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Yan V Fyodorov

Publications and source records attributed to Yan V Fyodorov.

At least 19 recordsLinked to original sources
arXiv2026

Kac-Rice inspired approach to non-Hermitian random matrices

We suggest a method of analyzing the joint probability density (JPD) ${\cal P}_N(z,{\bf v})$ of an eigenvalue $z$ and the associated right eigenvector ${\bf v}$ (normalized with ${\bf v}^*{\bf v}=1$) for non-Hermitian random matrices of a given size $N\times N$. The approach is essentially based on the Kac-Rice counting formula applied to the associated characteristic polynomial combined with a certain integral identity for the Dirac delta function of such a polynomial. To illustrate utility of the general method we derive ${\cal P}_N(z,{\bf v})$ in the two particular cases: (i) one-parameter family of matrices interpolating between complex Ginibre and real Ginibre ensembles and (ii) a complex Ginibre matrix additively perturbed by a general fixed matrix. In particular, in the former case we analyze the formation of an excess of eigenvalues in the vicinity of the real axis on approaching the real Ginibre limit, which eventually gives rise to the existence of a new scaling regime of "weak non-reality" as $N\to \infty$. In the second case we further analyze non-Hermitian Rosenzweig-Porter model which recently attracted considerable interest in physics literature. In addition, we provide new insights into eigenvalue and eigenvector distribution for a general rank one perturbation of complex Ginibre matrices of finite size $N$, and in the structure of an outlier emerging as $N\gg 1$. Finally we discuss a generalization of the proposed method which is expected to be suitable for analysis of JPD involving both left- and right eigenvectors.

math-ph↗
arXiv2026

Microscopic Parametric Correlations and Spectral Rigidity in the Non-Hermitian Threefold Way

We study microscopic spectral correlations between two nearby parameter values in the three non-Hermitian Gaussian bulk classes: $A$ (unconstrained), $AI^\dagger$ (complex symmetric), and $AII^\dagger$ (complex self-dual, with each degenerate doublet counted once). Parameter dependence is modelled by stationary matrix Ornstein--Uhlenbeck evolution, whose microscopic time scale is $N^{-1}$. For every positive integer replica number $n$, we derive exact finite-$N$ auxiliary-field integral representations for joint characteristic-polynomial moments and obtain their bulk asymptotics as $N\to\infty$ with $n$ fixed. With the spectrum normalized to the unit disk, correlations near a bulk point $z_0$, between spectral positions $z_1,z_2$ at times $t_1,t_2$, depend only on $s=N|z_1-z_2|^2+N(1-|z_0|^2)|t_1-t_2|$. Adopting the static Hermitian/non-Hermitian replica continuation yields explicit two-time kernels; for $AI^\dagger$ and $AII^\dagger$ these are replica conjectures and, to our knowledge, the first microscopic two-time kernels proposed for these classes. They determine density correlations, cross-time eigenvalue-count covariances, and equal-time number variances. For a disk of mean eigenvalue count $y$, the latter obey $\mbox{Var}\mathcal N_X(D_y)=κ_X\sqrt y+ β_X/\sqrt y+o(y^{-1/2})$, with explicitly given constants for each class. Exact perturbation theory further identifies eigenvector nonorthogonality as the mechanism of short-time spectral diffusion and yields a basis-independent condition number for a two-dimensional Kramers eigenspace in class $AII^\dagger$. Direct simulations agree quantitatively with its conjectured inverse-gamma law.

math-ph↗
arXiv2022

Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates

We consider a nonlinear autonomous random dynamical system of $N$ degrees of freedom coupled by Gaussian random interactions and characterized by a continuous spectrum $n_μ(λ)$ of real positive relaxation rates. Using Kac-Rice formalism, the computation of annealed complexities (both of stable equilibria and of all types of equilibria) is reduced to evaluating the averages involving the modulus of the determinant of the random Jacobian matrix. In the limit of large system $N\gg 1$ we derive exact analytical results for the complexities for short-range correlated coupling fields, extending results previously obtained for the "homogeneous" relaxation spectrum characterised by a single relaxation rate. We show the emergence of a "topology trivialisation" transition from a complex phase with exponentially many equilibria to a simple phase with a single equilibrium as the magnitude of the random field is decreased. Within the complex phase the complexity of stable equilibria undergoes an additional transition from a phase with exponentially small probability to find a single equilibrium to a phase with exponentially many stable equilibria as the fraction of gradient component of the field is increased. The behaviour of the complexity at the transition is found only to depend on the small $λ$ behaviour of the spectrum of relaxation rates $n_μ(λ)$ and thus conjectured to be universal. We also provide some insights into a counting problem motivated by a paper by B. Spivak and A. Zyuzin of 2004 about wave scattering in a disordered nonlinear medium.

