arXiv subjects

Yan V. Fyodorov

Publications and source records attributed to Yan V. Fyodorov.

At least 19 recordsLinked to original sources

arXiv2026

Reflection Resonances in the One-Dimensional Anderson Localization: Finite-Length Statistics, Wigner Time Delay, and Boundary Eigenfunctions

We study reflection-resonance poles $Z_j=E_j-iΓ_j$, $Γ_j>0$, of a finite one-dimensional disordered sample of length $L$, coupled at one end to a semi-infinite lead, in the regime $L\gg\ell_L\gg k^{-1}$, where $\ell_L$ is the localization length and $k=\sqrt{E}$. The key step is to relate the resonance density to the reflection coefficient at the complex energy $E+iΓ$, corresponding to uniform absorption. Exact finite-chain Kac--Rice and Poincaré--Lelong identities reduce pole counting to a finite-length diffusion of the reflected intensity. For the perfect contact transparency the density crosses over from the localization-controlled $Γ^{-1}$ law to the broad-resonance $Γ^{-2}$ law at $Γ_L=k/\ell_L$, in agreement with $L\to\infty$ result of Fyodorov and Meibohm. Finite length cuts off the $Γ^{-1}$ regime at $Γ_{\rm ultra}=\frac{e^{1-γ_{\rm E}}}{2}Γ_L e^{-L/\ell_L}$. We derive the ultranarrow resonances crossover shape as an explicit moving front; for contact transparency $\mathcal T<1$ this scale shifts to $\mathcal TΓ_{\rm ultra}$. The same reflection process yields the finite-length Wigner time-delay statistics and, in the weak-absorption limit, the Comtet--Texier distribution. We show that at eigenvalues of the corresponding closed Dirichlet sample the Wigner delay is inversely proportional to the squared boundary derivative of the normalized eigenfunction. This quantity also gives the eigenvalue response to displacement of the Dirichlet boundary at the lead-contact end, and hence the force exerted by the eigenmode on that boundary; we obtain its finite-length distribution. Finally, the finite-transparency resonance density obeys the single-channel Moldauer--Simonius sum rule, linking its perfect-coupling divergence to the $Γ^{-2}$ tail. Direct lattice and spectral computations test the crossover and front constant.

cond-mat.dis-nn↗

arXiv2026

Parametric correlations in non-Hermitian quantum chaos: random matrix approach

Motivated by the surge of interest in statistics of non-Hermitian random matrices as a framework for description of universal characteristics of dissipative chaotic quantum many-body systems, we address the problem of characterizing the parametric correlations of spectral densities. Considering parameter-dependent ensemble of complex Ginibre matrices we derive an explicit, closed-form expression for the parametric number covariance in the systems of symmetry class $\mathbf{A}$ for eigenvalues in a circular domain containing on average a finite number of eigenvalues in the spectral bulk. This behavior is expected to be universal, as further supported by numerical evidence for the real Ginibre ensemble, non-Hermitian Bernoulli Wigner matrices and bi-unitarily invariant ensembles. We also discuss a relation between parametric correlations of spectral densities and the distribution of the so-called eigenvector non-orthogonality factor, which attracted considerable interest in recent years.

quant-ph↗

arXiv2026

Random Matrix Theory for Chaotic Wave Scattering and Transport

We review random matrix approaches to chaotic wave scattering and transport in open systems. Starting from the effective non-Hermitian Hamiltonian formulation, we discuss the scattering matrix, reaction matrix, time delays, and complex resonances as complementary probes of open chaotic dynamics. We emphasize universal statistics governed by symmetry, openness, and channel coupling. Topics include the maximum-entropy description of fixed-energy scattering and its applications to quantum transport, energy correlations, resonance and eigenfunction statistics, and selected wave-chaotic phenomena induced by finite absorption. The focus throughout is on non-perturbative methods and universal structures underlying open quantum and wave chaotic systems.

