arXiv subjects

Pierpaolo Vivo

Publications and source records attributed to Pierpaolo Vivo.

At least 73 records · Page 4Linked to original sources

arXiv2016

Large deviations of radial statistics in the two-dimensional one-component plasma

The two-dimensional one-component plasma is an ubiquitous model for several vortex systems. For special values of the coupling constant $βq^2$ (where $q$ is the particles charge and $β$ the inverse temperature), the model also corresponds to the eigenvalues distribution of normal matrix models. Several features of the system are discussed in the limit of large number $N$ of particles for generic values of the coupling constant. We show that the statistics of a class of radial observables produces a rich phase diagram, and their asymptotic behaviour in terms of large deviation functions is calculated explicitly, including next-to-leading terms up to order 1/N. We demonstrate a split-off phenomenon associated to atypical fluctuations of the edge density profile. We also show explicitly that a failure of the fluid phase assumption of the plasma can break a genuine $1/N$-expansion of the free energy. Our findings are corroborated by numerical comparisons with exact finite-N formulae valid for $βq^2=2$.

math-ph↗

arXiv2016

Random pure states: quantifying bipartite entanglement beyond the linear statistics

We analyze the properties of entangled random pure states of a quantum system partitioned into two smaller subsystems of dimensions $N$ and $M$. Framing the problem in terms of random matrices with a fixed-trace constraint, we establish, for arbitrary $N \leq M$, a general relation between the $n$-point densities and the cross-moments of the eigenvalues of the reduced density matrix, i.e. the so-called Schmidt eigenvalues, and the analogous functionals of the eigenvalues of the Wishart-Laguerre ensemble of the random matrix theory. This allows us to derive explicit expressions for two-level densities, and also an exact expression for the variance of von Neumann entropy at finite $N,M$. Then we focus on the moments $\mathbb{E}\{K^a\}$ of the Schmidt number $K$, the reciprocal of the purity. This is a random variable supported on $[1,N]$, which quantifies the number of degrees of freedom effectively contributing to the entanglement. We derive a wealth of analytical results for $\mathbb{E}\{K^a\}$ for $N = 2$ and $N=3$ and arbitrary $M$, and also for square $N = M$ systems by spotting for the latter a connection with the probability $P(x_{min}^{GUE} \geq \sqrt{2N}ξ)$ that the smallest eigenvalue $x_{min}^{GUE}$ of a $N\times N$ matrix belonging to the Gaussian Unitary Ensemble is larger than $\sqrt{2N}ξ$. As a byproduct, we present an exact asymptotic expansion for $P(x_{min}^{GUE} \geq \sqrt{2N}ξ)$ for finite $N$ as $ξ\to \infty$. Our results are corroborated by numerical simulations whenever possible, with excellent agreement.

cond-mat.stat-mech↗

arXiv2016

Large deviations of spread measures for Gaussian matrices

For a large $n\times m$ Gaussian matrix, we compute the joint statistics, including large deviation tails, of generalized and total variance - the scaled log-determinant $H$ and trace $T$ of the corresponding $n\times n$ covariance matrix. Using a Coulomb gas technique, we find that the Laplace transform of their joint distribution $\mathcal{P}_n(h,t)$ decays for large $n,m$ (with $c=m/n\geq 1$ fixed) as $\hat{\mathcal{P}}_n(s,w)\approx \exp\left(-βn^2 J(s,w)\right)$, where $β$ is the Dyson index of the ensemble and $J(s,w)$ is a $β$-independent large deviation function, which we compute exactly for any $c$. The corresponding large deviation functions in real space are worked out and checked with extensive numerical simulations. The results are complemented with a finite $n,m$ treatment based on the Laguerre-Selberg integral. The statistics of atypically small log-determinants is shown to be driven by the split-off of the smallest eigenvalue, leading to an abrupt change in the large deviation speed.