cond-mat.stat-mech↗
arXiv2021

Counting equilibria of large complex systems by instability index

We consider a nonlinear autonomous system of $N\gg 1$ degrees of freedom randomly coupled by both relaxational ('gradient') and non-relaxational ('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 non-trivial 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. We further calculate the mean proportion of equilibria which have a fixed fraction of unstable directions.

math-ph↗
arXiv2021

Random Matrix Theory of Resonances: an Overview

Scattering of electromagnetic waves in billiard-like systems has become a standard experimental tool of studying properties associated with Quantum Chaos. Random Matrix Theory (RMT) describing statistics of eigenfrequencies and associated eigenfunctions remains one of the pillars of theoretical understanding of quantum chaotic systems. In a scattering system coupling to continuum via antennae converts real eigenfrequencies into poles of the scattering matrix in the complex frequency plane and the associated eigenfunctions into decaying resonance states. Understanding statistics of these poles, as well as associated non-orthogonal resonance eigenfunctions within RMT approach is still possible, though much more challenging task.

cond-mat.dis-nn↗
arXiv2021

Eigenfunction non-orthogonality factors and the shape of CPA-like dips in a single-channel reflection from lossy chaotic cavities

Motivated by the phenomenon of Coherent Perfect Absorption, we study the shape of the deepest minima in the frequency-dependent single-channel reflection of waves from a cavity with spatially uniform losses. We show that it is largely determined by non-orthogonality factors $O_{nn}$ of the eigenmodes associated with the non-selfadjoint effective Hamiltonian. For cavities supporting chaotic ray dynamics we then use random matrix theory to derive, fully non-perturbatively, the explicit probability density ${\cal P}(O_{nn})$ of the non-orthogonality factors for systems with both broken and preserved time reversal symmetry. The results imply that $O_{nn}$ are heavy-tail distributed, with the universal tail ${\cal P}(O_{nn}\gg 1)\sim O_{nn}^{-3}$.

cond-mat.dis-nn↗
arXiv2020

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. We also find that beyond the resilience radius the number of fixed points in a ball surrounding the original point of equilibrium grows exponentially with N, making systems dynamics highly sensitive to far enough displacements from the origin.

cond-mat.dis-nn↗
arXiv2020

Manifolds in 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 $N \gg 1$ 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. The complexities vanish respectively quadratically (stationary points) and cubically (minima) at $μ_c^-$. For $d\geq 1$ they admit a finite "massless" limit $μ=0$ which is used to provide an upper bound for the depinning threshold under an applied force.

cond-mat.dis-nn↗
arXiv2018

Exponential number of equilibria and depinning threshold for a directed polymer in a random potential

By extending the Kac-Rice approach to manifolds of finite internal dimension, we show that the mean number $\left\langle\mathcal{N}_\mathrm{tot}\right\rangle$ of all possible equilibria (i.e. force-free configurations, a.k.a. equilibrium points) of an elastic line (directed polymer), confined in a harmonic well and submitted to a quenched random Gaussian potential in dimension $d=1+1$, grows exponentially $\left\langle\mathcal{N}_\mathrm{tot}\right\rangle\sim\exp{(r\,L)}$ with its length $L$. The growth rate $r$ is found to be directly related to the generalised Lyapunov exponent (GLE) which is a moment-generating function characterising the large-deviation type fluctuations of the solution to the initial value problem associated with the random Schrödinger operator of the 1D Anderson localization problem. For strong confinement, the rate $r$ is small and given by a non-perturbative (instanton, Lifshitz tail-like) contribution to GLE. For weak confinement, the rate $r$ is found to be proportional to the inverse Larkin length of the pinning theory. As an application, identifying the depinning with a landscape "topology trivialization" phenomenon, we obtain an upper bound for the depinning threshold $f_c$, in the presence of an applied force, for elastic lines and $d$-dimensional manifolds, expressed through the mean modulus of the spectral determinant of the Laplace operators with a random potential. We also discuss the question of counting of stable equilibria. Finally, we extend the method to calculate the asymptotic number of equilibria at fixed energy (elastic, potential and total), and obtain the (annealed) distribution of the energy density over these equilibria (i.e. force-free configurations). Some connections with the Larkin model are also established.

