Outliers in the Single Ring Theorem

This text is about spiked models of non Hermitian random matrices. More specifically, we consider matrices of the type $A+P$, where the rank of $P$ stays bounded as the dimension goes to infinity and where the matrix $A$ is a non Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if $P$ has some eigenvalues out of the maximal circle of the single ring, then $A+P$ has some eigenvalues (called outliers) in the neighborhood of those of $P$, which is not the case for the eigenvalues of $P$ in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of $A$ around the eigenvalues of $P$ and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such facts had already been noticed for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of $P$) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in the Hermitian case, but only in the case of non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a previous result by Tao for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new.Comment: Version v2 contains a major improvement with respect to the first one: we now consider the general case for fluctuations of the outliers. In version v4, we slightly weakened the hypotheses. In v5, we simplified notation and added a remark about the real case. 42 pages, 4 figures, to appear in Probab. Theory Related Field

Fluctuations for analytic test functions in the Single Ring Theorem

We consider a non-Hermitian random matrix $A$ whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the empirical eigenvalue distribution of $A$ converges to a limit measure supported by a ring $S$. In this text, we establish the convergence in distribution of random variables of the type $Tr (f(A)M)$ where $f$ is analytic on $S$ and the Frobenius norm of $M$ has order $\sqrt{N}$. As corollaries, we obtain central limit theorems for linear spectral statistics of $A$ (for analytic test functions) and for finite rank projections of $f(A)$ (like matrix entries). As an application, we locate outliers in multiplicative perturbations of $A$.Comment: 29 pages, 1 figure. In Version v2, we slightly modified the assumptions, in order to fix a problem un the control of the tails (see Assumption 2.3). In v3, some minors typos were corrected. In v4, some explanations were added in the introduction and some typos were corrected. To appear in Indiana Univ. Math.

Note on 2-rational fields

We compute the Galois group of the maximal 2-ramified and complexified pro-2-extension of any 2-rational number field.Comment: This Note is motivated by the paper ``Galois 2-extensions unramified outside 2'' of J. Jossey. We bring into focus some classical technics on abelian $\ell$-ramification which considerably simplify proofs in such subjects; for instance, the main Theorem 2, due to J-F. Jaulent, generalizes the purpose of Jossey's pape

On Debt Service and Renegotiation when Debt-holders Are More Strategic

The contingent claims analysis of the firm financing often presents a debt renegotiation game with a passive bank which does not use strategically its capability to force liquidation, contrary towhat is observed in practice. The first purpose of this paper is to introduce more strategic bank behaviour into the continuous-time model developed by Mella-Barral and Perraudin (1997) and Hackbarth, Hennessy, and Leland (2007). Its second purpose is to account for variations in the information obtained by the parties during the contract period. We show that with public information and private debt only, the optimal probability of debt renegotiation is fixed by the firm's anticipated liquidation value. When we add public debt and asymmetric information, the good-type firm may be tempted to mimic the bad-type to reduce its debt service. We show that to deter such mimicking, banks may sometimes refuse to renegotiate with strong firms having a low liquidation value. Our results are in line with the empirical observation that recovery rate at emergence of bankruptcy is function of the share of private debt in all the firm's debt and is relatively low.Debt service, debt renegotiation, recovery rate, strategic bank, bankruptcy, contingent claim

The role of storm flows in concentration of pesticides associated with particulate and dissolved fractions as a threat to aquatic ecosystems - Case study: the agricultural watershed of Save river (Southwest of France)

Measurement of the fluxes of pesticides was carried out for a year, ending in March 2009, in the Save catchment, in the vicinity of Toulouse. The hydrograph separation technique was used to evaluate the respective contribution of stormflow and baseflow in transport of 12 pesticide molecules. Transport of over 59% of pesticides and their controlling factors such as total suspended matter (TSM), particulate organic carbon (POC) and dissolved organic carbon (DOC) occurred during storm periods. Hysteresis patterns could be observed in the concentration-discharge relationships only for some molecules between rising and falling periods of the storm hydrograph. Clockwise hysteresis was noticed for low to moderately soluble pesticide molecules and for particulate fractions, which explains the role of surface runoff in pesticide displacement. In contrast, anticlockwise hysteresis was registered for soluble molecules and dissolved fractions, explaining the role of subsurface flows and soil leaching processes. The important role of TSM, POC and DOC in pesticide transport was clearly established. We also came to the conclusion that the role of stormy periods in pesticide movement and their controlling factors worked as a threat to aquatic ecosystems. And there was a positive relation between riverine TSM, POC, DOC and pesticides according to pesticide properties

