arXiv subjects

Pierpaolo Vivo

Publications and source records attributed to Pierpaolo Vivo.

At least 91 records · Page 5Linked to original sources

arXiv2012

Large deviations of the top eigenvalue of large Cauchy random matrices

We compute analytically the probability density function (pdf) of the largest eigenvalue $λ_{\max}$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($β= 2$), orthogonal ($β=1$) and symplectic ($β=4$) ensembles of such heavy-tailed random matrices. We show that a central non-Gaussian regime for $λ_{\max} \sim \mathcal{O}(N)$ is flanked by large deviation tails on both sides which we compute here exactly for any value of $β$. By matching these tails with the central regime, we obtain the exact leading asymptotic behaviors of the pdf in the central regime, which generalizes the Tracy-Widom distribution known for Gaussian ensembles, both at small and large arguments and for any $β$. Our analytical results are confirmed by numerical simulations.

cond-mat.stat-mech↗

arXiv2012

Invariant $β$-Wishart ensembles, crossover densities and asymptotic corrections to the Marchenko-Pastur law

We construct a diffusive matrix model for the $β$-Wishart (or Laguerre) ensemble for general continuous $β\in [0,2]$, which preserves invariance under the orthogonal/unitary group transformation. Scaling the Dyson index $β$ with the largest size $M$ of the data matrix as $β=2c/M$ (with $c$ a fixed positive constant), we obtain a family of spectral densities parametrized by $c$. As $c$ is varied, this density interpolates continuously between the Mar\vcenko-Pastur ($c\to \infty$ limit) and the Gamma law ($c\to 0$ limit). Analyzing the full Stieltjes transform (resolvent) equation, we obtain as a byproduct the correction to the Mar\vcenko-Pastur density in the bulk up to order 1/M for all $β$ and up to order $1/M^2$ for the particular cases $β=1,2$.

cond-mat.stat-mech↗

arXiv2012

Equilibrium strategy and population-size effects in lowest unique bid auctions

In lowest unique bid auctions, $N$ players bid for an item. The winner is whoever places the \emph{lowest} bid, provided that it is also unique. We use a grand canonical approach to derive an analytical expression for the equilibrium distribution of strategies. We then study the properties of the solution as a function of the mean number of players, and compare them with a large dataset of internet auctions. The theory agrees with the data with striking accuracy for small population size $N$, while for larger $N$ a qualitatively different distribution is observed. We interpret this result as the emergence of two different regimes, one in which adaptation is feasible and one in which it is not. Our results question the actual possibility of a large population to adapt and find the optimal strategy when participating in a collective game.

cs.GT↗

arXiv2011

Number of relevant directions in Principal Component Analysis and Wishart random matrices

We compute analytically, for large $N$, the probability $\mathcal{P}(N_+,N)$ that a $N\times N$ Wishart random matrix has $N_+$ eigenvalues exceeding a threshold $Nζ$, including its large deviation tails. This probability plays a benchmark role when performing the Principal Component Analysis of a large empirical dataset. We find that $\mathcal{P}(N_+,N)\approx\exp(-βN^2 ψ_ζ(N_+/N))$, where $β$ is the Dyson index of the ensemble and $ψ_ζ(κ)$ is a rate function that we compute explicitly in the full range $0\leq κ\leq 1$ and for any $ζ$. The rate function $ψ_ζ(κ)$ displays a quadratic behavior modulated by a logarithmic singularity close to its minimum $κ^\star(ζ)$. This is shown to be a consequence of a phase transition in an associated Coulomb gas problem. The variance $Δ(N)$ of the number of relevant components is also shown to grow universally (independent of $ζ)$ as $Δ(N)\sim (βπ^2)^{-1}\ln N$ for large $N$.

cond-mat.stat-mech↗

arXiv2011

Phase transitions in the distribution of the Andreev conductance of superconductor-metal junctions with multiple transverse modes

We compute analytically the full distribution of Andreev conductance $G_{\mathrm{NS}}$ of a metal-superconductor interface with a large number $N_c$ of transverse modes, using a random matrix approach. The probability distribution $\mathcal{P}(G_{\mathrm{NS}},N_c)$ in the limit of large $N_c$ displays a Gaussian behavior near the average value $ = (2-\sqrt{2}) N_c$ and asymmetric power-law tails in the two limits of very small and very large $G_{\mathrm{NS}}$. In addition, we find a novel third regime sandwiched between the central Gaussian peak and the power law tail for large $G_{\mathrm{NS}}$. Weakly non-analytic points separate these four regimes---these are shown to be consequences of three phase transitions in an associated Coulomb gas problem.