cond-mat.stat-mech↗

arXiv2016

Correlators for the Wigner-Smith time-delay matrix of chaotic cavities

We study the Wigner-Smith time-delay matrix $Q$ of a ballistic quantum dot supporting $N$ scattering channels. We compute the $v$-point correlators of the power traces $\mathrm{Tr} Q^κ$ for arbitrary $v\geq1$ at leading order for large $N$ using techniques from the random matrix theory approach to quantum chromodynamics. We conjecture that the cumulants of the $\mathrm{Tr} Q^κ$'s are integer-valued at leading order in $N$ and include a MATHEMATICA code that computes their generating functions recursively.

math-ph↗

arXiv2016

Universal covariance formula for linear statistics on random matrices

We derive an analytical formula for the covariance $\mathrm{Cov}(A,B)$ of two smooth linear statistics $A=\sum_i a(λ_i)$ and $B=\sum_i b(λ_i)$ to leading order for $N\to\infty$, where $\{λ_i\}$ are the $N$ real eigenvalues of a general one-cut random-matrix model with Dyson index $β$. The formula, carrying the universal $1/β$ prefactor, depends on the random-matrix ensemble only through the edge points $[λ_-,λ_+]$ of the limiting spectral density. For $A=B$, we recover in some special cases the classical variance formulas by Beenakker and Dyson-Mehta, clarifying the respective ranges of applicability. Some choices of $a(x)$ and $b(x)$ lead to a striking \emph{decorrelation} of the corresponding linear statistics. We provide two applications - the joint statistics of conductance and shot noise in ideal chaotic cavities, and some new fluctuation relations for traces of powers of random matrices.

cond-mat.stat-mech↗

arXiv2016

A shortcut through the Coulomb gas method for spectral linear statistics on random matrices

In the last decade, spectral linear statistics on large dimensional random matrices have attracted significant attention. Within the physics community, a privileged role has been played by invariant matrix ensembles for which a two dimensional Coulomb gas analogy is available. We present a critical revision of the Coulomb gas method in Random Matrix Theory (RMT) borrowing language and tools from Large Deviations Theory. This allows us to formalize an equivalent, but more effective and quicker route toward RMT free energy calculations. Moreover, we argue that this more modern viewpoint is likely to shed further light on the interesting issues of weak phase transitions and evaporation phenomena recently observed in RMT.

math-ph↗

arXiv2016

Number statistics for $β$-ensembles of random matrices: applications to trapped fermions at zero temperature

Let $\mathcal{P}_β^{(V)} (N_{\cal I})$ be the probability that a $N\times N$ $β$-ensemble of random matrices with confining potential $V(x)$ has $N_{\cal I}$ eigenvalues inside an interval ${\cal I}=[a,b]$ of the real line. We introduce a general formalism, based on the Coulomb gas technique and the resolvent method, to compute analytically $\mathcal{P}_β^{(V)} (N_{\cal I})$ for large $N$. We show that this probability scales for large $N$ as $\mathcal{P}_β^{(V)} (N_{\cal I})\approx \exp\left(-βN^2 ψ^{(V)}(N_{\cal I} /N)\right)$, where $β$ is the Dyson index of the ensemble. The rate function $ψ^{(V)}(k_{\cal I})$, independent of $β$, is computed in terms of single integrals that can be easily evaluated numerically. The general formalism is then applied to the classical $β$-Gaussian (${\cal I}=[-L,L]$), $β$-Wishart (${\cal I}=[1,L]$) and $β$-Cauchy (${\cal I}=[-L,L]$) ensembles. Expanding the rate function around its minimum, we find that generically the number variance ${\rm Var}(N_{\cal I})$ exhibits a non-monotonic behavior as a function of the size of the interval, with a maximum that can be precisely characterized. These analytical results, corroborated by numerical simulations, provide the full counting statistics of many systems where random matrix models apply. In particular, we present results for the full counting statistics of zero temperature one-dimensional spinless fermions in a harmonic trap.

cond-mat.stat-mech↗

arXiv2015

Large deviations of the maximum of independent and identically distributed random variables

