arXiv subjects

Yan Fyodorov

Publications and source records attributed to Yan Fyodorov.

4 recordsLinked to original sources

arXiv2025

Large deviations of spectral determinants of matrix-valued random Schr\"odinger operators and Dyson Brownian motion in cubic potentials

We study the moments of $\overline{|\det(H-E)|^q}$ and the associated large deviations of $\log |\det(H-E)|$ where $H$ are random matrix operators involving Laplace operators and random potentials. This includes as a special case Hessians of random elastic manifolds at a generic energy configuration. In one dimension $d=1$ these are $N \times N$ matrix valued random Schr\"odinger operators and $\log | \det(H-E) | $ is the sum of the $N$ associated Lyapunov exponents. Using a mapping to a stochastic matrix Ricatti equation we make a connection between the spectral properties of these operators and the total $N$ particle current of a Dyson Brownian motion (DBM) in a cubic potential. The latter model was studied by Allez and Dumaz [1] who showed that for $N=+\infty$ it exhibits a sharp transition between a phase with non-zero current and a confined (zero current) phase. We compute the barrier-crossing probability of the DBM at large but finite $N$, which gives an estimate of the exponential tail of the average density of states of a matrix Schrodinger operator below the edge of its spectrum. The barrier behaves as $\sim N (-E)^{3/2}$ at large negative energy and vanishes as $\sim N(E^*-E)^{5/4}$ near the edge. For $q=1$ the present work provides an independent derivation of the total complexity of stationary points for an elastic string embedded in $N$ dimension in presence of disorder.

math-ph↗

arXiv2015

On the number of connected components of random algebraic hypersurfaces

We study the expectation of the number of components $b_0(X)$ of a random algebraic hypersurface $X$ defined by the zero set in projective space $\mathbb{R}P^n$ of a random homogeneous polynomial $f$ of degree $d$. Specifically, we consider "invariant ensembles", that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. The classification due to E. Kostlan shows that specifying an invariant ensemble is equivalent to assigning a weight to each eigenspace of the spherical Laplacian. Fixing $n$, we consider a family of invariant ensembles (choice of eigenspace weights) depending on the degree $d$. Under a rescaling assumption on the eigenspace weights (as $d \rightarrow \infty$), we prove that the order of growth of $\mathbb{E} b_0(X)$ satisfies: $$\mathbb{E} b_{0}(X)=Θ\left(\left[ \mathbb{E} b_0(X\cap \mathbb{R}P^1) \right]^{n} \right). $$ This relates the average number of components of $X$ to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial $f|_{\mathbb{R}P^1}.$ The proof requires an upper bound for $\mathbb{E} b_0(X)$, which we obtain by counting extrema using Random Matrix Theory methods from recent work of the first author, and it also requires a lower bound, which we obtain by a modification of the barrier method. We also provide a quantitative upper bound for the implied constant in the above asymptotic; for the real Fubini-Study model these estimates reveal super-exponential decay of the leading coefficient (in $d$) of $\mathbb{E} b_0(X)$ (as $n \rightarrow \infty$).

math.AG↗

arXiv2014

High values of disorder-generated multifractals and logarithmically correlated processes

In the introductory section of the article we give a brief account of recent insights into statistics of high and extreme values of disorder-generated multifractals following a recent work by the first author with P. Le Doussal and A. Rosso (FLR) employing a close relation between multifractality and logarithmically correlated random fields. We then substantiate some aspects of the FLR approach analytically for multifractal eigenvectors in the Ruijsenaars-Schneider ensemble (RSE) of random matrices introduced by E. Bogomolny and the second author by providing an ab initio calculation that reveals hidden logarithmic correlations at the background of the disorder-generated multifractality. In the rest we investigate numerically a few representative models of that class, including the study of the highest component of multifractal eigenvectors in the Ruijsenaars-Schneider ensemble.

cond-mat.dis-nn↗

arXiv1999

Time Delay Correlations and Resonances in 1D Disordered Systems

The frequency dependent time delay correlation function $K(Ω)$ is studied analytically for a particle reflected from a finite one-dimensional disordered system. In the long sample limit $K(Ω)$ can be used to extract the resonance width distribution $ρ(Γ)$. Both quantities are found to decay algebraically as $Γ^{-ν}$, and $Ω^{-ν}$, $ν\simeq 1.25$ in a large range of arguments. Numerical calculations for the resonance width distribution in 1D non-Hermitian tight-binding model agree reasonably with the analytical formulas.

cond-mat.dis-nn↗