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Random walks on rank one symmetric spaces of noncompact type
Gnetov F., Konakov V.
We establish a central limit theorem, a local limit theorem, and a law of large numbers for a natural
random walk on a symmetric space M of non-compact type and rank one. This class of spaces, which
includes the complex and quaternionic hyperbolic spaces and the Cayley hyperbolic plane, generalizes
the real hyperbolic space Hn. Our approach introduces a unified algebraic framework that generalizes
the M¨obius addition, previously used for the constant curvature case, to define the random walk via a
non-Euclidean summation of variables. We demonstrate that the renormalized walk converges to the
heat kernel associated with the Laplace-Beltrami operator on M, which plays the role of the limiting
normal law. The proofs leverage the harmonic analysis of spherical functions on symmetric spaces. To
the best of our knowledge, these results are new in the context of rank one symmetric spaces.
Dorovskiy A., / Series arXiv "math". 2026.
In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere is proved. A classification of these families up to moderate equivalence in neighborhoods of their large bifurcation supports is presented, based on such invariants as the configuration and the characteristic set. The realization lemma is proved. ...
Added: May 14, 2026
Taletskii D., / Series arXiv "math". 2026.
A vertex subset of a graph is called a \textit{distance-$k$ independent set} if the distance between any two of its distinct vertices is at least $k + 1$. For all $n,k \geq 1$, we determine the minimum possible number of inclusion-wise maximal distance-$k$ independent sets among all $n$-vertex trees. It equals~$n$ if $n \leq k ...
Added: May 1, 2026
Ovcharenko M., / Series arXiv "math". 2026.
We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in n⩽4 variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, ...
Added: April 30, 2026
Zlotnik Alexander, / Series arXiv "math". 2026. No. 2602.03481v1.
We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(\eta,u,\theta)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(\eta^0,u^0,e^0)$ in a ...
Added: April 18, 2026
Medvedev V., / Series arXiv "math". 2026.
We investigate the interplay between the dimension of the space of static potentials and the geometric and topological structure of the underlying static three-manifold. A partial classification of boundaryless static manifolds is obtained in terms of this dimension. We also treat the case of static manifolds with boundary. In particular, we prove that if a ...
Added: April 3, 2026
Gabdullin N., Androsov I., / Series Computer Science "arxiv.org". 2026.
Label prediction in neural networks (NNs) has O(n) complexity proportional to the number of classes. This holds true for classification using fully connected layers and cosine similarity with some set of class prototypes. In this paper we show that if NN latent space (LS) geometry is known and possesses specific properties, label prediction complexity can ...
Added: April 2, 2026
Kolesnikov A., / Series arXiv "math". 2025.
We study Blaschke--Santal{ó}-type inequalities for N>=2 sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{ó}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of N>2 sets.
We also discuss links to the ...
Added: February 13, 2026
Sorokin K., Beketov M., Онучин А. et al., / arxiv.org. Серия cs.SI "Social and Information Networks ". 2025.
Community detection in complex networks is a fundamental problem, open to new approaches in various scientific settings. We introduce a novel community detection method, based on Ricci flow on graphs. Our technique iteratively updates edge weights (their metric lengths) according to their (combinatorial) Foster version of Ricci curvature computed from effective resistance distance between the ...
Added: January 15, 2026
Gaianov N., Parusnikova A., / Cornell University. Серия math "arxiv.org". 2025.
An algebraic q-difference equation is considered. A sufficient condition for the existence of a formal power-logarithmic expansion of a solution to such an equation in the neighborhood of zero is proposed. An example of applying this sufficient condition for constructing a formal expansion of a solution to a certain q-difference analogue of the fifth Painlevé equation ...
Added: December 25, 2025
Popov V., / Series arXiv "math". 2025. No. 2502.01539.
We prove that the variety of flexes of algebraic curves
of degree 3 in the projective plane is an ideal theoretic complete
intersection in the product of a two-dimensional and a nine-dimensional projective spaces. ...
Added: December 16, 2025
Kazakov A., Koryakin V., Safonov K. et al., / Series arXiv "math". 2025.
The Lorenz attractor is the first example of a robustly chaotic non-hyperbolic attractor. Each orbit of such an attractor has a positive top Lyapunov exponent, and this property persists under small perturbations despite possible bifurcations of the attractor. In this paper, we study the boundary of the Lorenz attractor existence region in the Shimizu-Morioka model. ...
Added: December 4, 2025
Bitter I., Konakov V., / Cornell University. Серия arXiv "math". 2025. № 2505.24548.
В работе приводится обобщение локальной предельной теоремы о сходимости неоднородных цепей Маркова к диффузионному пределу на случай, когда соответ- ствующие коэффициенты процессов удовлетворяют слабым условиям регулярности и совпадают лишь асимптотически. В частности, рассматриваемые нами коэффици- енты сноса могут быть неограниченными с не более чем линейным ростом, а оценки отражают перенос терминального состояния неограниченным трендом через ...