A pedagogical account of some aspects of Extreme Value Statistics (EVS) is presented from the somewhat non-standard viewpoint of Large Deviation Theory. We address the following problem: given a set of $N$ i.i.d. random variables $\{X_1,\ldots,X_N\}$ drawn from a parent probability density function (pdf) $p(x)$, what is the probability that the maximum value of the set $X_{\mathrm{max}}=\max_i X_i$ is "atypically larger" than expected? The cases of exponential and Gaussian distributed variables are worked out in detail, and the right rate function for a general pdf in the Gumbel basin of attraction is derived. The Gaussian case convincingly demonstrates that the full rate function cannot be determined from the knowledge of the limiting distribution (Gumbel) alone, thus implying that it indeed carries additional information. Given the simplicity and richness of the result and its derivation, its absence from textbooks, tutorials and lecture notes on EVS for physicists appears inexplicable.

cond-mat.stat-mech↗

arXiv2015

Invariant sums of random matrices and the onset of level repulsion

We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary) and yet the interaction between eigenvalues is not Vandermondian. The ensemble contains real symmetric or complex hermitian matrices $\mathbf{S}$ of the form $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{O}_i \mathbf{D}_i\mathbf{O}_i^{\mathrm{T}}\rangle$ or $\mathbf{S}=\sum_{i=1}^M \langle \mathbf{U}_i \mathbf{D}_i\mathbf{U}_i^\dagger\rangle$ respectively. The diagonal matrices $\mathbf{D}_i=\mathrm{diag}\{λ_1^{(i)},\ldots,λ_N^{(i)}\}$ are constructed from real eigenvalues drawn \emph{independently} from distributions $p^{(i)}(x)$, while the matrices $\mathbf{O}_i$ and $\mathbf{U}_i$ are all orthogonal or unitary. The average $\langle\cdot\rangle$ is simultaneously performed over the symmetry group and the joint distribution of $\{λ_j^{(i)}\}$. We focus on the limits i.) $N\to\infty$ and ii.) $M\to\infty$, with $N=2$. In the limit i.), the resulting sum $\mathbf{S}$ develops level repulsion even though the original matrices do not feature it, and classical RMT universality is restored asymptotically. In the limit ii.) the spacing distribution attains scaling forms that are computed exactly: for the orthogonal case, we recover the $β=1$ Wigner's surmise, while for the unitary case an entirely new universal distribution is obtained. Our results allow to probe analytically the microscopic statistics of the sum of random matrices that become asymptotically free. We also give an interpretation of this model in terms of radial random walks in a matrix space. The analytical results are corroborated by numerical simulations.

cond-mat.stat-mech↗

arXiv2015

Joint statistics of quantum transport in chaotic cavities

We study the joint statistics of conductance $G$ and shot noise $P$ in chaotic cavities supporting a large number $N$ of open electronic channels in the two attached leads. We determine the full phase diagram in the $(G,P)$ plane, employing a Coulomb gas technique on the joint density of transmission eigenvalues, as dictated by Random Matrix Theory. We find that in the region of typical fluctuations, conductance and shot noise are uncorrelated and jointly Gaussian, and away from it they fluctuate according to a different joint rate function in each phase of the $(G,P)$ plane. Different functional forms of the rate function in different regions emerge as a direct consequence of third order phase transitions in the associated Coulomb gas problem.

cond-mat.stat-mech↗

arXiv2015

A unified fluctuation formula for one-cut $β$-ensembles of random matrices

Using a Coulomb gas approach, we compute the generating function of the covariances of power traces for one-cut $β$-ensembles of random matrices in the limit of large matrix size. This formula depends only on the support of the spectral density, and is therefore universal for a large class of models. This allows us to derive a closed-form expression for the limiting covariances of an arbitrary one-cut $β$-ensemble. As particular cases of the main result we consider the classical $β$-Gaussian, $β$-Wishart and $β$-Jacobi ensembles, for which we derive previously available results as well as new ones within a unified simple framework. We also discuss the connections between the problem of trace fluctuations for the Gaussian Unitary Ensemble and the enumeration of planar maps.