nlin.CD↗

arXiv2026

Gradual eigenvector ergodization in coupled Ginibre matrices

Non-Hermitian random matrices provide a useful framework for understanding universal characteristics of dissipative quantum chaotic systems with loss or gain. We consider a model of two such system represented by two independent $N\times N$ complex Ginibre matrices interacting via a deterministic matrix $c{\bf 1}_N$, where $c$ is the complex coupling parameter whose magnitude $|c|$ controls the interaction strength. We characterize quantitatively how the eigenvectors of the whole system, initially localized in one of the individual subsystems for $|c|=0$, eventually spread over the full system with growing interaction strength. The resulting asymptotic formula describing such spread in the limit $N\to \infty$ is very explicit and provides a full picture of the gradual ergodization of eigenvectors as a function of the coupling parameter $|c|$ in the whole transition regime. As a by-product of our method we also compute the mean eigenvalue density for our model at the origin of the spectral bulk $z=0$ in the fully ergodic regime, when the coupling is scaled with the matrix size as $c=\sqrt{N}\tilde{c}$. We find that as $N\to \infty$ the limiting density at the origin vanishes beyond the critical value $|\tilde{c}|=1$, signalling of a split of the density support in the complex plane into two disjoint domains.

math-ph↗

arXiv2026

Density of reflection resonances in one-dimensional disordered Schrödinger operators

We develop an analytic approach to evaluating the density $ρ({\cal E},Γ)$ of complex resonance poles with real energies $\mathcal{E}$ and widths $Γ$ in the pure reflection problem from a one-dimensional disordered sample with white-noise random potential. We start with establishing a general link between the density of resonances and the distribution of the reflection coefficient $r=|R(E,L)|^2$, where $R(E,L)$ is the reflection amplitude, at {\it complex} energies $E = {\cal E} +iη$, identifying the parameter $η>0$ with the uniform rate of absorption within the disordered medium. We show that leveraging this link allows for a detailed analysis of the resonance density in the weak disorder limit. In particular, for a (semi)infinite sample, it yields an explicit formula for $ρ({\cal E},Γ)$, describing the crossover from narrow to broad resonances in a unified way. Similarly, our approach yields a limiting formula for $ρ({\cal E},Γ)$ in the opposite case of a short disordered sample, with size much smaller than the localization length. This regime seems to have not been systematically addressed in the literature before, with the corresponding analysis requiring an accurate and rather non-trivial implementation of WKB-like asymptotics in the scattering problem. Finally, we study the resonance statistics numerically for the one-dimensional Anderson tight-binding model and compare the results with our analytic expressions.

cond-mat.dis-nn↗

arXiv2025

Ground state energy fluctuations of pinned elastic manifolds

We describe the atypical fluctuations of the ground state energy of the random elastic manifold, a disordered model defined on a lattice of linear size $L$ with internal dimension $0\leq d<4$ embedded in a medium of dimension $N\gg 1$. The ground-state energy results from a competition between confinement, elasticity and disorder. We obtain an exact description of the large deviation rate function with speed $NL^d$ and its different phases, corresponding to different patterns of replica symmetry breaking (RSB). Our results show that the ground-state energy satisfies a central limit theorem and we obtain an explicit expression for the rescaled variance. In the (massless) limit of zero confinement, this variance vanishes for short-range disorder and the ground-state energy displays super-concentration. From our results on the large deviation function, we characterise explicitly the left tail of the distribution of the typical fluctuations of the ground state energy. It displays an exponential tail for a one step RSB pattern while for a full RSB pattern it decays super-exponentially with a non trivial exponent $ξ$ that we compute explicitly.

cond-mat.stat-mech↗

arXiv2025

Spectral Density and Eigenvector Nonorthogonality in Complex Symmetric Random Matrices