cond-mat.dis-nn↗
arXiv2018

Hessian spectrum at the global minimum of high-dimensional random landscapes

Using the replica method we calculate the mean spectral density of the Hessian matrix at the global minimum of a random $N \gg 1$ dimensional isotropic, translationally invariant Gaussian random landscape confined by a parabolic potential with fixed curvature $μ>0$. Simple landscapes with generically a single minimum are typical for $μ>μ_{c}$, and we show that the Hessian at the global minimum is always {\it gapped}, with the low spectral edge being strictly positive. When approaching from above the transitional point $μ= μ_{c}$ separating simple landscapes from 'glassy' ones, with exponentially abundant minima, the spectral gap vanishes as $(μ-μ_c)^2$. For $μ<μ_c$ the Hessian spectrum is qualitatively different for 'moderately complex' and 'genuinely complex' landscapes. The former are typical for short-range correlated random potentials and correspond to 1-step replica-symmetry breaking mechanism. Their Hessian spectra turn out to be again gapped, with the gap vanishing on approaching $μ_c$ from below with a larger critical exponent, as $(μ_c-μ)^4$. At the same time in the 'most complex' landscapes with long-ranged power-law correlations the replica symmetry is completely broken. We show that in that case the Hessian remains gapless for all values of $μ<μ_c$, indicating the presence of 'marginally stable' spatial directions. Finally, the potentials with {\it logarithmic} correlations share both 1RSB nature and gapless spectrum. The spectral density of the Hessian always takes the semi-circular form, up to a shift and an amplitude that we explicitly calculate.

cond-mat.dis-nn↗
arXiv2018

A spin glass model for reconstructing nonlinearly encrypted signals corrupted by noise

An encryption of a signal ${\bf s}\in\mathbb{R^N}$ is a random mapping ${\bf s}\mapsto \textbf{y}=(y_1,\ldots,y_M)^T\in \mathbb{R}^M$ which can be corrupted by an additive noise. Given the Encryption Redundancy Parameter (ERP) $μ=M/N\ge 1$, the signal strength parameter $R=\sqrt{\sum_i s_i^2/N}$, and the ('bare') noise-to-signal ratio (NSR) $γ\ge 0$, we consider the problem of reconstructing ${\bf s}$ from its corrupted image by a Least Square Scheme for a certain class of random Gaussian mappings. The problem is equivalent to finding the configuration of minimal energy in a certain version of spherical spin glass model, with squared Gaussian-distributed random potential. We use the Parisi replica symmetry breaking scheme to evaluate the mean overlap $p_{\infty}\in [0,1]$ between the original signal and its recovered image (known as 'estimator') as $N\to \infty$, which is a measure of the quality of the signal reconstruction. We explicitly analyze the general case of linear-quadratic family of random mappings and discuss the full $p_{\infty} (γ)$ curve. When nonlinearity exceeds a certain threshold but redundancy is not yet too big, the replica symmetric solution is necessarily broken in some interval of NSR. We show that encryptions with a nonvanishing linear component permit reconstructions with $p_{\infty}>0$ for any $μ>1$ and any $γ<\infty$, with $p_{\infty}\sim γ^{-1/2}$ as $γ\to \infty$. In contrast, for the case of purely quadratic nonlinearity, for any ERP $μ>1$ there exists a threshold NSR value $γ_c(μ)$ such that $p_{\infty}=0$ for $γ>γ_c(μ)$ making the reconstruction impossible. The behaviour close to the threshold is given by $p_{\infty}\sim (γ_c-γ)^{3/4}$ and is controlled by the replica symmetry breaking mechanism.

cond-mat.dis-nn↗
arXiv2018

On statistics of bi-orthogonal eigenvectors in real and complex Ginibre ensembles: combining partial Schur decomposition with supersymmetry

