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Yan V. Fyodorov

Publications and source records attributed to Yan V. Fyodorov.

At least 55 records · Page 3Linked to original sources
arXiv2016

Moments of the position of the maximum for GUE characteristic polynomials and for log-correlated Gaussian processes

We study three instances of log-correlated processes on the interval: the logarithm of the Gaussian unitary ensemble (GUE) characteristic polynomial, the Gaussian log-correlated potential in presence of edge charges, and the Fractional Brownian motion with Hurst index $H \to 0$ (fBM0). In previous collaborations we obtained the probability distribution function (PDF) of the value of the global minimum (equivalently maximum) for the first two processes, using the {\it freezing-duality conjecture} (FDC). Here we study the PDF of the position of the maximum $x_m$ through its moments. Using replica, this requires calculating moments of the density of eigenvalues in the $β$-Jacobi ensemble. Using Jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary $β>0$ and positive integer $n$ in terms of sums over partitions. For positive moments, this expression agrees with a very recent independent derivation by Mezzadri and Reynolds. We check our results against a contour integral formula derived recently by Borodin and Gorin (presented in the Appendix A from these authors). The duality necessary for the FDC to work is proved, and on our expressions, found to correspond to exchange of partitions with their dual. Performing the limit $n \to 0$ and to negative Dyson index $β\to -2$, we obtain the moments of $x_m$ and give explicit expressions for the lowest ones. Numerical checks for the GUE polynomials, performed independently by N. Simm, indicate encouraging agreement. Some results are also obtained for moments in Laguerre, Hermite-Gaussian, as well as circular and related ensembles. The correlations of the position and the value of the field at the minimum are also analyzed.

math-ph↗
arXiv2016

Large time zero temperature dynamics of the spherical p=2-spin glass model of finite size

We revisit the long time dynamics of the spherical fully connected $p = 2$-spin glass model when the number of spins $N$ is large but {\it finite}. At $T=0$ where the system is in a (trivial) spin-glass phase, and on long time scale $t \gtrsim {\cal O}{(N^{2/3})}$ we show that the behavior of physical observables, like the energy, correlation and response functions, is controlled by the density of near-extreme eigenvalues at the edge of the spectrum of the coupling matrix $J$, and are thus non self-averaging. We show that the late time decay of these observables, once averaged over the disorder, is controlled by new universal exponents which we compute exactly.

cond-mat.dis-nn↗
arXiv2015

On the distribution of maximum value of the characteristic polynomial of GUE random matrices

Motivated by recently discovered relations between logarithmically correlated Gaussian processes and characteristic polynomials of large random $N \times N$ matrices $H$ from the Gaussian Unitary Ensemble (GUE), we consider the problem of characterising the distribution of the global maximum of $D_{N}(x):=-\log|\det(xI-H)|$ as $N \to \infty$ and $x\in (-1,1)$. We arrive at an explicit expression for the asymptotic probability density of the (appropriately shifted) maximum by combining the rigorous Fisher-Hartwig asymptotics due to Krasovsky \cite{K07} with the heuristic {\it freezing transition} scenario for logarithmically correlated processes. Although the general idea behind the method is the same as for the earlier considered case of the Circular Unitary Ensemble, the present GUE case poses new challenges. In particular we show how the conjectured {\it self-duality} in the freezing scenario plays the crucial role in our selection of the form of the maximum distribution. Finally, we demonstrate a good agreement of the found probability density with the results of direct numerical simulations of the maxima of $D_{N}(x)$.

math-ph↗
arXiv2015

Resonance width distribution in RMT: Weak coupling regime beyond Porter-Thomas

We employ the random matrix theory (RMT) framework to revisit the distribution of resonance widths in quantum chaotic systems weakly coupled to the continuum via a finite number M of open channels. In contrast to the standard first-order perturbation theory treatment we do not a priory assume the resonance widths being small compared to the mean level spacing. We show that to the leading order in weak coupling the perturbative $χ^2_M$ distribution of the resonance widths (in particular, the Porter-Thomas distribution at M=1) should be corrected by a factor related to a certain average of the ratio of square roots of the characteristic polynomial ("spectral determinant") of the underlying RMT Hamiltonian. A simple single-channel expression is obtained that properly approximates the width distribution also at large resonance overlap, where the Porter-Thomas result is no longer applicable.

