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Yan V. Fyodorov

Publications and source records attributed to Yan V. Fyodorov.

At least 37 records · Page 2Linked to original sources
arXiv2023

Extreme eigenvalues and the emerging outlier in rank-one non-Hermitian deformations of the Gaussian Unitary Ensemble

Complex eigenvalues of random matrices $J=\text{GUE }+ iγ\diag (1, 0, \ldots, 0)$ 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 $N\gg 1$ the eigenvalue density of $J$ 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 $N$ 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. In the critical regime we determine the density of such extreme eigenvalues, and show how the outlier gradually separates itself from the rest of the extreme eigenvalues. Finally, we describe the fluctuations in the height of the eigenvalue outlier for large but finite $N$ in terms of the associated large deviation function.

math-ph↗
arXiv2022

Superposition of Random Plane Waves in High Spatial Dimensions: Random Matrix Approach to Landscape Complexity

Motivated by current interest in understanding statistical properties of random landscapes in high-dimensional spaces, we consider a model of the landscape in $\mathbb{R}^N$ obtained by superimposing $M>N$ plane waves of random wavevectors and amplitudes. For this landscape we show how to compute the "annealed complexity" controlling the asymptotic growth rate of the mean number of stationary points as $N\to \infty$ at fixed ratio $α=M/N>1$. The framework of this computation requires us to study spectral properties of $N\times N$ matrices $W=KTK^T$, where $T$ is diagonal with $M$ mean zero i.i.d. real normally distributed entries, and all $MN$ entries of $K$ are also i.i.d. real normal random variables. We suggest to call the latter Gaussian Marchenko-Pastur Ensemble, as such matrices appeared in the seminal 1967 paper by those authors. We compute the associated mean spectral density and evaluate some moments and correlation functions involving products of characteristic polynomials for such and related matrices.

cond-mat.stat-mech↗
arXiv2021

Optimization landscape in the simplest constrained random least-square problem

We analyze statistical features of the ``optimization landscape'' in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of an overcomplete system of $M>N$ linear equations $({\bf a}_k,{\bf x})=b_k, \, k=1,\ldots,M$ on the $N-$sphere ${\bf x}^2=N$. We treat both the $N-$component vectors ${\bf a}_k$ and parameters $b_k$ as independent mean zero real Gaussian random variables. First, we derive the exact expressions for the mean number of stationary points of the least-square loss function in the framework of the Kac-Rice approach combined with the Random Matrix Theory for Wishart Ensemble, and then perform its asymptotic analysis as $N\to \infty$ at a fixed $α=M/N>1$ in various regimes. In particular, this analysis allows to extract the Large Deviation Function for the density of the smallest Lagrange multiplier $λ_{min}$ associated with the problem, and in this way to find its most probable value. This can be further used to predict the asymptotic minimal value ${\cal E}_{min}$ of the loss function as $N\to \infty$. Finally, we develop an alternative approach based on the replica trick to conjecture the form of the Large Deviation function for the density of ${\cal E}_{min}$ at $N\gg 1$. As a by-product, we find the value of the {\it compatibility threshold} $α_c$ which is the minimal value of the asymptotic ratio $M/N$ such that the random linear system on the $N-$sphere is typically compatible.

math-ph↗
arXiv2021

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 $τ_\text{W}$ for sub-unitary wave chaotic scattering systems. We first demonstrate theoretically that the mean value of the $\text{Re}[τ_\text{W}]$ 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. The experimental results are in excellent agreement with the developed theory. The tails of the distributions of both real and imaginary time delay are measured and are also found to agree with theory. The results are applicable to any practical realization of a wave chaotic scattering system in the short-wavelength limit.

cond-mat.dis-nn↗
arXiv2021

Generalization of Wigner Time Delay to Sub-Unitary Scattering Systems

We introduce a complex generalization of Wigner time delay $τ$ for sub-unitary scattering systems. Theoretical expressions for complex time delay as a function of excitation energy, uniform and non-uniform 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. We find that time delay and the determinant of the scattering matrix share a common feature in that the resonant behavior in $\text{Re}[τ]$ and $\text{Im}[τ]$ serves as a reliable indicator of the condition for Coherent Perfect Absorption (CPA). This work opens a new window on time delay in lossy systems and provides a means to identify the poles and zeros of the scattering matrix from experimental data. The results also enable a new approach to achieving CPA at an arbitrary energy/frequency in complex scattering systems.

