Correlation kernels for sums and products of random matrices

Let $X$ be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of $GX$ and $TX$, where $G$ is a complex Ginibre matrix and $T$ is a truncated unitary matrix. We also consider the product of $X$ and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum $H + M$ where $H$ is a GUE matrix and $M$ is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of $H + M$ follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.Comment: 33 pages, some changes suggested by the referee is made and some references are adde

Asymptotics for a special solution to the second member of the Painleve I hierarchy

We study the asymptotic behavior of a special smooth solution y(x,t) to the second member of the Painleve I hierarchy. This solution arises in random matrix theory and in the study of Hamiltonian perturbations of hyperbolic equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is known and relatively simple, but it turns out to be more subtle when x and t tend to infinity simultaneously. We distinguish a region of algebraic asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain rigorous asymptotics in both regions. We also discuss two critical transitional asymptotic regimes.Comment: 19 page

Is the COVID-19 crisis an opportunity to boost the euro as a global currency? Bruegel Policy Contribution Issue n ̊11 | June 2020.

The euro became an international currency when it was created two decades ago. However, the euro's internationalisation peaked as early as 2005 and it was never comparable to the US dollar. Its international status declined with the euro crisis. Faced with a US administration willing to use its hegemonic currency to extend its domestic policies beyond its borders, Europe is reflecting on how to promote actively the internationalisation of the euro, to help ensure its autonomy. But promoting a more prominent role for the euro is difficult and would involve far-reaching changes to the fabric of the monetary union. Historically, countries issuing dominant currencies have been characterised by: a large and growing economy, free movement of capital, a willingness to play an international role, stability, an ability to provide a large and elastic supply of safe assets, developed financial markets, and significant geopolitical and/or military power. The monetary union does not meet all these criteria. The only way for the euro to play a major international role is to improve the institutional setup of the monetary union. First, the supply of euro-denominated safe assets from the monetary union should be increased. To avoid a COVID-19 depression, euro-area countries have increased massively the supply of their debt securities in the last two months. With its new purchase programme, the European Central Bank has ensured that euro-area sovereign bonds retain their safe asset status. Decisions by the Eurogroup also increase the supply of common European safe assets. The European Commission’s proposal to issue up to €750 billion in EU debt to finance its recovery plan is a step in the right direction In federations, joint issuance typically goes hand-in-hand with federal and central control of spending and a strong grip on revenues. To be politically sustainable, similar central control would be needed in the EU. The treaty-based EU framework is the closest to fulfilling these criteria with political accountability through the European Parliament, political control via the Commission, and a court of auditors and an anti-fraud office, but ultimately the treaty base is insufficient for a true quantum leap. It is essential to ensure a strong recovery for all countries and thus make the euro area an attractive destination for investment. A strong recovery will also be fundamental to preserve or even improve the supply of safe assets, as growth is crucial for debt sustainability

Benefits and drawbacks of European Unemployment Insurance. Bruegel Policy Brief 2014/06, September 2014

The issue: Unemployment in Europe has increased to high levels and economic growth has remained subdued. A debate on additional policy instruments to address the situation is therefore warranted. Fiscal stabilisation mechanisms have not provided adequate fiscal stabilisation during the crisis in some countries nor in the euro area as a whole. Different preferences and historical developments mean that national labour markets are differently organised, which sometimes hinders the efficient working of the monetary union. European Unemployment Insurance (EUI) has been proposed as a measure to contribute to fiscal policy management and improve labour markets. | Read more at Bruegel http://www.bruegel.org/publications/publication-detail/publication/847-benefits-and-drawbacks-of-european-unemployment-insurance

Universality of a double scaling limit near singular edge points in random matrix models

We consider unitary random matrix ensembles Z_{n,s,t}^{-1}e^{-n tr V_{s,t}(M)}dM on the space of Hermitian n x n matrices M, where the confining potential V_{s,t} is such that the limiting mean density of eigenvalues (as n\to\infty and s,t\to 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P_I^2 equation, which is a fourth order analogue of the Painleve I equation. In order to prove our result, we use the well-known connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P_I^2 equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights e^{-nV_{s,t}} on \mathbb{R}. The special solution of the P_I^2 equation pops up in the n^{-2/7}-term of the asymptotics.Comment: 32 pages, 3 figure

Addressing weak inflation: The European Central Bank's shopping list. Bruegel Policy Contribution 2014/05, May 2014

There are clear benefits to price stability. High inflation can distort corporate investment decisions and the consumption behaviour of households. Changes to inflation redistribute real wealth and income between different segments of society, such as savers and borrowers, or young and old. Price stability is therefore a fundamental public good and it became a fundamental principle of European Economic and Monetary Union. But the European Treaties do not define price stability. It was left to the Governing Council of the European Central Bank (ECB) to quantify it: "Price stability is defined as a year-on-year increase in the Harmonised Index of Consumer Prices (HICP) for the euro area of below 2%"[1]. The Governing Council has also clarified that it aims to maintain inflation below, but close to, two percent over the medium term, though it has not quantified what 'closeness' means, nor has it given a precise definition of the 'medium term'[2]. The clarification has been widely interpreted to mean that the actual target of the ECB is close to, but below, two percent inflation in the medium term

Double scaling limits of random matrices and minimal (2m,1) models: the merging of two cuts in a degenerate case

In this article, we show that the double scaling limit correlation functions of a random matrix model when two cuts merge with degeneracy $2m$ (i.e. when $y\sim x^{2m}$ for arbitrary values of the integer $m$) are the same as the determinantal formulae defined by conformal $(2m,1)$ models. Our approach follows the one developed by Berg\`{e}re and Eynard in \cite{BergereEynard} and uses a Lax pair representation of the conformal $(2m,1)$ models (giving Painlev\'e II integrable hierarchy) as suggested by Bleher and Eynard in \cite{BleherEynard}. In particular we define Baker-Akhiezer functions associated to the Lax pair to construct a kernel which is then used to compute determinantal formulae giving the correlation functions of the double scaling limit of a matrix model near the merging of two cuts.Comment: 37 pages, 4 figures. Presentation improved, typos corrected. Published in Journal Of Statistical Mechanic