We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated 'non-orthogonality overlap factor' (also known as the 'eigenvalue condition number') of the left and right eigenvectors for non-selfadjoint Gaussian random matrices of size $N\times N$. First we derive the general finite $N$ expression for the JPD of a real eigenvalue $λ$ and the associated non-orthogonality factor in the real Ginibre ensemble, and then analyze its 'bulk' and 'edge' scaling limits. The ensuing distribution is maximally heavy-tailed, so that all integer moments beyond normalization are divergent. A similar calculation for a complex eigenvalue $z$ and the associated non-orthogonality factor in the complex Ginibre ensemble is presented as well and yields a distribution with the finite first moment. Its 'bulk' scaling limit yields a distribution whose first moment reproduces the well-known result of Chalker and Mehlig \cite{ChalkerMehlig1998}, and we provide the 'edge' scaling distribution for this case as well. Our method involves evaluating the ensemble average of products and ratios of integer and half-integer powers of characteristic polynomials for Ginibre matrices, which we perform in the framework of a supersymmetry approach. Our paper complements recent studies by Bourgade and Dubach \cite{BourgadeDubach}.

math-ph↗
arXiv2018

On characteristic polynomials for a generalized chiral random matrix ensemble with a source

We evaluate averages involving characteristic polynomials, inverse characteristic polynomials and ratios of characteristic polynomials for a $N\times N$ random matrix taken from a $L$-deformed Chiral Gaussian Unitary Ensemble with an external source $Ω$. Relation to a recently studied statistics of bi-orthogonal eigenvectors in the complex Ginibre ensemble, see Y.V. Fyodorov arXiv:1710.04699, is briefly discussed as a motivation to study asymptotics of these objects in the case of external source proportional to the identity matrix. In particular, for an associated 'complex bulk/chiral edge' scaling regime we retrieve the kernel related to Bessel/Macdonald functions.

math-ph↗
arXiv2016

Topology trivialization transition in random non-gradient autonomous ODE's on a sphere

We calculate the mean total number of equilibrium points in a system of $N$ random autonomous ODE's introduced by Cugliandolo et al. in 1997 to describe non-relaxational glassy dynamics on the high-dimensional sphere. In doing it we suggest a new approach which allows such a calculation to be done most straightforwardly, and is based on efficiently incorporating the Langrange multiplier into the Kac-Rice framework. Analysing the asymptotic behaviour for large $N$ we confirm that the phenomenon of 'topology trivialization' revealed earlier for other systems holds also in the present framework with nonrelaxational dynamics. Namely, by increasing the variance of the random 'magnetic field' term in dynamical equations we find a 'phase transition' from the exponentially abundant number of equilibria down to just two equilibria. Classifying the equilibria in the nontrivial phase by stability remains an open problem.

math-ph↗
arXiv2013

High-Dimensional Random Fields and Random Matrix Theory

Our goal is to discuss in detail the calculation of the mean number of stationary points and minima for random isotropic Gaussian fields on a sphere as well as for stationary Gaussian random fields in a background parabolic confinement. After developing the general formalism based on the high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory (RMT) techniques to perform analysis of the random energy landscape of $p-$spin spherical spinglasses and a related glass model, both displaying a zero-temperature one-step replica symmetry breaking glass transition as a function of control parameters (e.g. a magnetic field or curvature of the confining potential). A particular emphasis of the presented analysis is on understanding in detail the picture of "topology trivialization" (in the sense of drastic reduction of the number of stationary points) of the landscape which takes place in the vicinity of the zero-temperature glass transition in both models. We will reveal the important role of the GOE "edge scaling" spectral region and the Tracy-Widom distribution of the maximal eigenvalue of GOE matrices for providing an accurate quantitative description of the universal features of the topology trivialization scenario.

math-ph↗
arXiv2013

Topology trivialization and large deviations for the minimum in the simplest random optimization

Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N-1)- dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the "trust region subproblem" or "constraint least square problem". When both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field. We first identify and study two distinct large-N scaling regimes in which the linear term (magnetic field) leads to a gradual topology trivialization, i.e. reduction in the total number N_{tot} of critical (stationary) points in the cost function landscape. In the first regime N_{tot} remains of the order $N$ and the cost function (energy) has generically two almost degenerate minima with the Tracy-Widom (TW) statistics. In the second regime the number of critical points is of the order of unity with a finite probability for a single minimum. In that case the mean total number of extrema (minima and maxima) of the cost function is given by the Laplace transform of the TW density, and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the TW law. Though the full form of that distribution is not yet known to us, one of its far tails can be inferred from the large deviation theory for the global minimum. In the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large-deviation approximation by finding both the rate function and the leading pre-exponential factor.

cond-mat.dis-nn↗