cond-mat.mes-hall↗
arXiv2015

On Random Matrix Averages Involving Half-Integer Powers of GOE Characteristic Polynomials

Correlation functions involving products and ratios of half-integer powers of characteristic polynomials of random matrices from the Gaussian Orthogonal Ensemble (GOE) frequently arise in applications of Random Matrix Theory (RMT) to physics of quantum chaotic systems, and beyond. We provide an explicit evaluation of the large-$N$ limits of a few non-trivial objects of that sort within a variant of the supersymmetry formalism, and via a related but different method. As one of the applications we derive the distribution of an off-diagonal entry $K_{ab}$ of the resolvent (or Wigner $K$-matrix) of GOE matrices which, among other things, is of relevance for experiments on chaotic wave scattering in electromagnetic resonators.

math-ph↗
arXiv2013

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 p_N(θ) of large N\times N random unitary (CUE) matrices; i.e. the extreme value statistics of p_N(θ) when N \rightarrow\infty. In addition, we argue that it leads to multifractal-like behaviour in the total length μ_N(x) of the intervals in which |p_N(θ)|>N^x, x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta-function ζ(s) over stretches of the critical line s=1/2+it of given constant length, and present the results of numerical computations of the large values of ζ(1/2+it). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

math-ph↗
arXiv2012

Counting function fluctuations and extreme value threshold in multifractal patterns: the case study of an ideal $1/f$ noise

To understand the sample-to-sample fluctuations in disorder-generated multifractal patterns we investigate analytically as well as numerically the statistics of high values of the simplest model - the ideal periodic $1/f$ Gaussian noise. By employing the thermodynamic formalism we predict the characteristic scale and the precise scaling form of the distribution of number of points above a given level. We demonstrate that the powerlaw forward tail of the probability density, with exponent controlled by the level, results in an important difference between the mean and the typical values of the counting function. This can be further used to determine the typical threshold $x_m$ of extreme values in the pattern which turns out to be given by $x_m^{(typ)}=2-c\ln{\ln{M}}/\ln{M}$ with $c=3/2$. Such observation provides a rather compelling explanation of the mechanism behind universality of $c$. Revealed mechanisms are conjectured to retain their qualitative validity for a broad class of disorder-generated multifractal fields. In particular, we predict that the typical value of the maximum $p_{max}$ of intensity is to be given by $-\ln{p_{max}} = α_{-}\ln{M} + \frac{3}{2f'(α_{-})}\ln{\ln{M}} + O(1)$, where $f(α)$ is the corresponding singularity spectrum vanishing at $α=α_{-}>0$. For the $1/f$ noise we also derive exact as well as well-controlled approximate formulas for the mean and the variance of the counting function without recourse to the thermodynamic formalism.

cond-mat.stat-mech↗
arXiv2012

Critical Behaviour 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 behaviour of $N_{m}$ in the simplest glass-like transition occuring in a toy model of a single particle in $N$-dimensional random environment, with $N\gg 1$. Varying the control parameter $μ$ through the critical value $μ_c$ we analyse in detail how $N_{m}(μ)$ drops from being exponentially large in the glassy phase to $N_{m}(μ)\sim 1$ on the other side of the transition. We also extract a subleading behaviour of $N_{m}(μ)$ in both glassy and simple phases. The width $δμ/μ_c$ of the critical region is found to scale as $N^{-1/3}$ and inside that region $N_{m}(μ)$ converges to a limiting shape expressed in terms of the Tracy-Widom distribution.

cond-mat.stat-mech↗
arXiv2012

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 x N random unitary (CUE) matrices. We postulate that our results extend to the extreme values taken by the Riemann zeta-function zeta(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.

math-ph↗
arXiv2010

Introduction to the Random Matrix Theory: Gaussian Unitary Ensemble and Beyond

These lectures provide an informal introduction into the notions and tools used to analyze statistical properties of eigenvalues of large random Hermitian matrices. After developing the general machinery of orthogonal polynomial method, we study in most detail Gaussian Unitary Ensemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotach asymptotics of Hermite polynomials in various regimes and employ it in spectral analysis of the GUE. In the last part of the course we discuss general relations between orthogonal polynomials and characteristic polynomials of random matrices which is an active area of current research.

math-ph↗
arXiv2007

On an explicit construction of Parisi landscapes in finite dimensional Euclidean spaces

