arXiv subjects

Anna Maltsev

Publications and source records attributed to Anna Maltsev.

16 recordsLinked to original sources

arXiv2026

Mesoscopic Linear Statistics for Two Ensembles of Quantum Graphs

We study mesoscopic linear spectral statistics for two ensembles of random quantum graphs. These are defined by a discrete graph $G$ and a unitary-matrix-valued function $U(k)$ indexed by directed edges of $G$. The matrix function $U(k)$ is constructed from unitary matrices $U^{(v)}$ indexed by the neighbours of each vertex $v$. The first ensemble is obtained by sampling the underlying discrete graph uniformly from the set of $d$-regular graphs. The second ensemble is obtained by sampling $U^{(v)}$ uniformly from the Haar measure, independently for each vertex. We prove that the variance of a linear spectral statistic in the large graph limit on polynomial mesoscopic scales coincides with that of the Gaussian Orthogonal/Unitary Ensemble.

math-ph↗

arXiv2026

Moments at the hard edge and Rayleigh functions

Motivated by the analogy between spectral moments of random matrices and associated zeta functions, we study inverse power trace moments of the Laguerre ensemble of dimension $N$ and inverse temperature parameter $\beta>0$. We consider a large $N$ regime determined by the low-lying eigenvalues of the ensemble known as the hard edge. In the classical cases $\beta \in \{1,2,4\}$, we obtain explicit results for the inverse moments and extend these to formulae for the corresponding Mellin transforms. In the case of general $\beta>0$, by a result of Fyodorov and Le Doussal, we obtain a different formula for the moments given as a sum over partitions. We use this to consider a low temperature limit where $\beta \to \infty$ as $N \to \infty$. In this limit, we show that the moments are given in terms of the Bessel zeta function.

math-ph↗

arXiv2024

Eigenvalue statistics of Elliptic Volatility Model with power-law tailed volatility

In this paper we study an ensemble of random matrices called Elliptic Volatility Model, which arises in finance as models of stock returns. This model consists of a product of independent matrices $X = ΣZ $ where $Z$ is a $T$ by $S$ matrix of i.i.d. light-tailed variables with mean 0 and variance 1 and $Σ$ is a diagonal matrix. In this paper, we take the randomness of $Σ$ to be i.i.d. heavy tailed. We obtain an explicit formula for the empirical spectral distribution of $X^*X$ in the particular case when the elements of $Σ$ are distributed as Student's t with parameter 3. We furthermore obtain the distribution of the largest eigenvalue in more general case, and we compare our results to financial data.

math.PR↗

arXiv2023

Bulk Universality for Complex non-Hermitian Matrices with Independent and Identically Distributed Entries

We consider N x N matrices with complex entries that are perturbed by a complex Gaussian matrix with small variance. We prove that if the unperturbed matrix satisfies certain local laws then the bulk correlation functions are universal in the large N limit. Assuming the entries are independent and identically distributed with a common distribution that has finite moments, the Gaussian component is removed by the four moment theorem of Tao and Vu.

math.PR↗

arXiv2022

Local Marchenko-Pastur law at the hard edge of the Sample Covariance ensemble

Consider an $N$ by $N$ matrix $X$ of complex entries with iid real and imaginary parts. We show that the local density of eigenvalues of $X^*X$ converges to the Marchenko-Pastur law on the optimal scale with probability $1$. We also obtain rigidity of the eigenvalues in the bulk and near both hard and soft edges. Here we avoid logarithmic and polynomial corrections by working directly with high powers of expectation of the Stieltjes transforms. We work under the assumption that the entries have a finite 4th moment and are truncated at $N^{1/4}$. In this work we simplify and adapt the methods from prior papers of Götze-Tikhomirov and Cacciapuoti-Maltsev-Schlein to covariance matrices.

math.PR↗

arXiv2022

Covariance Kernel of Linear Spectral Statistics for Half-Heavy tailed Wigner Matrices

In this paper we analyze the covariance kernel of the Gaussian process that arises as the limit of fluctuations of linear spectral statistics for Wigner matrices with a few moments. More precisely, the process we study here corresponds to Hermitian matrices with independent entries that have $α$ moments for $2<α< 4$. We obtain a closed form $α$-dependent expression for the covariance of the limiting process resulting from fluctuations of the Stieltjes transform by explicitly integrating the known double Laplace transform integral formula obtained in the literature. We then express the covariance as an integral kernel acting on bounded continuous test functions. The resulting formulation allows us to offer a heuristic interpretation of the impact the typical large eigenvalues of this matrix ensemble have on the covariance structure.

