Showing papers by "Van Vu published in 2020"

Random matrix products: Universality and least singular values

Abstract: We establish, under a moment matching hypothesis, the local universality of the correlation functions associated with products of $M$ independent i.i.d. random matrices, as $M$ is fixed, and the sizes of the matrices tend to infinity. This generalizes an earlier result of Tao and the third author for the case $M=1$. We also prove Gaussian limits for the centered linear spectral statistics of products of $M$ independent i.i.d. random matrices. This is done in two steps. First, we establish the result for product random matrices with Gaussian entries, and then extend to the general case of non-Gaussian entries by another moment matching argument. Prior to our result, Gaussian limits were known only for the case $M=1$. In a similar fashion, we establish Gaussian limits for the centered linear spectral statistics of products of independent truncated random unitary matrices. In both cases, we are able to obtain explicit expressions for the limiting variances. The main difficulty in our study is that the entries of the product matrix are no longer independent. Our key technical lemma is a lower bound on the least singular value of the translated linearization matrix associated with the product of $M$ normalized independent random matrices with independent and identically distributed sub-Gaussian entries. This lemma is of independent interest.

Recent progress in combinatorial random matrix theory

Abstract: We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.

Random Eigenfunctions on Flat Tori: Universality for the Number of Intersections

Abstract: We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues with a given smooth curve in flat tori are universal under various families of randomness.

Central limit theorems for the real zeros of Weyl polynomials

Abstract: We establish the central limit theorem for the number of real roots of the Weyl polynomial $P_n(x)=xi_0 + xi_1 x+ ... + xi_n (n!)^{(-1/2)} x^n$, where $xi_i$ are iid Gaussian random variables. The main ingredients in the proof are new estimates for the correlation functions of the real roots of $P_n$ and a comparison argument exploiting local laws and repulsion properties of these real roots.

Do banks learn from financial markets? Evidence from loan contract pricing

Abstract: This paper examines the feedback effect between trading in financial markets and bank loan contracting. We find that banks charge higher loan rates for borrowers with higher short selling activities. This result is robust to various identification tests and robustness checks. We further find that this effect is not driven by bear raid risk but rather the information content of short selling. Supporting this argument, our result is stronger when the information asymmetry between the bank and the borrowing firm is greater. Overall, our findings suggest that the information content of short selling has important implications for bank loan costs.

Sparse random matrices have simple spectrum

Abstract: On definit une classe $M_{n}$ de matrices symetriques clairsemees, a coefficients independants, en posant $M_{ij}=\delta _{ij}\xi _{ij}$ pour $i\leq j$, ou les $\delta _{ij}$ sont des variables aleatoires de Bernoulli i.i.d. prenant la valeur $1$ avec probabilite $p\geq n^{-1+\delta }$ pour une constante $\delta >0$ arbitraire, et les $\xi _{ij}$ sont des variables aleatoires sous-gaussiennes i.i.d. centrees. Nous montrons qu’avec une grande probabilite, cette classe de matrices aleatoires a un spectre simple, c’est-a-dire que les valeurs propres sont de multiplicite $1$. Une legere modification de la demonstration de ce resultat permet de montrer montrer que la matrice d’adjacence d’un graphe d’Erdős–Renyi clairseme a un spectre simple pour $n^{-1+\delta }\leq p\leq 1-n^{-1+\delta}$. Ces resultats sont optimaux en les exposants. Le resultat pour les graphes a des liens avec le celebre probleme de l’isomorphisme de graphe.

Random orthonormal polynomials: local universality and expected number of real roots

Abstract: We consider random orthonormal polynomials $$ F_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\epsilon)$ moments, and $(p_n)_{n=0}^{\infty}$ is the system of orthonormal polynomials with respect to a fixed compactly supported measure on the real line. Under mild technical assumptions satisfied by many classes of classical polynomial systems, we establish universality for the leading asymptotics of the average number of real roots of $F_n$, both globally and locally. Prior to this paper, these results were known only for random orthonormal polynomials with Gaussian coefficients using the Kac-Rice formula, a method that does not extend to the generality of our paper. The main ingredients in the proof and also the main novelty of this paper are local universality results for the distribution of the roots of $F_n$, where we prove that many local statistics are essentially independent of the distribution of the coefficients.

Do environmental and social risks affect corporate financial policies

Abstract: We investigate the effects of heightened corporate social responsibility (CSR) risk, through environmental and social (E&S) incidents, on corporate financial policies. We test for this relation by investigating changes in corporate financial policies of unaffected but at-risk firms after an E&S incident. We find that firms draw down on cash holding, cut capital expenditure and borrow more debt. The changes in financial policies coincide with an increase in CSR investment, which in turn helps preventing future E&S incidents. The reaction is more pronounced in firms with weaker reputational capital and corporate governance due to their inability to absorb adverse CSR shocks. Overall, our analysis suggests that firms are responsive to heightened CSR risk.