Pieces of nilpotent cones for classical groups

We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato's exotic nilpotent cone. We prove that the number of \F_q-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig's result that corresponding special pieces in types $B_n$ and $C_n$ have the same number of \F_q-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic.Comment: 32 page

Finance and Fear: Sentiment, Media, and Financial Markets During the COVID-19 Pandemic

This thesis aims to build on existing research of market psychology and the effect of sentiment on financial markets. The main objective of this study is to determine the ability of investors to make rational decisions during the most recent period of high sentiment. The anomalies that have occurred in the stock market can be better understood by market psychology which focuses on the biases and social factors that influence investors. The media is a newly relevant factor impacting the volume of sentiment present in the market. A review of literature reveals that many studies of sentiment and financial market’s conclude that emotion has a promitment influence on investors decision. The current pandemic has had detrimental effects on human life and human livelihood. A unique economic situation has emerged as uncertainty increased and resulted in extreme volatility in financial markets that cannot be explained by mainstream financial theories and rational decision making alone. This thesis expands on recent literature about the pandemic and attempts to understand the market movements by looking at a range of explanatory data reflecting panic, sentiment, fake news, and infodemics in the media alongside measures of fundamental economic conditions to assess irrational decisions during the pandemic. I will use these measures to test sentiment and the media\u27s significance on the S&P500 and the 10 year treasury yield in the US for 2020 and 2021. This model identifies when sentiment has the most influence on investment decisions and how. Furthermore, I evaluate how investors treat the bond market differently than the stock market

Distributions of Conductance and Shot Noise and Associated Phase Transitions

For a chaotic cavity with two indentical leads each supporting N channels, we compute analytically, for large N, the full distribution of the conductance and the shot noise power and show that in both cases there is a central Gaussian region flanked on both sides by non-Gaussian tails. The distribution is weakly singular at the junction of Gaussian and non-Gaussian regimes, a direct consequence of two phase transitions in an associated Coulomb gas problem.Comment: 5 pages, 3 figures include

Distribution of reflection eigenvalues in many-channel chaotic cavities with absorption

The reflection matrix R=S^{\dagger}S, with S being the scattering matrix, differs from the unit one, when absorption is finite. Using the random matrix approach, we calculate analytically the distribution function of its eigenvalues in the limit of a large number of propagating modes in the leads attached to a chaotic cavity. The obtained result is independent on the presence of time-reversal symmetry in the system, being valid at finite absorption and arbitrary openness of the system. The particular cases of perfectly and weakly open cavities are considered in detail. An application of our results to the problem of thermal emission from random media is briefly discussed.Comment: 4 pages, 2 figures; (Ref.[5b] added, appropriate modification in text

How often is a random quantum state k-entangled?

The set of trace preserving, positive maps acting on density matrices of size d forms a convex body. We investigate its nested subsets consisting of k-positive maps, where k=2,...,d. Working with the measure induced by the Hilbert-Schmidt distance we derive asymptotically tight bounds for the volumes of these sets. Our results strongly suggest that the inner set of (k+1)-positive maps forms a small fraction of the outer set of k-positive maps. These results are related to analogous bounds for the relative volume of the sets of k-entangled states describing a bipartite d X d system.Comment: 19 pages in latex, 1 figure include

Statistics of S-matrix poles for chaotic systems with broken time reversal invariance: a conjecture

In the framework of a random matrix description of chaotic quantum scattering the positions of $S-$matrix poles are given by complex eigenvalues $Z_i$ of an effective non-Hermitian random-matrix Hamiltonian. We put forward a conjecture on statistics of $Z_i$ for systems with broken time-reversal invariance and verify that it allows to reproduce statistical characteristics of Wigner time delays known from independent calculations. We analyze the ensuing two-point statistical measures as e.g. spectral form factor and the number variance. In addition we find the density of complex eigenvalues of real asymmetric matrices generalizing the recent result by Efetov\cite{Efnh}.Comment: 4 page

Trace distance from the viewpoint of quantum operation techniques

In the present paper, the trace distance is exposed within the quantum operations formalism. The definition of the trace distance in terms of a maximum over all quantum operations is given. It is shown that for any pair of different states, there are an uncountably infinite number of maximizing quantum operations. Conversely, for any operation of the described type, there are an uncountably infinite number of those pairs of states that the maximum is reached by the operation. A behavior of the trace distance under considered operations is studied. Relations and distinctions between the trace distance and the sine distance are discussed.Comment: 26 pages, no figures. The bibliography is extended, explanatory improvement

Almost-Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre eigenvalue statistics

By using the method of orthogonal polynomials we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner- Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures (as e.g. spectral form factor, number variance and small distance behavior of the nearest neighbor distance distribution $p(s)$) are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior $p(s)\propto s^{5/2}$ for some parameter values.Comment: 4 pages, RevTE

Integrable theory of quantum transport in chaotic cavities

The problem of quantum transport in chaotic cavities with broken time-reversal symmetry is shown to be completely integrable in the universal limit. This observation is utilised to determine the cumulants and the distribution function of conductance for a cavity with ideal leads supporting an arbitrary number $n$ of propagating modes. Expressed in terms of solutions to the fifth Painlev\'e transcendent and/or the Toda lattice equation, the conductance distribution is further analysed in the large-$n$ limit that reveals long exponential tails in the otherwise Gaussian curve.Comment: 4 pages; final version to appear in Physical Review Letter