math.PR↗

arXiv2022

Fluctuations of linear statistics of half-heavy-tailed random matrices

We consider a Wigner matrix $A$ with entries tail decaying as $x^{-α}$ with $2<α<4$ for large $x$ and study fluctuations of linear statistics $N^{-1}\operatorname{Tr}φ(A)$. The behavior of such fluctuations has been understood for both heavy-tailed matrices (i.e. $α< 2$) and light-tailed matrices (i.e. $α> 4$). This paper fills in the gap of understanding for $2<α<4$. We find that while linear spectral statistics for heavy-tailed matrices have fluctuations of order $N^{-1/2}$ and those for light-tailed matrices have fluctuations of order $N^{-1}$, the linear spectral statistics for half-heavy-tailed matrices exhibit an intermediate $α$-dependent order of $N^{-α/4}$.

math.PR↗

arXiv2019

Mechanisms of Calcium Leak from Cardiac Sarcoplasmic Reticulum Revealed by Statistical Mechanics

Heart muscle contraction is normally activated by a synchronized Ca release from sarcoplasmic reticulum (SR), a major intracellular Ca store. However, under abnormal conditions Ca leaks from the SR, decreasing heart contraction amplitude and increasing risk of life-threatening arrhythmia. The mechanisms and regimes of SR operation generating the abnormal Ca leak remain unclear. Here we employed both numerical and analytical modeling to get mechanistic insights into the emergent Ca leak phenomenon. Our numerical simulations using a detailed realistic model of Ca release unit (CRU) reveal sharp transitions resulting in Ca leak. The emergence of leak is closely mapped mathematically to the Ising model from statistical mechanics. The system steady-state behavior is determined by two aggregate parameters: the analogues of magnetic field ($h$) and the inverse temperature ($β$) in the Ising model, for which we have explicit formulas in terms of SR Ca and release channel opening/closing rates. The classification of leak regimes takes the shape of a phase $β$-$h$ diagram, with the regime boundaries occurring at $h$=0 and a critical value of $β$ ($β*$) which we estimate using a classical Ising model and mean field theory. Our theory predicts that a synchronized Ca leak will occur when $h$>0 and $β>β*$ and a disordered leak occurs when $β<β*$ and $h$ is not too negative. The disorder leak is distinguished from synchronized leak (in long-lasting sparks) by larger Peierls contour lengths, an output parameter reflecting degree of disorder. Thus, in addition to our detailed numerical model approach we also offer an instantaneous computational tool using analytical formulas of the Ising model for respective RyR parameters and SR Ca load that describe and classify phase transitions and leak emergence.

q-bio.SC↗

arXiv2017

Clusters of calcium release channels harness the Ising phase transition to confine their elementary intracellular signals

Intracellular Ca signals represent a universal mechanism of cell function. Messages carried by Ca are local, rapid, and powerful enough to be delivered over the thermal noise. A higher signal to noise ratio is achieved by a cooperative action of Ca release channels such as IP3 receptors or ryanodine receptors arranged in clusters or release units containing a few to several hundred release channels. The release channels synchronize their openings via Ca-induced-Ca-release, generating high-amplitude local Ca signals known as puffs in neurons or sparks in muscle cells. Despite the high release amplitude and positive feedback nature of the activation, Ca signals are strictly confined in time and space by an unexplained termination mechanism. Here we show that the collective transition of release channels from an open to a closed state is identical to the phase transition associated with the reversal of magnetic field in an Ising ferromagnet. We demonstrate this mechanism using numerical model simulations of Ca sparks over a wide range of cluster sizes from 25 to 169 release channels. While prior studies suggested contributions of stochastic attrition and Ca store depletion, our new simple quantitative criterion closely predicts the depletion level required for spark termination for each cluster size. We further formulate exact requirements for a cluster of release channels to follow the Ising model in any cell type. Thus we describe deterministically the behaviour of a system on a coarser scale (release unit) which is random on a finer scale (release channels), bridging the gap between scales. Our results provide the first exact mapping of a nanoscale biological signalling model to an interacting particle system in statistical physics, making the extensive mathematical apparatus available to quantitative biology.

