Yan Fyodorov

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We reconsider the problem of calculating arbitrary negative integer moments of the (regularized) characteristic polynomial for N×N random matrices taken from the Gaussian Unitary Ensemble (GUE). A very compact and convenient integral representation is found via the use of a matrix integral close to that considered by Ingham and Siegel. We find the(More)
Abstract We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials; b) constructed from Cauchy transforms of the same orthogonal polynomials and finally c) those constructed(More)
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(More)
We calculate the negative integer moments of the (regularized) characteristic polynomials of N × N random matrices taken from the Gaussian Orthogonal Ensemble (GOE) in the limit as N → ∞. The results agree nontrivially with a recent conjecture of Berry & Keating motivated by techniques developed in the theory of singularity-dominated strong fluctuations.(More)
We investigate some implications of the freezing scenario proposed by Carpentier and Le Doussal (CLD) for a Random Energy Model (REM) with logarithmically correlated random potential. We introduce a particular (circular) variant of the model, and show that the integer moments of the partition function in the hightemperature phase are given by the well-known(More)
We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N × N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard ”supersymmetry” approach, we integrate out Grassmann variables at the early stage and(More)
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(More)
We consider an ensemble of large non-Hermitian random matrices of the form Ĥ + iÂs, where Ĥ and Âs are Hermitian statistically independent random N × N matrices. We demonstrate the existence of a new nontrivial regime of weak non-Hermiticity characterized by the condition that the average of NTrÂs is of the same order as that of TrĤ 2 when N → ∞. We find(More)
We calculate a general spectral correlation function of products and ratios of characteristic polynomials for a N × N random matrix taken from the chiral Gaussian Unitary Ensemble (chGUE). Our derivation is based upon finding a Itzykson-Zuber type integral for matrices from the non-compact manifold Gl(n, C)/U(1)× ... × U(1) (matrix Macdonald function). The(More)
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(More)