cond-mat.dis-nn↗
arXiv2020

The band structure of a model of spatial random permutation

We study a random permutation of a lattice box in which each permutation is given a Boltzmann weight with energy equal to the total Euclidean displacement. Our main result establishes the band structure of the model as the box-size $N$ tends to infinity and the inverse temperature~$β$ tends to zero; in particular, we show that the mean displacement is of order $\min \{ 1/β, N\}$. In one dimension our results are more precise, specifying leading-order constants and giving bounds on the rates of convergence. Our proofs exploit a connection, via matrix permanents, between random permutations and Gaussian fields; although this connection is well-known in other settings, to the best of our knowledge its application to the study of random permutations is novel. As a byproduct of our analysis, we also provide asymptotics for the permanents of Kac-Murdock-Szego (KMS) matrices.

math.PR↗
arXiv2020

Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble

We study the distribution of the eigenvalue condition numbers $κ_i=\sqrt{ (\mathbf{l}_i^* \mathbf{l}_i)(\mathbf{r}_i^* \mathbf{r}_i)}$ associated with real eigenvalues $λ_i$ of partially asymmetric $N\times N$ random matrices from the real Elliptic Gaussian ensemble. The large values of $κ_i$ signal the non-orthogonality of the (bi-orthogonal) set of left $\mathbf{l}_i$ and right $\mathbf{r}_i$ eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite $N$ expression for the joint density function(JDF) ${\cal P}_N(z,t)$ of $t=κ_i^2-1$ and $λ_i$ taking value $z$, and investigate its several scaling regimes in the limit $N\to \infty$. When the degree of asymmetry is fixed as $N\to \infty$, the number of real eigenvalues is $O(\sqrt{N})$, and in the bulk of the real spectrum $t_i=O(N)$, while on approaching the spectral edges the non-orthogonality is weaker: $t_i=O(\sqrt{N})$. In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of $N$ eigenvalues remain real as $N\to \infty$. In such a regime eigenvectors are weakly non-orthogonal, $t=O(1)$, and we derive the associated JDF, finding that the characteristic tail ${\cal P}(z,t)\sim t^{-2}$ survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.

math-ph↗
arXiv2020

Separability gap and large deviation entanglement criterion

For a given Hamiltonian $H$ on a multipartite quantum system, one is interested in finding the energy $E_0$ of its ground state. In the separability approximation, arising as a natural consequence of measurement in a separable basis, one looks for the minimal expectation value $λ_{\rm min}^{\otimes}$ of $H$ among all product states. For several concrete model Hamiltonians, we investigate the difference $λ_{\rm min}^{\otimes}-E_0$, called separability gap, which vanishes if the ground state has a product structure. In the generic case of a random Hermitian matrix of the Gaussian orthogonal ensemble, we find explicit bounds for the size of the gap which depend on the number of subsystems and hold with probability one. This implies an effective entanglement criterion applicable for any multipartite quantum system: If an expectation value of a typical observable among a given state is sufficiently distant from the average value, the state is almost surely entangled.

quant-ph↗
arXiv2020

Statistics of extremes in eigenvalue-counting staircases

We consider the number ${\cal N}_{θ_A}(θ)$ of eigenvalues $e^{i θ_j}$ of a random unitary matrix, drawn from CUE$_β(N)$, in the interval $θ_j \in [θ_A,θ]$. The deviations from its mean, ${\cal N}_{θ_A}(θ) - \mathbb{E}({\cal N}_{θ_A}(θ))$, 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. By using an extended Fisher-Hartwig conjecture for Toeplitz determinants, supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any $β>0$. It exhibits combined features of standard counting statistics of fermions (free for $β=2$ and with Sutherland-type interaction for $β\ne 2$) in an interval and extremal statistics of the fractional Brownian motion with Hurst index $H=0$. The $β=2$ results are expected to apply to the statistics of zeroes of the Riemann Zeta function

cond-mat.stat-mech↗
arXiv2020

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 Refection 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 study various asymptotics and special cases of the associated kernel. The time-correlation function of the RTD is then evaluated and compared with numerical simulations. This allows to identify a cubic tail in the distribution of RTD, which we conjecture to be a superuniversal characteristic valid for all symmetry classes. We also discuss two methods for possible extraction of S-matrix zeroes from scattering data by harmonic inversion.

cond-mat.dis-nn↗
arXiv2019

Optimal Wave Fields for Micro-manipulation in Complex Scattering Environments

The manipulation of small objects with light has become an indispensable tool in many areas of research ranging from physics to biology and medicine. Here we demonstrate how to implement micro-manipulation at the optimal level of efficiency for targets of arbitrary shape and inside complex environments such as disordered media. Our approach is to design wave-fronts in the far-field that have optimal properties in the near-field of the target such as to apply to it the strongest possible force, pressure or torque as well as to achieve the most efficient focus at the target position. Free of any iterative optimization, our approach only relies on a simple eigenvalue problem established from the scattering matrix of the system and its dependence on the target parameters. To illustrate this theoretical concept, we perform a proof-of-principle experiment in the microwave regime, which fully confirms our predictions.

