Intensity statistics inside an open wave-chaotic cavity with broken time-reversal invariance.

Using the supersymmetric method of random matrix theory within the Heidelberg approach framework we provide statistical description of stationary intensity sampled in locations inside an open wave-chaotic cavity, assuming that the time-reversal invariance inside the cavity is fully broken. In particular, we show that when incoming waves are fed via a finite number M of open channels the probability density P(I) for the single-point intensity I decays as a power law for large intensities: P(I)∼I^{-(M+2)}, provided there is no internal losses. This behavior is in marked difference with the Rayleigh law P(I)∼exp(-I/I[over ¯]), which turns out to be valid only in the limit M→∞.

Extreme Eigenvalues and the Emerging Outlier in Rank-One Non-Hermitian Deformations of the Gaussian Unitary Ensemble.

Complex eigenvalues of random matrices J=GUE+iγdiag(1,0,…,0) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions N≫1 the eigenvalue density of undergoes an abrupt restructuring at γ=1, the critical threshold beyond which a single eigenvalue outlier ("broad resonance") appears. We provide a detailed description of this restructuring transition, including the scaling with of the width of the critical region about the outlier threshold γ=1 and the associated scaling for the real parts ("resonance positions") and imaginary parts ("resonance widths") of the eigenvalues which are farthest away from the real axis.

Statistics of Complex Wigner Time Delays as a Counter of S-Matrix Poles: Theory and Experiment.

We study the statistical properties of the complex generalization of Wigner time delay τ_{W} for subunitary wave-chaotic scattering systems. We first demonstrate theoretically that the mean value of the Re[τ_{W}] distribution function for a system with uniform absorption strength η is equal to the fraction of scattering matrix poles with imaginary parts exceeding η. The theory is tested experimentally with an ensemble of microwave graphs with either one or two scattering channels and showing broken time-reversal invariance and variable uniform attenuation.

Counting equilibria of large complex systems by instability index.

We consider a nonlinear autonomous system of [Formula: see text] degrees of freedom randomly coupled by both relaxational ("gradient") and nonrelaxational ("solenoidal") random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of "absolute instability" where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal.

Generalization of Wigner time delay to subunitary scattering systems.

We introduce a complex generalization of the Wigner time delay τ for subunitary scattering systems. Theoretical expressions for complex time delays as a function of excitation energy, uniform and nonuniform loss, and coupling are given. We find very good agreement between theory and experimental data taken on microwave graphs containing an electronically variable lumped-loss element.

Nonlinearity-generated resilience in large complex systems.

We consider a generic nonlinear extension of May's 1972 model by including all higher-order terms in the expansion around the chosen fixed point (placed at the origin) with random Gaussian coefficients. The ensuing analysis reveals that as long as the origin remains stable, it is surrounded by a "resilience gap": there are no other fixed points within a radius r_{*}>0 and the system is therefore expected to be resilient to a typical initial displacement small in comparison to r_{*}. The radius r_{*} is shown to vanish at the same threshold where the origin loses local stability, revealing a mechanism by which systems close to the tipping point become less resilient.

Chaotic scattering with localized losses: S-matrix zeros and reflection time difference for systems with broken time-reversal invariance.

Motivated by recent studies of the phenomenon of coherent perfect absorption, we develop the random matrix theory framework for understanding statistics of the zeros of the (subunitary) scattering matrices in the complex energy plane, as well as of the recently introduced reflection time difference (RTD). The latter plays the same role for S-matrix zeros as the Wigner time delay does for its poles. For systems with broken time-reversal invariance, we derive the n-point correlation functions of the zeros in a closed determinantal form, and we study various asymptotics and special cases of the associated kernel.

Statistics of Extremes in Eigenvalue-Counting Staircases.

We consider the number N_{θ_{A}}(θ) of eigenvalues e^{iθ_{j}} of a random unitary matrix, drawn from CUE_{β}(N), in the interval θ_{j}∈[θ_{A},θ]. The deviations from its mean, N_{θ_{A}}(θ)-E[N_{θ_{A}}(θ)], form a random process as function of θ. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes.

Manifolds in a high-dimensional random landscape: Complexity of stationary points and depinning.

We obtain explicit expressions for the annealed complexities associated, respectively, with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension d<4 embedded in a random medium of dimension N≫1 and confined by a parabolic potential with the curvature parameter μ. These complexities are found to both vanish at the critical value μ_{c} identified as the Larkin mass. For μ<μ_{c} the system is in complex phase corresponding to the replica symmetry breaking in its T=0 thermodynamics.

Log-correlated random-energy models with extensive free-energy fluctuations: Pathologies caused by rare events as signatures of phase transitions.

We address systematically an apparent nonphysical behavior of the free-energy moment generating function for several instances of the logarithmically correlated models: the fractional Brownian motion with Hurst index H=0 (fBm0) (and its bridge version), a one-dimensional model appearing in decaying Burgers turbulence with log-correlated initial conditions and, finally, the two-dimensional log-correlated random-energy model (logREM) introduced in Cao et al. [Phys. Rev.

Nonlinear analogue of the May-Wigner instability transition.

We study a system of [Formula: see text] degrees of freedom coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling, the system is exponentially relaxing to an equilibrium with rate μ We show that, while increasing the ratio of the coupling strength to the relaxation rate, the system experiences an abrupt transition from a topologically trivial phase portrait with a single equilibrium into a topologically nontrivial regime characterized by an exponential number of equilibria, the vast majority of which are expected to be unstable. It is suggested that this picture provides a global view on the nature of the May-Wigner instability transition originally discovered by local linear stability analysis.