A Structural Balance Sheet Model of Sovereign Credit Risk

This article studies sovereign credit spreads using a contingent claims model and a balance sheet representation of the sovereign economy. Analytical formulae for domestic and external debt values as well as for the financial guarantee are derived in a framework where recovery rate is endogenously determined as the solution of a strategic bargaining game. The approach allows to relate sovereign credit spreads to observable macroeconomic factors, and in particular accounts for contagion effects through the corporate and banking sectors. Pricing performance as well as predictions about credit spread determinants are successfully tested on the Brazilian economy.Sovereign credit spread, Balance sheet, Recovery rate, Contingent claims analysis, Contagion effects

Modeling Stiffness and Damping in Rotational Degrees of Freedom Using Multibond Graphs

A contribution is proposed for the modeling of mechanical systems using multibond graphs. When modeling a physical system, it may be needed to catch the dynamic behavior contribution of the joints between bodies of the system and therefore to characterize the stiffness and damping of the links between them. The visibility of where dissipative or capacitive elements need to be implemented to represent stiffness and damping in multibond graphs is not obvious and will be explained. A multibond graph architecture is then proposed to add stiffness and damping in hree rotational degrees of freedom. The resulting joint combines the spherical joint multibond graph relaxed causal constraints while physically representing three concatenated revolute joints. The mathematical foundations are presented, and then illustrated through the modeling and simulation of an inertial navigation system; in which stiffness and damping between the gimbals are taken into account. This method is particularly useful when modeling and simulating multibody systems using Newton-Euler formalism in multibond graphs. Future work will show how this method can be extended to more complex systems such as rotorcraft blades' connections with its rotor hub.Fondation Airbus Grou

Structural changes in lipid-free humic acids during composting of sewage sludge

Structural changes in humic acids (HAs), extracted after lipid removal from sewage sludge during composting, were investigated using various chemical methods (elemental analysis, Fourier transform infrared spectroscopy and 13C-nuclear magnetic resonance (NMR) spectroscopy). Compared to non-purified HAs, lipid-free HAs (LFHAs) exhibit higher C and N contents and high absorbance around 1652, 1540 and 1230 cm1, which indicates the intensity of the etherified aromatic structures and nitrogencontaining components. Less absorbance around 2920, 1600, 1414 and 1100 cm1 could be assigned to their low level of aliphatic compounds, mainly those with a carboxyl group. According to 13C-NMR spectroscopy, almost 45% of aliphatic structures are removed by lipid extraction and these correspond mainly to long-chain fatty acids. During composting, significant decomposition of non-substituted alkyl structures and N-containing components occurred, increasing the relative intensity of etherified aromatic structures

Point-record incentives, asymmetric information and dynamic data

Les politiques de sécurité routière utilisent souvent des mécanismes incitatifs basés sur les infractions pour améliorer le comportement des conducteurs. Ces mécanismes sont soit monétaires (amendes, primes d'assurance), soit non monétaires (permis à points). Nous utilisons des données québécoises couvrant une période allant de 1983 à 1996 pour analyser l'efficacité incitative de ces mécanismes. Nous analysons leurs propriétés théoriques par rapport au nombre de points associés aux infractions et par rapport au temps contrat. Ces propriétés sont ensuite testées empiriquement. Nous comparons l'efficacité globale des différents mécanismes incitatifs et nous relions les résultats obtenus avec les propriétés de la relation entre l'effort de conduite prudente et le risque d'infractions. Nous concluons à la présence d'aléa moral dans les données. Par ailleurs, la prime indicée sur les points introduite en 1992 a réduit de 15% la fréquence d'infractions.