Non-Hermitian random matrices with statistical spectral characteristics beyond the standard Ginibre ensembles have recently emerged in the description of dissipative quantum many-body systems as well as in non-ergodic wave transport in complex media. We investigate the class AI$^†$ of complex symmetric random matrices, for which available analytic results remain scarce. Using a recently proposed framework by one of the authors, we analyze this class for Gaussian entries and derive an explicit, closed-form expression for the joint distribution of a complex eigenvalue and its right eigenvector for arbitrary matrix size $N\ge 2$ in the entire complex plane. From this, we obtain the distribution of the eigenvector non-orthogonality overlap and the mean eigenvalue density, both for finite $N$ and in the large-$N$ limit. Notably, at the spectral edge both the eigenvalue density and eigenvector statistics exhibits a limiting behavior that differs from the Ginibre universtality class. This behavior is expected to be universal, as further supported by numerical evidence for Bernoulli random matrices.

math-ph↗

arXiv2025

Non-orthogonal eigenvectors, fluctuation-dissipation relations and entropy production

Celebrated fluctuation-dissipation theorem (FDT) linking the response function to time dependent correlations of observables measured in the reference unperturbed state is one of the central results in equilibrium statistical mechanics. In this Letter we discuss an extension of the standard FDT to the case when multidimensional matrix representing transition probabilities is strictly non-normal. This feature dramatically modifies the dynamics, by incorporating the effect of eigenvector nonorthogonality via the associated overlap matrix of Chalker-Mehlig type. In particular, the rate of entropy production per unit time is strongly enhanced by that matrix. We suggest, that this mechanism has an impact on the studies of collective phenomena in neural matrix models, leading, via transient behavior, to such phenomena as synchronization and emergence of the memory. We also expect, that the described mechanism generating the entropy production is generic for wide class of phenomena, where dynamics is driven by non-normal operators. For the case of driving by a large Ginibre matrix the entropy production rate is evaluated analytically, as well as for the Rajan-Abbott model for neural networks.

cond-mat.stat-mech↗

arXiv2025

Statistics of the non-zero eigenvalues and singular values of low-rank random matrices with non-negative entries

We compute analytically the probability distribution and moments of the sum and product of the non-zero eigenvalues and singular values of random matrices with (i) non-negative entries, (ii) fixed rank, and (iii) prescribed sums of the entries in each row. Applications of such matrices are discussed in the context of Markov chains, economics and social networks to name a few. All results are valid at finite matrix size and are given in terms of the statistics of vectors of general Dirichlet random variables. Analytical results are corroborated by numerical simulations throughout with excellent agreement.

cond-mat.stat-mech↗

arXiv2024

Zeros of conditional Gaussian analytic functions, random sub-unitary matrices and q-series

We investigate radial statistics of zeros of hyperbolic Gaussian Analytic Functions (GAF) of the form $φ(z) = \sum_{k\ge 0} c_k z^k$ given that $|φ(0)|^2=t$ and assuming coefficients $c_k$ to be independent standard complex normals. We obtain the full conditional distribution of $N_q$, the number of zeros of $φ(z)$ within a disk of radius $\sqrt{q}$ centred at the origin, and prove its asymptotic normality in the limit when $q\to 1^{-}$, the limit that captures the entire zero set of $φ(z)$. In the same limit we also develop precise estimates for conditional probabilities of moderate to large deviations from normality. Finally, we determine the asymptotic form of $P_k(t;q)=\mathrm{Prob} \{ N_q= k | |φ(0)|^2=t \}$ in the limit when $k$ is kept fixed whilst $q$ approaches 1. To leading order, the hole probability $P_0(t;q)$ does not depend on $t$ for $t>0$ but yet is different from that of $P_0(t=0;q)$ and coincides with the hole probability for unconditioned hyperbolic GAF of the form $\sum_{k\ge 0} \sqrt{k+1}\, c_k z^k$. We also find that asymptotically as $q \to 1^{-}$, $P_k(t;q)= e^t P_{k}(0;q)$ for every fixed $k \ge 1$ with $P_{k}(0;q)= \mathrm{Prob} \{ N_q =k-1 \}$.

