constructing discrete harmonic functions in wedges

We investigate the asymptotic behavior of random paths on a graded graph which describes the subword order for words in two letters. This graph, denoted by $\mathcal{Z}$, has been introduced by Viennot, who also discovered a remarkable bijection between paths on $\mathcal{Z}$ and sequences of permutations. Later on, Gnedin and Olshanski used this bijection to describe the set of Gibbs measures on this graph. Both authors also conjectured that the Martin boundary of $\mathcal{Z}$ should coincide with its minimal boundary. We give here a proof of this conjecture by describing the distribution of a large random path conditioned on having a prescribed endpoint. We also relate paths on the graph $\mathcal{Z}$ with paths on the Young lattice, and we finally give a central limit theorem for the Plancherel measure on the set of paths in $\mathcal{Z}$.

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Zigzag diagrams and Martin boundary

The present paper implements a complex analytic method to recover the spectrum of a matrix perturbed by either the addition or the multiplication of a random matrix noise, under the assumption that the distribution of the noise is unitarily invariant. This method, introduced by Arizmendi, Tarrago and Vargas in arXiv:1711.08871, is done in two steps : the first step consists in a fixed point method to compute the Stieltjes transform of the desired distribution in a certain domain, and the second step is a classical deconvolution by a Cauchy distribution, whose parameter depends on the intensity of the noise. We also provide explicit bounds for the mean squared error of the first step.

Spectral deconvolution of unitarily invariant matrix models

We study the asymptotic behavior of zero-drift random walks confined to multidimensional convex cones, when the endpoint is close to the boundary. We derive a local limit theorem in the fluctuation regime.

Pierre Tarrago

Papers

Martin boundary of random walks in convex cones

Constructing discrete harmonic functions in wedges

Zigzag diagrams and Martin boundary

Spectral deconvolution of unitarily invariant matrix models

Boundary behavior of random walks in cones