math-ph↗

arXiv2014

Phase transitions and edge scaling of number variance in Gaussian random matrices

We consider $N\times N$ Gaussian random matrices, whose average density of eigenvalues has the Wigner semi-circle form over $[-\sqrt{2},\sqrt{2}]$. For such matrices, using a Coulomb gas technique, we compute the large $N$ behavior of the probability $\mathcal{P}_{\scriptscriptstyle N,L}(N_L)$ that $N_L$ eigenvalues lie within the box $[-L,L]$. This probability scales as $\mathcal{P}_{\scriptscriptstyle N,L}(N_L=κ_L N)\approx\exp\left(-β N^2 ψ_L(κ_L)\right)$, where $β$ is the Dyson index of the ensemble and $ψ_L(κ_L)$ is a $β$-independent rate function that we compute exactly. We identify three regimes as $L$ is varied: (i) $\, N^{-1}\ll L<\sqrt{2}$ (bulk), (ii) $\ L\sim\sqrt{2}$ on a scale of $\mathcal{O}(N^{-{2}/{3}})$ (edge) and (iii) $\ L > \sqrt{2}$ (tail). We find a dramatic non-monotonic behavior of the number variance $V_N(L)$ as a function of $L$: after a logarithmic growth $\propto \ln (N L)$ in the bulk (when $L \sim {\cal O}(1/N)$), $V_N(L)$ decreases abruptly as $L$ approaches the edge of the semi-circle before it decays as a stretched exponential for $L > \sqrt{2}$. This "drop-off" of $V_N(L)$ at the edge is described by a scaling function $\tilde V_β$ which smoothly interpolates between the bulk (i) and the tail (iii). For $β= 2$ we compute $\tilde V_2$ explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for $β=2$ the full statistics of particle-number fluctuations at zero temperature of 1d spinless fermions in a harmonic trap.

cond-mat.stat-mech↗

arXiv2014

Phase transitions in the condition number distribution of Gaussian random matrices

We study the statistics of the condition number $κ=λ_{\mathrm{max}}/λ_{\mathrm{min}}$ (the ratio between largest and smallest squared singular values) of $N\times M$ Gaussian random matrices. Using a Coulomb fluid technique, we derive analytically and for large $N$ the cumulative $\mathcal{P}[κ x]$ distributions of $κ$. We find that these distributions decay as $\mathcal{P}[κ x]\approx\exp\left(-βN Φ_{+}(x)\right)$, where $β$ is the Dyson index of the ensemble. The left and right rate functions $Φ_{\pm}(x)$ are independent of $β$ and calculated exactly for any choice of the rectangularity parameter $α=M/N-1>0$. Interestingly, they show a weak non-analytic behavior at their minimum $\langleκ\rangle$ (corresponding to the average condition number), a direct consequence of a phase transition in the associated Coulomb fluid problem. Matching the behavior of the rate functions around $\langleκ\rangle$, we determine exactly the scale of typical fluctuations $\sim\mathcal{O}(N^{-2/3})$ and the tails of the limiting distribution of $κ$. The analytical results are in excellent agreement with numerical simulations.

cond-mat.stat-mech↗

arXiv2014

A singular-potential random matrix model arising in mean-field glassy systems

We consider an invariant random matrix model where the standard Gaussian potential is distorted by an additional single pole of order $m$. We compute the average or macroscopic spectral density in the limit of large matrix size, solving the loop equation with the additional constraint of vanishing trace on average. The density is generally supported on two disconnected intervals lying on the two sides of the pole. In the limit of having no pole, we recover the standard semicircle. Obtained in the planar limit, our results apply to matrices with orthogonal, unitary or symplectic symmetry alike. The orthogonal case with $m=2$ is motivated by an application to spin glass physics. In the Sherrington-Kirkpatrick mean-field model, in the paramagnetic phase and for sufficiently large systems the spin glass susceptibility is a random variable, depending on the realization of disorder. It is essentially given by a linear statistics on the eigenvalues of the coupling matrix. As such its large deviation function can be computed using standard Coulomb fluid techniques. The resulting free energy of the associated fluid precisely corresponds to the partition function of our random matrix model. Numerical simulations provide an excellent confirmation of our analytical results.