q-bio.CB↗

arXiv2017

Density and spacings for the energy levels of quadratic Fermi operators

The work presents a proof of convergence of the density of energy levels to a Gaussian distribution for a wide class of quadratic forms of Fermi operators. This general result applies also to quadratic operators with disorder, e.g., containing random coefficients. The spacing distribution of the unfolded spectrum is investigated numerically. For generic systems the level spacings behave as the spacings in a Poisson process. Level clustering persists in presence of disorder.

math-ph↗

arXiv2015

Fluctuations in the two-dimensional one-component plasma and associated fourth-order phase transition

We study the distribution of the mean radial displacement of charges of a 2D one-component plasma in the thermodynamic limit $N\to\infty$ at finite temperature $β>0$. We compute explicitly the large deviation functions showing the emergence of a fourth-order phase transition as a consequence of a change of topology in the plasma distribution. This weak phase transition occurs exactly at the ground state of the plasma. These results have been compared with the integrable case (finite $N$) of plasma parameter $βq^2=2$. In this case the problem can be mapped to the stationary properties of 2D Dyson Brownian particles and to a non-Hermitian matrix model.

math-ph↗

arXiv2014

Bounds for the Stieltjes Transform and the Density of States of Wigner Matrices

We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from [22,10], by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate $(\log N)/N$ and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of [22,10,23].

math-ph↗

arXiv2012

Local Marchenko-Pastur Law at the Hard Edge of Sample Covariance Matrices

Let $X_N$ be a $N\times N$ matrix whose entries are i.i.d. complex random variables with mean zero and variance $\frac{1}{N}$. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix $X_N^*X_N$ for $N\to\infty$. We prove that the empirical density of eigenvalues in an interval $[E,E+η]$ converges to the Marchenko-Pastur law locally on the optimal scale, $N η/\sqrt{E} \gg (\log N)^b$, and in any interval up to the hard edge, $\frac{(\log N)^b}{N^2}\lesssim E \leq 4-κ$, for any $κ>0$. As a consequence, we show the complete delocalization of the eigenvectors.

math-ph↗

arXiv2011

Average Density of States for Hermitian Wigner Matrices

We consider ensembles of $N \times N$ Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density function of the entries, we show that the expectation of the density of states on {\it arbitrarily} small intervals converges to the semicircle law, as $N$ tends to infinity.

math-ph↗

arXiv2011

A Wegner estimate for Wigner matrices

In the first part of these notes, we review some of the recent developments in the study of the spectral properties of Wigner matrices. In the second part, we present a new proof of a Wegner estimate for the eigenvalues of a large class of Wigner matrices. The Wegner estimate gives an upper bound for the probability to find an eigenvalue in an interval $I$, proportional to the size $|I|$ of the interval.

math-ph↗

arXiv2009

Universality Limits of a Reproducing Kernel for a Half-Line Schrödinger Operator and Clock Behavior of Eigenvalues

We extend some recent results of Lubinsky, Levin, Simon, and Totik from measures with compact support to spectral measures of Schrödinger operators on the half-line. In particular, we define a reproducing kernel $S_L$ for Schrödinger operators and we use it to study the fine spacing of eigenvalues in a box of the half-line Schrödinger operator with perturbed periodic potential. We show that if solutions $u(ξ, x)$ are bounded in $x$ by $e^{εx}$ uniformly for $ξ$ near the spectrum in an average sense and the spectral measure is positive and absolutely continuous in a bounded interval $I$ in the interior of the spectrum with $ξ_0\in I$, then uniformly in $I$ $$\frac{S_L(ξ_0 + a/L, ξ_0 + b/L)}{S_L(ξ_0, ξ_0)} \to \frac{\sin(πρ(ξ_0)(a - b))}{πρ(ξ_0)(a - b)},$$ where $ρ(ξ)dξ$ is the density of states. We deduce that the eigenvalues near $ξ_0$ in a large box of size $L$ are spaced asymptotically as $\frac{1}{Lρ}$. We adapt the methods used to show similar results for orthogonal polynomials.

math-ph↗