math.PR↗

arXiv2024

Superposition of plane waves in high spatial dimensions: from landscape complexity to the deepest minimum value

In this article, we introduce and analyse some statistical properties of a class of models of random landscapes of the form ${\cal H}({\bf x})=\fracμ{2}{\bf x}^2+\sum_{l=1}^M ϕ_l({\bf k}_l\cdot {\bf x}), \, \, {\bf x}\in \mathbb{R}^N,\,\, μ>0 $ where both the functions $ϕ_l(z)$ and vectors ${\bf k}_l$ are random. An important example of such landscape describes superposition of $M$ plane waves with random amplitudes, directions of the wavevectors, and phases, further confined by a parabolic potential of curvature $μ$. Our main efforts are directed towards analysing the landscape features in the limit $N\to \infty, M\to \infty$ keeping $α=M/N$ finite. In such a limit we find (i) the rates of asymptotic exponential growth with $N$ of the mean number of all critical points and of local minima known as the annealed complexities and (ii) the expression for the mean value of the deepest landscape minimum (the ground-state energy). In particular, for the latter we derive the Parisi-like optimisation functional and analyse conditions for the optimiser to reflect various phases for different values of $μ$ and $α$: replica-symmetric, one-step and full replica symmetry broken, as well as criteria for continuous, Gardner and random first order transitions between different phases.

cond-mat.dis-nn↗

arXiv2024

Mean eigenvector self-overlap in the real and complex elliptic Ginibre ensembles at strong and weak non-Hermiticity

We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in $N\times N$ non-Hermitian random Gaussian matrices. In well known works by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as $N\to \infty$. In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles characterized by correlations between off-diagonal entries controlled by a parameter $τ\in[0,1]$, with $τ=1$ corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite $N$, for any eigenvalue off the real axis. We further investigate several scaling regimes as $N\rightarrow \infty$, both in the limit of strong non-Hermiticity keeping a fixed $τ\in[0,1)$ and in the weak non-Hermiticity limit, with $τ$ approaching unity in such a way that $N(1-τ)$ remains finite.

math-ph↗

arXiv2023

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 ${\cal P}(I)$ for the single-point intensity $I$ decays as a power law for large intensities: ${\cal P}(I)\sim I^{-(M+2)}$, provided there is no internal losses. This behaviour is in marked difference with the Rayleigh law ${\cal P}(I)\sim \exp(-I/\overline{I})$ which turns out to be valid only in the limit $M\to \infty$. We also find the joint probability density of intensities $I_1, \ldots, I_L$ in $L>1$ observation points, and then extract the corresponding statistics for the maximal intensity in the observation pattern. For $L\to \infty$ the resulting limiting extreme value statistics (EVS) turns out to be different from the classical EVS distributions.

cond-mat.dis-nn↗

arXiv2023

Mean left-right eigenvector self-overlap in the real Ginibre ensemble

We study analytically the Chalker-Mehlig mean diagonal overlap $\mathcal{O}(z)$ between left and right eigenvectors associated with a complex eigenvalue $z$ of $N\times N$ matrices in the real Ginibre ensemble (GinOE). We first derive a general finite $N$ expression for the mean overlap and then investigate several scaling regimes in the limit $N\rightarrow \infty$. While in the generic spectral bulk and edge of the GinOE the limiting expressions for $\mathcal{O}(z)$ are found to coincide with the known results for the complex Ginibre ensemble (GinUE), in the region of eigenvalue depletion close to the real axis the asymptotic for the GinOE is considerably different. We also study numerically the distribution of diagonal overlaps and conjecture that it is the same in the bulk and at the edge of both the GinOE and GinUE, but essentially different in the depletion region of the GinOE.

math-ph↗

arXiv2023

On the density of complex eigenvalues of Wigner reaction matrix in a disordered or chaotic system with absorption