math-ph↗

arXiv2013

Index Distribution of Cauchy Random Matrices

Using a Coulomb gas technique, we compute analytically the probability $\mathcal{P}_β^{(C)}(N_+,N)$ that a large $N\times N$ Cauchy random matrix has $N_+$ positive eigenvalues, where $N_+$ is called the index of the ensemble. We show that this probability scales for large $N$ as $\mathcal{P}_β^{(C)}(N_+,N)\approx \exp\left[-βN^2 ψ_C(N_+/N)\right]$, where $β$ is the Dyson index of the ensemble. The rate function $ψ_C(κ)$ is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis. We find that the rate function, around its minimum at $κ=1/2$, has a quadratic behavior modulated by a logarithmic singularity. As a consequence, the variance of the index scales for large $N$ as $\mathrm{Var}(N_+)\sim σ_C\ln N$, where $σ_C=2/(βπ^2)$ is twice as large as the corresponding prefactor in the Gaussian and Wishart cases. The analytical results are checked by numerical simulations and against an exact finite $N$ formula which, for $β=2$, can be derived using orthogonal polynomials.

cond-mat.stat-mech↗

arXiv2013

Outage Capacity for the Optical MIMO Channel

MIMO processing techniques in fiber optical communications have been proposed as a promising approach to meet increasing demand for information throughput. In this context, the multiple channels correspond to the multiple modes and/or multiple cores in the fiber. In this paper we characterize the distribution of the mutual information with Gaussian input in a simple channel model for this system. Assuming significant cross talk between cores, negligible backscattering and near-lossless propagation in the fiber, we model the transmission channel as a random complex unitary matrix. The loss in the transmission may be parameterized by a number of unutilized channels in the fiber. We analyze the system in a dual fashion. First, we evaluate a closed-form expression for the outage probability, which is handy for small matrices. We also apply the asymptotic approach, in particular the Coulomb gas method from statistical mechanics, to obtain closed-form results for the ergodic mutual information, its variance as well as the outage probability for Gaussian input in the limit of large number of cores/modes. By comparing our analytic results to simulations, we see that, despite the fact that this method is nominally valid for large number of modes, our method is quite accurate even for small to modest number of channels.

cs.IT↗

arXiv2013

Statistics of quantum transport in weakly non-ideal chaotic cavities

We consider statistics of electronic transport in chaotic cavities where time-reversal symmetry is broken and one of the leads is weakly non-ideal, i.e. it contains tunnel barriers characterized by tunneling probabilities $Γ_i$. Using symmetric function expansions and a generalized Selberg integral, we develop a systematic perturbation theory in $1-Γ_i$ valid for arbitrary number of channels, and obtain explicit formulas up to second order for the average and variance of the conductance, and for the average shot-noise. Higher moments of the conductance are considered to leading order.

cond-mat.mes-hall↗

arXiv2013

Entanglement production in non-ideal cavities and optimal opacity

We compute analytically the distributions of concurrence $\bm{\mathcal{C}}$ and squared norm $\bm{\mathcal{N}}$ for the production of electronic entanglement in a chaotic quantum dot. The dot is connected to the external world via one ideal and one partially transparent lead, characterized by the opacity $γ$. The average concurrence increases with $γ$ while the average squared norm of the entangled state decreases, making it less likely to be detected. When a minimal detectable norm $\bm{\mathcal{N}}_0$ is required, the average concurrence is maximal for an optimal value of the opacity $γ^\star(\bm{\mathcal{N}}_0)$ which is explicitly computed as a function of $\bm{\mathcal{N}}_0$. If $\bm{\mathcal{N}}_0$ is larger than the critical value $\bm{\mathcal{N}}_0^\star\simeq 0.3693\dots$, the average entanglement production is maximal for the completely ideal case, a direct consequence of an interesting bifurcation effect.

cond-mat.stat-mech↗