cond-mat.mes-hall↗

arXiv2011

Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace

The degree of entanglement of random pure states in bipartite quantum systems can be estimated from the distribution of the extreme Schmidt eigenvalues. For a bipartition of size M\geq N, these are distributed according to a Wishart-Laguerre ensemble (WL) of random matrices of size N x M, with a fixed-trace constraint. We first compute the distribution and moments of the smallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitrary M\geq N. Our method is based on a Laplace inversion of the recursive results for the corresponding orthogonal WL ensemble by Edelman. Explicit examples are given for fixed N and M, generalizing and simplifying earlier results. In the microscopic large-N limit with M-N fixed, the orthogonal and unitary WL distributions exhibit universality after a suitable rescaling and are therefore independent of the constraint. We prove that very recent results given in terms of hypergeometric functions of matrix argument are equivalent to more explicit expressions in terms of a Pfaffian or determinant of Bessel functions. While the latter were mostly known from the random matrix literature on the QCD Dirac operator spectrum, we also derive some new results in the orthogonal symmetry class.

math-ph↗

arXiv2011

Moments of Wishart-Laguerre and Jacobi ensembles of random matrices: application to the quantum transport problem in chaotic cavities

We collect explicit and user-friendly expressions for one-point densities of the real eigenvalues $\{λ_i\}$ of $N\times N$ Wishart-Laguerre and Jacobi random matrices with orthogonal, unitary and symplectic symmetry. Using these formulae, we compute integer moments $τ_n=<\sum_{i=1}^Nλ_i^n>$ for all symmetry classes without any large $N$ approximation. In particular, our results provide exact expressions for moments of transmission eigenvalues in chaotic cavities with time-reversal or spin-flip symmetry and supporting a finite and arbitrary number of electronic channels in the two incoming leads.

cond-mat.stat-mech↗

arXiv2010

Largest Schmidt eigenvalue of entangled random pure states and conductance distribution in chaotic cavities

A strategy to evaluate the distribution of the largest Schmidt eigenvalue for entangled random pure states of bipartite systems is proposed. We point out that the multiple integral defining the sought quantity for a bipartition of sizes N, M is formally identical (upon simple algebraic manipulations) to the one providing the probability density of Landauer conductance in open chaotic cavities supporting N and M electronic channels in the two leads. Known results about the latter can then be straightforwardly employed in the former problem for both systems with broken (β = 2) and preserved (β = 1) time reversal symmetry. The analytical results, yielding a continuous but not everywhere analytic distribution, are in excellent agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2010

How many eigenvalues of a Gaussian random matrix are positive?

We study the probability distribution of the index ${\mathcal N}_+$, i.e., the number of positive eigenvalues of an $N\times N$ Gaussian random matrix. We show analytically that, for large $N$ and large $\mathcal{N}_+$ with the fraction $0\le c=\mathcal{N}_+/N\le 1$ of positive eigenvalues fixed, the index distribution $\mathcal{P}({\mathcal N}_+=cN,N)\sim\exp[-βN^2 Φ(c)]$ where $β$ is the Dyson index characterizing the Gaussian ensemble. The associated large deviation rate function $Φ(c)$ is computed explicitly for all $0\leq c \leq 1$. It is independent of $β$ and displays a quadratic form modulated by a logarithmic singularity around $c=1/2$. As a consequence, the distribution of the index has a Gaussian form near the peak, but with a variance $Δ(N)$ of index fluctuations growing as $Δ(N)\sim \log N/βπ^2$ for large $N$. For $β=2$, this result is independently confirmed against an exact finite $N$ formula, yielding $Δ(N)= \log N/2π^2 +C+\mathcal{O}(N^{-1})$ for large $N$, where the constant $C$ has the nontrivial value $C=(γ+1+3\log 2)/2π^2\simeq 0.185248...$ and $γ=0.5772...$ is the Euler constant. We also determine for large $N$ the probability that the interval $[ζ_1,ζ_2]$ is free of eigenvalues. Part of these results have been announced in a recent letter [\textit{Phys. Rev. Lett.} {\bf 103}, 220603 (2009)].

cond-mat.stat-mech↗

arXiv2010

Entangled random pure states with orthogonal symmetry: exact results

We compute analytically the density $\varrho_{N,M}(λ)$ of Schmidt eigenvalues, distributed according to a fixed-trace Wishart-Laguerre measure, and the average Rényi entropy $\langle\mathcal{S}_q\rangle$ for reduced density matrices of entangled random pure states with orthogonal symmetry $(β=1)$. The results are valid for arbitrary dimensions $N=2k,M$ of the corresponding Hilbert space partitions, and are in excellent agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2010

Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

We investigate the asymptotic behavior of the Selberg-like integral $$ \frac1{N!}\int_{[0,1]^N}x_1^p\prod_{i<j}(x_i-x_j)^2\prod_ix_i^{a-1}(1-x_i)^{b-1}dx_i$$, as $N\to\infty$ for different scalings of the parameters $a$ and $b$ with $N$. Integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting $N$ channels in the two attached leads. Making use of Newton's interpolation formula, we show that an asymptotic limit exists and we compute it explicitly.