Freezing transitions and extreme values: random matrix theory, ζ (1/2 + it) and disordered landscapes.

We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when N → ∞.

Critical behavior of the number of minima of a random landscape at the glass transition point and the Tracy-Widom distribution.

We exploit a relation between the mean number N(m) of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behavior of N(m) in the simplest glasslike transition occuring in a toy model of a single particle in an N-dimensional random environment, with N>>1. Varying the control parameter μ through the critical value μ(c) we analyze in detail how N(m)(μ) drops from being exponentially large in the glassy phase to N(m)(μ)~1 on the other side of the transition. We also extract a subleading behavior of N(m)(μ) in both glassy and simple phases.

Statistics of resonance width shifts as a signature of eigenfunction nonorthogonality.

We consider an open (scattering) quantum system under the action of a perturbation of its closed counterpart. It is demonstrated that the resulting shift of resonance widths is a sensitive indicator of the nonorthogonality of resonance wave functions, being zero only if those were orthogonal. Focusing further on chaotic systems, we employ random matrix theory to introduce a new type of parametric statistics in open systems and derive the distribution of the resonance width shifts in the regime of weak coupling to the continuum.

Freezing transition, characteristic polynomials of random matrices, and the Riemann zeta function.

We argue that the freezing transition scenario, previously explored in the statistical mechanics of 1/f-noise random energy models, also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large N×N random unitary matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta function ζ(s) over sections of the critical line s=1/2+it of constant length and present the results of numerical computations in support. Our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes, random-matrix theory, and the theory of the Riemann zeta function.

Exact relations between multifractal exponents at the Anderson transition.

Two exact relations between mutlifractal exponents are shown to hold at the critical point of the Anderson localization transition. The first relation implies a symmetry of the multifractal spectrum linking the exponents with indices q<1/2 to those with q>1/2. The second relation connects the wave-function multifractality to that of Wigner delay times in a system with a lead attached.

Complexity of random energy landscapes, glass transition, and absolute value of the spectral determinant of random matrices.

Finding the mean of the total number N(tot) of stationary points for N-dimensional random energy landscapes is reduced to averaging the absolute value of the characteristic polynomial of the corresponding Hessian. For any finite N we provide the exact solution to the problem for a class of landscapes corresponding to the "toy model" of manifolds in a random environment. For N>>1 our asymptotic analysis reveals a phase transition at some critical value mu(c) of a control parameter mu from a phase with a finite landscape complexity: N(tot) approximately e(N Sigma), Sigma(mu0 to the phase with vanishing complexity: Sigma(mu>mu(c))=0.

Variance of transmitted power in multichannel dissipative ergodic structures invariant under time reversal.

We use random matrix theory (RMT) to study the first two moments of the wave power transmitted in time-reversal invariant systems having ergodic motion. Dissipation is modeled by a number of loss channels of variable coupling strength. To make a connection with ultrasonic experiments on ergodic elastodynamic billiards, the channels injecting and collecting the waves are assumed to be negligibly coupled to the medium and to contribute essentially no dissipation.

Distribution of the local density of states, reflection coefficient, and Wigner delay time in absorbing ergodic systems at the point of chiral symmetry.

Employing the chiral Gaussian unitary ensemble of random matrices, we calculate the probability distribution of the local density of states for zero-dimensional ("quantum chaotic") two-sublattice systems at the point of chiral symmetry E=0 and in the presence of uniform absorption. The obtained result can be used to find the distributions of the reflection coefficient and of the Wigner time delay for such systems.

Level curvature distribution in a model of two uncoupled chaotic subsystems.

We study distributions of eigenvalue curvatures for a block-diagonal random matrix perturbed by a full random matrix. The most natural physical realization of this model is a quantum chaotic system with some inherent symmetry, such that its energy levels form two independent subsequences, subject to a generic perturbation which does not respect the symmetry. We describe analytically a crossover in the form of a curvature distribution with a tunable parameter, namely, the ratio of intersubsystem/intrasubsystem coupling strengths.

Statistics of transmitted power in multichannel dissipative ergodic structures.

We use the random matrix theory (RMT) to study the probability distribution function and moments of the wave power transmitted inside systems with ergodic wave motion. The results describe either open multichannel systems or their closed counterparts with local-in-space internal dissipation. We concentrate on the regime of broken time-reversal invariance and employ two different analytical approaches: the exact supersymmetry method and a simpler technique that uses RMT eigenstatistics for closed nondissipative systems as an input.

Random symmetric matrices with a constraint: the spectral density of random impedance networks.

We derive the mean eigenvalue density for symmetric Gaussian random N x N matrices in the limit of large N, with a constraint implying that the row sum of matrix elements should vanish. The result is shown to be equivalent to a result found recently for the average density of resonances in random impedance networks [Y.V.

Statistics of resonances and nonorthogonal eigenfunctions in a model for single-channel chaotic scattering.

We describe analytical and numerical results on the statistical properties of complex eigenvalues and the corresponding nonorthogonal eigenvectors for non-Hermitian random matrices modeling one-channel quantum-chaotic scattering in systems with broken time-reversal invariance.

Reducing nonideal to ideal coupling in random matrix description of chaotic scattering: application to the time-delay problem.

We write explicitly a transformation of the scattering phases reducing the problem of quantum chaotic scattering for systems with M statistically equivalent channels at nonideal coupling to that for ideal coupling. Unfolding the phases by their local density leads to universality of their local fluctuations for large M. A relation between the partial time delays and diagonal matrix elements of the Wigner-Smith matrix is revealed for ideal coupling.