arXiv subjects

Anastasiia Trofimova

Publications and source records attributed to Anastasiia Trofimova.

3 recordsLinked to original sources

arXiv2026

Probabilistic Analysis of the Random Spectral Radius for a Matrix Family

We investigate joint spectral characteristics of a family of matrices $\mathcal F $, associated with products in the semigroup generated by $\mathcal F$. In the literature, extremal measures such as the well-known joint spectral radius and the lower spectral radius have been extensively studied. However, these measures fail to capture the typical growth rate of matrix products, focusing instead on the worst and best-case scenarios. Nevertheless, when examining, for instance, a switching dynamical system, a probabilistic rate of growth, which characterizes typical trajectories, emerges as a highly intriguing and significant measure. In this article, we study the random spectral radius, defined as the spectral radius of a length-$n$ product sampled at random from the semigroup according to a given probability measure. We establish asymptotic results, namely a Law of Large Numbers and a Central Limit Theorem, for diagonal (equivalently, commuting), upper- or lower-triangular, and small perturbations of diagonal matrices. Beyond recovering the correct scaling, we obtain exact closed-form expressions for the limiting value and variance, which may be useful in numerical applications requiring precise constants. In the coalescence regime, where the leading eigenvalues merge, the limiting distribution is non-Gaussian: it is given by the maximum of a correlated Gaussian vector with explicit covariance structure. This phenomenon governs phase transitions between distinct growth regimes in switching systems.

math.DS↗

arXiv2025

Approximation bounds for norm constrained deep neural networks

This paper studies the approximation capacity of neural networks with an arbitrary activation function and with norm constraint on the weights. Upper and lower bounds on the approximation error of these networks are computed for smooth function classes. The upper bound is proven by first approximating high-degree monomials and then generalizing it to functions via a partition of unity and Taylor expansion. The lower bound is derived through the Rademacher complexity of neural networks. A probabilistic version of the upper bound is also provided by considering neural networks with randomly sampled weights and biases. Finally, it is shown that the assumption on the regularity of the activation function can be significantly weakened without worsening the approximation error, and the approximation upper bound is validated with numerical experiments.

math.NA↗

arXiv2025

Scaling Properties of Current Fluctuations in Periodic TASEP

We study current fluctuations in the Totally Asymmetric Simple Exclusion Process (TASEP) on a ring with $N$ sites and $p$ particles. By introducing a deformation parameter $γ$, we analyze the tilted operator that governs the statistics of the time-integrated current. Employing the coordinate Bethe ansatz, we derive implicit expressions for the scaled cumulant generating function (SCGF), i.e. the largest eigenvalue, and the spectral gap, both in terms of Bethe roots. Their asymptotic behaviour is characterized by using the geometric structure of Cassini oval. In the thermodynamic limit at fixed particle density, we identify a dynamical phase transition separating fluctuation regimes. For $γ>0$, the SCGF exhibits ballistic growth with system size, $λ_1 \sim N$. In contrast, for $γ<0$, the SCGF converges to $-1$ as $N\to\infty$. This transition is reflected in the spectral gap, which controls the system's relaxation timescale. For $γ>0$, the gap closes at polynomial speed, $Δ\sim N^{-1}$, consistent with rapid relaxation with enhanced current. For $γ<0$, the gap vanishes exponentially, $Δ\sim \exp(-cN)$, signaling metastability with diminished current. Our non-perturbative results provide insights into large deviations and the relaxation dynamics in driven particle systems.

cond-mat.stat-mech↗