physics.optics↗
arXiv2019

Manifolds pinned by a high-dimensional random landscape: Hessian at the global energy minimum

We consider an elastic manifold of internal dimension $d$ and length $L$ pinned in a $N$ dimensional random potential and confined by an additional parabolic potential of curvature $μ$. We are interested in the mean spectral density $ρ(λ)$ of the Hessian matrix $K$ at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for $ρ(λ)$ for a fixed $L^d$ in the $N \to \infty$ limit extending $d=0$ results of our previous work. A particular attention is devoted to analyzing the limit of extended lattice systems by letting $L\to \infty$. In all cases we show that for a confinement curvature $μ$ exceeding a critical value $μ_c$, the so-called "Larkin mass", the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at $μ\to μ_c$. For $μ<μ_c$ the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all $d$ as $(μ_c-μ)^4$ near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called "marginal cases" in $d=1,2$ which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning.

cond-mat.dis-nn↗
arXiv2018

Extreme values of CUE characteristic polynomials: a numerical study

We present the results of systematic numerical computations relating to the extreme value statistics of the characteristic polynomials of random unitary matrices drawn from the Circular Unitary Ensemble (CUE) of Random Matrix Theory. In particular, we investigate a range of recent conjectures and theoretical results inspired by analogies with the theory of logarithmically-correlated Gaussian random fields. These include phenomena related to the conjectured freezing transition. Our numerical results are consistent with, and therefore support, the previous conjectures and theory. We also go beyond previous investigations in several directions: we provide the first quantitative evidence in support of a correlation between extreme values of the characteristic polynomials and large gaps in the spectrum, we investigate the rate of convergence to the limiting formulae previously considered, and we extend the previous analysis of the CUE to the C$β$E which corresponds to allowing the degree of the eigenvalue repulsion to become a parameter.

cond-mat.stat-mech↗
arXiv2017

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 non-physical 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 1D model appearing in decaying Burgers turbulence with log-correlated initial conditions, and finally, the two-dimensional logREM introduced in [Cao et al., Phys.Rev.Lett.,118,090601] based on the 2D Gaussian free field (GFF) with background charges and directly related to the Liouville field theory. All these models share anomalously large fluctuations of the associated free energy, with a variance proportional to the log of the system size. We argue that a seemingly non-physical vanishing of the moment generating function for some values of parameters is related to the termination point transition (a.k.a pre-freezing). We study the associated universal log corrections in the frozen phase, both for log-REMs and for the standard REM, filling a gap in the literature. For the above mentioned integrable instances of logREMs, we predict the non-trivial free energy cumulants describing non-Gaussian fluctuations on the top of the Gaussian with extensive variance. Some of the predictions are tested numerically.

cond-mat.stat-mech↗
arXiv2017

Distribution of zeros of the S-matrix of chaotic cavities with localized losses and Coherent Perfect Absorption: non-perturbative results

We employ the Random Matrix Theory framework to calculate the density of zeroes of an $M$-channel scattering matrix describing a chaotic cavity with a single localized absorber embedded in it. Our approach extends beyond the weak-coupling limit of the cavity with the channels and applies for any absorption strength. Importantly it provides an insight for the optimal amount of loss needed to realize a chaotic coherent perfect absorbing (CPA) trap. Our predictions are tested against simulations for two types of traps: a complex network of resonators and quantum graphs.

cond-mat.dis-nn↗
arXiv2016

One step replica symmetry breaking and extreme order statistics of logarithmic REMs

Building upon the one-step replica symmetry breaking formalism, duly understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. The distribution of the random shift is determined solely by the large-distance ("infra-red", IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely by the small-distance ("ultraviolet", UV) limit, in terms of the biased minimal process. Our approach provides connections of the replica framework to results in the probability literature and sheds further light on the freezing/duality conjecture which was the source of many previous results for log-REMs. In this way we derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higher-order generalizations) in terms of model-specific contributions from UV as well as IR limits. In particular, we show that the second min statistics is largely independent of details of UV data, whose influence is seen only through the mean value of the gap. For a given log-correlated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of $1/f$-noise.

cond-mat.stat-mech↗
arXiv2016

Nonlinear Analogue of the May-Wigner Instability Transition

We study a system of $N\gg 1$ 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 non-trivial regime characterised 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.

cond-mat.dis-nn↗