Added: December 3, 2025
Bogomolov F. A., Schrandt S., / Series arXiv "math". 2025.
We discuss phenomena of stabilization for direct images of line bundles over projective curves mapping onto the projective line, for maps of sufficiently big degree. ...
Added: December 1, 2025
Deviatov R., Baek S., / Series arXiv "math". 2025.
The torsion index of split simple groups has been extensively studied, notably by Totaro, who calculated the torsion indexes of the spin groups and $E_{8}$ in [5] and [6], respectively. The aim of this paper is to provide upper bounds for the torsion index of half-spin groups, the only remaining case in the calculation of ...
Added: December 1, 2025
Hessian-based lightweight neural network for brain vessel segmentation on a minimal training dataset
Меньшиков И. А., Бернадотт А. К., Elvimov N. S., / Series arXie "Statistical mechanics". 2025.
Accurate segmentation of blood vessels in brain magnetic resonance angiography (MRA) is essential for successful surgical procedures, such as aneurysm repair or bypass surgery. Currently, annotation is primarily performed through manual segmentation or classical methods, such as the Frangi filter, which often lack sufficient accuracy. Neural networks have emerged as powerful tools for medical image ...
Added: December 1, 2025
Prokhorov Y., / Series arXiv "math". 2025.
A $\mathbf{Q}$-conic bundle is a contraction $f: X\to Z$ of a three-dimensional algebraic variety $X$ to a surface~$Z$ such that the variety~$X$ has only terminal $\mathbf{Q}$-factorial singularities, the anticanonical divisor $-K_X$ is~$f$-ample, and $\uprho(X/Z)=1$. We provide an algorithm to transform a $\mathbf{Q}$-conic bundle to its standard form. ...
Added: December 1, 2025
Amerik E., Verbitsky M., Soldatenkov A., / Series arXiv "math". 2025.
Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They defined ``Mukai's elementary transformation'' as the blow-up of a subvariety ruled by complex projective spaces, composed with the contraction ...
Added: December 1, 2025
Kuznetsova A., / Series arXiv "math". 2025.
We study birational automorphisms of algebraic varieties of bounded growth, i.e. such that the norms of the inverse images ${(f^n)}^* \colon \mathrm{NS}(X)\to \mathrm{NS}(X)$ of the powers of the automorphism $f\in\mathrm{Bir}(X)$ are bounded above for $n\geqslant 0$. We prove that some power of an infinite order automorphism of a variety $X$ with such property factors either ...
Added: December 1, 2025
Kratz M., Prokopenko E., Extremes 2023 Vol. 26 No. 3 P. 509–544
We build a sharp approximation of the whole distribution of the sum of iid heavy-tailed random vectors, combining mean and extreme behaviors. It extends the so-called ’normex’ approach from a univariate to a multivariate framework. We propose two possible multi-normex distributions, named d-Normex and MRV-Normex. Both rely on the Gaussian distribution for describing the mean behavior, ...
Added: February 20, 2025
Glinskiy V., Artem Logachov, Logachova O. et al., Mathematics 2024 Vol. 12 No. 21 Article 3319
We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. However, we propose a one-sample estimator where the condition results from a ...
Added: February 19, 2025
Logachov A.V., Mogulskii A. A., Yambartsev A. A., Siberian Electronic Mathematical Reports 2024 Vol. 21 No. 2 P. 914–926
We obtain a bound for the convergence rate in the central limit theorem for the number of triangles in a heterogeneous Erdos-Renyi graphs. Our approach is reminiscent of Hoeffding decomposition (a common technique in the theory of U-statistics). We show that the centered and normalized number of triangles asymptotically behaves as the normalized sum of ...
Added: February 19, 2025
A. Logachov, Logachova O., Yambartsev A., Stochastic Processes and their Applications 2024 Vol. 176 Article 104447
In this paper, we propose a new definition of catastrophes and present our results on large deviations for Poisson processes with catastrophes that satisfy this definition. Our earlier work focused on (almost) uniformly distributed catastrophes, but the current paper extends the results to a larger class of catastrophes. We show that the rate function remains ...
Added: December 7, 2024
Ulyanov V. V., Mathematics 2024 Vol. 12 No. 14 Article 2276
In 1733, de Moivre, investigating the limit distribution of the binomial distribution, was the first to discover the existence of the normal distribution and the central limit theorem (CLT). In this review article, we briefly recall the history of classical CLT and martingale CLT, and introduce new directions of CLT, namely Peng’s nonlinear CLT and ...
Added: October 31, 2024
Lyulko Y., Theory of Probability and Its Applications 2010 Vol. 54 No. 3 P. 516–525
In this paper we consider the problems of finding stochastic representations of functional F = F(ω) of a random walk. We obtain both single and multiple representations of functionals FN = max k ≦ N Sk, Fτ-a = max k ≦ τ-a Sk, where τ-a is the time of the first reaching the level-a, a ∈ N, by a random walk. ...
Added: August 28, 2024