In an absorptive system the Wigner reaction $K-$matrix (directly related to the impedance matrix in acoustic or electromagnetic wave scattering) is non-selfadjoint, hence its eigenvalues are complex. The most interesting regime arises when the absorption, taken into account as an imaginary part of the spectral parameter, is of the order of the mean level spacing. I show how to derive the mean density of the complex eigenvalues for reflection problems in disordered or chaotic systems with broken time-reversal invariance. The computations are done in the framework of nonlinear $σ-$ model approach, assuming fixed $M$ and $N\to \infty$. Some explicit formulas are provided for zero-dimensional quantum chaotic system as well as for a semi-infinite quasi-1D system with fully operative Anderson localization.

cond-mat.dis-nn↗

arXiv2023

Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass

We derive, within the replica formalism, a generalisation of the Crisanti-Sommers formula to describe the large deviation function (LDF) ${\cal L}(e)$ for the speed-$N$ atypical fluctuations of the intensive ground-state energy $e$ of a generic spherical spin-glass in the presence of a random external magnetic field of variance $Γ$. We then analyse our exact formula for the LDF in much detail for the Replica symmetric, single step Replica Symmetry Breaking (1-RSB) and Full Replica Symmetry Breaking (FRSB) situations. Our main qualitative conclusion is that the level of RSB governing the LDF may be different from that for the typical ground state. We find that while the deepest ground-states are always controlled by a LDF of replica symmetric form, beyond a finite threshold $e\geq e_{t}$ a replica-symmetry breaking starts to be operative. These findings resolve the puzzling discrepancy between our earlier replica calculations for the $p=2$ spherical spin-glass and the rigorous results by Dembo and Zeitouni which we are able to reproduce invoking an 1-RSB pattern. Finally at an even larger critical energy $e_{c}\geq e_{t}$, acting as a "wall", the LDF diverges logarithmically, which we interpret as a change in the large deviation speed from $N$ to a faster growth. In addition, we show that in the limit $Γ\to 0$ the LDF takes non-trivial scaling forms (i) ${\cal L}(e) \sim G((e-e_c)/Γ)$ in the vicinity of the wall (ii) ${\cal L}(e) \sim Γ^{ην} F((e-e_{\rm typ})/Γ^ν)$ in the vicinity of the typical energy, characterised by two new exponents $η\geq 1$ and $ν$ characterising universality classes. Via matching the latter allows us to formulate several conjectures concerning the regime of {\it typical fluctuations}, identified as $e-e_{\rm typ} \sim N^{-1/η}$ and $Γ\sim N^{-1/(ην)}$.

cond-mat.stat-mech↗

arXiv2023

Resonances in a single-lead reflection from a disordered medium: $σ$-model approach

Using the framework of supersymmetric non-linear $σ$-model we develop a general non-perturbative characterisation of universal features of the density $ρ(Γ)$ of the imaginary parts (``width'') for $S$-matrix poles (``resonances'') describing waves incident and reflected from a disordered medium via $M$-channel waveguide/lead. Explicit expressions for $ρ(Γ)$ are derived for several instances of systems with broken time-reversal invariance, in particular for quasi-1D and 3D media. In the case of perfectly coupled lead with a few channels ($M\sim 1$) the most salient features are tails $ρ(Γ)\sim Γ^{-1}$ for narrow resonances reflecting exponential localization and $ρ(Γ)\sim Γ^{-2}$ for broad resonances reflecting states located in the vicinity of the attached wire. For multimode quasi 1D wires with $M\gg 1$, an intermediate asymptotics $ρ(Γ)\sim Γ^{-3/2}$ is shown to emerge reflecting diffusive nature of decay into wide enough contacts.

cond-mat.dis-nn↗

arXiv2023

The high-d landscapes paradigm: spin-glasses, and beyond

We review recent developments on the characterization of random landscapes in high-dimension. We focus in particular on the problem of characterizing the landscape topology and geometry, discussing techniques to count and classify its stationary points and stressing connections with the statistical physics of disordered systems and with random matrix theory.

cond-mat.dis-nn↗