We construct a N-dimensional Gaussian landscape with multiscale, translation invariant, logarithmic correlations and investigate the statistical mechanics of a single particle in this environment. In the limit of high dimension N>>1 the free energy of the system in the thermodynamic limit coincides with the most general version of Derrida's Generalized Random Energy Model. The low-temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape. We argue that our construction is in fact valid in any finite spatial dimensions, starting from N=1.

cond-mat.dis-nn↗
arXiv2007

Classical Particle in a Box with Random Potential: exploiting rotational symmetry of replicated Hamiltonian

We investigate thermodynamics of a single classical particle placed in a spherical box of a finite radius $R$ and subject to a superposition of a $N-$dimensional Gaussian random potential and the parabolic potential with the curvature $μ>0$. Earlier solutions of $R\to \infty$ version of this model were based on combining the replica trick with the Gaussian Variational Ansatz (GVA) for free energy, and revealed a possibility of a glassy phase at low temperatures. For a general $R$, we show how to utilize instead the underlying rotational symmetry of the replicated partition function and to arrive to a compact expression for the free energy in the limit $N\to \infty$ directly, without any need for intermediate variational approximations. This method reveals striking similarity with the much-studied spherical model of spin glasses. Depending on the value of $R$ and the three types of disorder - short-ranged, long-ranged, and logarithmic - the phase diagram of the system in the $(μ,T)$ plane undergoes considerable modifications. In the limit of infinite confinement radius our analysis confirms all previous results obtained by GVA.

cond-mat.dis-nn↗
arXiv2004

On Hubbard-Stratonovich Transformations over Hyperbolic Domains

We discuss and prove validity of the Hubbard-Stratonovich (HS) identities over hyperbolic domains which are used frequently in the studies on disordered systems and random matrices. We also introduce a counterpart of the HS identity arising in disordered systems with "chiral" symmetry. Apart from this we outline a way of deriving the nonlinear $σ$-model from the gauge-invariant Wegner $k-$orbital model avoiding the use of the HS transformations.

math-ph↗
arXiv2004

Statistics of Impedance, Local Density of States, and Reflection in Quantum Chaotic Systems with Absorption

We are interested in finding the joint distribution function of the real and imaginary parts of the local Green function for a system with chaotic internal wave scattering and a uniform energy loss (absorption). For a microwave cavity attached to a single-mode antenna the same quantity has a meaning of the complex cavity impedance. Using the random matrix approach, we relate its statistics to that of the reflection coefficient and scattering phase and provide exact distributions for systems with beta=2 and beta=4 symmetry class. In the case of beta=1 we provide an interpolation formula which incorporates all known limiting cases and fits excellently available experimental data as well as diverse numeric tests.

cond-mat.mes-hall↗
arXiv2004

Complexity of Random Energy Landscapes, Glass Transition and Absolute Value of Spectral Determinant of Random Matrices

Finding the mean of the total number N_{tot} of critical points for N-dimensional random energy landscapes is reduced to averaging the absolute value of 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 random environment. For N >>1 our asymptotic analysis reveals a phase transition at some critical value μ_c of a control parameter μfrom a phase with finite landscape complexity to the phase with vanishing complexity. The same value of the control parameter is known to correspond to an onset of glassy behaviour at zero temperature. Finally, we discuss a method of dealing with the modulus of the spectral determinant applicable to a broad class of problems.

cond-mat.dis-nn↗
arXiv2004

On the Largest Singular Values of Random Matrices with Independent Cauchy Entries

We apply the method of determinants to study the distribution of the largest singular values of large $ m \times n $ real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a factor of $ \frac{1}{m^2\*n^2}$)largest singular values agree in the limit with the statistics of the inhomogeneous Poisson random point process with the intensity $ \frac{1}π x^{-3/2} $ and, therefore, are different from the Tracy-Widom law. Among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of complex rectangular $ m \times n $ standard Wishart ensemble and real rectangular $ 2m \times 2n $ standard Wishart ensemble.

math.PR↗
arXiv2003

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. Within the RMT model we calculate the quantities of interest exactly, employing the supersymmetry technique. This approach is found to be more accurate than another method based on simplifying naive assumptions for the statistics of the eigenfrequencies and the eigenfunctions. The results of the supersymmetric method are confirmed by Monte Carlo numerical simulation and are used to reveal a possible source of the disagreement between the predictions of the naive theory and ultrasonic measurements.

cond-mat.mes-hall↗