math-ph↗

arXiv2010

Probability distributions of Linear Statistics in Chaotic Cavities and associated phase transitions

We establish large deviation formulas for linear statistics on the $N$ transmission eigenvalues $\{T_i\}$ of a chaotic cavity, in the framework of Random Matrix Theory. Given any linear statistics of interest $A=\sum_{i=1}^N a(T_i)$, the probability distribution $\mathcal{P}_A(A,N)$ of $A$ generically satisfies the large deviation formula $\lim_{N\to\infty}[-2\log\mathcal{P}_A(Nx,N)/βN^2]=Ψ_A(x)$, where $Ψ_A(x)$ is a rate function that we compute explicitly in many cases (conductance, shot noise, moments) and $β$ corresponds to different symmetry classes. Using these large deviation expressions, it is possible to recover easily known results and to produce new formulas, such as a closed form expression for $v(n)=\lim_{N\to\infty}\mathrm{var}(\mathcal{T}_n)$ (where $\mathcal{T}_n=\sum_{i}T_i^n$) for arbitrary integer $n$. The universal limit $v^\star=\lim_{n\to\infty} v(n)=1/2πβ$ is also computed exactly. The distributions display a central Gaussian region flanked on both sides by non-Gaussian tails. At the junction of the two regimes, weakly non-analytical points appear, a direct consequence of phase transitions in an associated Coulomb gas problem. Numerical checks are also provided, which are in full agreement with our asymptotic results in both real and Laplace space even for moderately small $N$. Part of the results have been announced in [P. Vivo, S.N. Majumdar and O. Bohigas, {\it Phys. Rev. Lett.} {\bf 101}, 216809 (2008)].

cond-mat.mes-hall↗

arXiv2010

Random Matrix approach to collective behavior and bulk universality in protein dynamics

Covariance matrices of amino acid displacements, commonly used to characterize the large-scale movements of proteins, are investigated through the prism of Random Matrix Theory. Bulk universality is detected in the local spacing statistics of noise-dressed eigenmodes, which is well described by a Brody distribution with parameter $β\simeq 0.8$. This finding, supported by other consistent indicators, implies a novel quantitative criterion to single out the collective degrees of freedom of the protein from the majority of high-energy, localized vibrations.

q-bio.QM↗

arXiv2009

Nonlinear Random Matrix Statistics, symmetric functions and hyperdeterminants

Nonlinear statistics (i.e. statistics of permanents) on the eigenvalues of invariant random matrix models are considered for the three Dyson's symmetry classes $β=1,2,4$. General formulas in terms of hyperdeterminants are found for $β=2$. For specific cases and all $β$s, more computationally efficient results are obtained, based on symmetric functions expansions. As an application, we consider the case of quantum transport in chaotic cavities extending results from [D.V. Savin, H.-J. Sommers and W. Wieczorek, {\it Phys. Rev. B} {\bf 77}, 125332 (2008)].

math-ph↗

arXiv2009

The Index Distribution of Gaussian Random Matrices

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N_{+}) of a random NxN matrix belonging to Gaussian orthogonal (β=1), unitary (β=2) or symplectic (β=4) ensembles. The distribution of the fraction of positive eigenvalues c=N_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-βN^2 Φ(c)] where the rate function Φ(c), symmetric around c=1/2 and universal (independent of $β$), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.

cond-mat.stat-mech↗

arXiv2009

Eroding market stability by proliferation of financial instruments

We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the development of financial instruments, with a dynamical picture of an interacting market, in a simple setting. The proliferation of financial instruments apparently provides more means for risk diversification, making the market more efficient and complete. In the simple market of interacting traders discussed here, the proliferation of financial instruments erodes systemic stability and it drives the market to a critical state characterized by large susceptibility, strong fluctuations and enhanced correlations among risks. This suggests that the hypothesis of APT may not be compatible with a stable market dynamics. In this perspective, market stability acquires the properties of a common good, which suggests that appropriate measures should be introduced in derivative markets, to preserve stability.

q-fin.TR↗

arXiv2009

Universal Correlations and Power-Law Tails in Financial Covariance Matrices

Signatures of universality are detected by comparing individual eigenvalue distributions and level spacings from financial covariance matrices to random matrix predictions. A chopping procedure is devised in order to produce a statistical ensemble of asset-price covariances from a single instance of financial data sets. Local results for the smallest eigenvalue and individual spacings are very stable upon reshuffling the time windows and assets. They are in good agreement with the universal Tracy-Widom distribution and Wigner surmise, respectively. This suggests a strong degree of robustness especially in the low-lying sector of the spectra, most relevant for portfolio selections. Conversely, the global spectral density of a single covariance matrix as well as the average over all unfolded nearest-neighbour spacing distributions deviate from standard Gaussian random matrix predictions. The data are in fair agreement with a recently introduced generalised random matrix model, with correlations showing a power-law decay.

q-fin.ST↗

arXiv2008

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.

cond-mat.mes-hall↗