Number of common sites visited by N random walkers

M. Tamm; Majumdar S.

?

Number of common sites visited by N random walkers

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 2012. Vol. 86. P. 021135.

Tamm M., Majumdar S.

We compute analytically the mean number of common sites, WN(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N−d) plane, there are three distinct regimes for the asymptotic large-t growth of WN(t). These three regimes are separated by two critical lines d=2 and d=dc(N)=2N/(N−1) in the (N-d) plane. For d<2, WN(t)∼td/2 for large t (the N dependence is only in the prefactor). For 2<d<dc(N), WN(t)∼tν where the exponent ν=N−d(N−1)/2 varies with N and d. Ford>dc(N), WN(t)→const as t→∞. Exactly at the critical dimensions there are logarithmic corrections: for d=2, we get WN(t)∼t/[lnt]N, while for d=dc(N), WN(t)∼lnt for large t. Our analytical predictions are verified in numerical simulations.

Research target: Mathematics Physics Computer Science

Priority areas: mathematics

Language: English

Full text

Keywords: random walks Brownian motion

Робототехника и искусственный интеллект: международные практики и возможности для России, Дальнего Востока и Арктики

Кузнецов М. Е., Полякова М., Лукьянович В. et al., ФАНУ "Востокгосплан", 2026.

Обзор международных практик развития робототехники и искусственного интеллекта и оценка возможностей их применения в условиях России, в первую очередь для Дальнего Востока и Арктической зоны РФ ...

Added: September 16, 2026

Universality of robust chaotic dynamics in a family of one-dimensional Lorenz maps

Kazakov A., Koryakin V., Safonov K. et al., Journal of Differential Equations 2026 Vol. 480 Article 114626

We study a family of one-dimensional maps that models the dynamics of a system of differential equations with a Lorenz attractor near a bifurcation curve where the system has a pair of homoclinic loops with zero separatrix value. Of particular interest is the region of the parameter plane where the map has a robust chaotic ...

Added: September 16, 2026

Continuously pleated torus or strange nonchaotic attractors seen as continuously fractalized quasiperiodic attractors

Letellier C., Stankevich N., Houri S. et al., Chaos 2026 Vol. 36 No. 9 Article 093128

Toroidal chaos designates a chaotic solution that is structured in the neighborhood of a torus. For being chaotic, such a torus needs to be discontinuously fractalized to present a Cantor set, ensuring the great sensitivity to initial conditions required for chaos. A fractal is related to self-similarity and a non-integer dimension. In fact, although clearly ...

Added: September 16, 2026

The Effect of the Gate Material in MIS Structures on the High-Field Charge Degradation in the Gate Dielectric

Andreev D. V., Kornev S. A., Bondarenko G.G. et al., Inorganic Materials: Applied Research 2026 Vol. 17 No. 5 P. 1226–1230

Comparative studies are carried out to investigate high-field charge degradation effects in the gate dielectric of metal—insulator—semiconductor (MIS) structures with aluminum and polysilicon gates. Charging phenomena in MIS structures are explored using high-field electron injection into the dielectric under conditions of gradually increasing voltage current density, with periodic short-term switching to a measurement mode at a constant low injection ...

Added: September 15, 2026

Vague stimulating ideas, images and metaphors in scientific thinking: researchers' work with horizons of unclear knowledge

Poddiakov A., / Series Social Science Research Network "Social Science Research Network". 2026. No. 7437658.

Clarity of knowledge and reasoning is necessary in many cases. Yet vagueness in scientific thinking related to surprise, curiosity, "ability to engage with not-knowing" (de Freitas) and abductive reasoning is also a crucially important source of scientific creativity which supplements combinatorial logic when dealing with the already known. Starting from studies by C. S. Peirce ...

Added: September 15, 2026

Water confined in heterogeneous clay–quartz pores: Structure and dynamics from atomistic simulations

Glushak A., Tararushkin E., Smirnov G., Colloids and Surfaces A: Physicochemical and Engineering Aspects 2027 Vol. 752 Article 141756

The molecular-scale behavior of water in heterogeneous clay–quartz systems is critical for understanding cementation processes in natural soils and compacted bentonite, which is crucial for oil recovery and nuclear waste disposal barriers construction. Here we investigate how heterogeneity of mixed Na-montmorillonite and α-quartz surfaces affect water structure and dynamics in nanopores of 2.5 to 60 Å ...

Added: September 15, 2026

Semi-fragile image watermarking: A comprehensive survey, classification, and research directions

Melman A., Evsyutin O., Computer Science Review 2027 Vol. 63 (A) Article 101066

One of the effective means of protecting digital images is invisible watermarking. Different scenarios have different requirements for watermark resistance to post-processing. Robust watermarking schemes are resistant to a wide range of modifications. Fragile watermarking schemes are designed for integrity control and are destroyed by even minor modifications of images. Semi-fragile watermarking schemes combine features ...

Added: September 15, 2026

Some Aspects of Remote State Restoring in State Transfer Governed by XXZ-Hamiltonian

Bochkin G., Doronin S., Fel’dman E. et al., Lobachevskii Journal of Mathematics 2024 Vol. 45 No. 3 P. 1188–1207

We consider the remote state restoring and perfect transfer of the zero-order coherence matrix (PTZ) in a spin system governed by the XXZ-Hamiltonian conserving the excitation number. The restoring tool is represented by several nonzero Larmor frequencies in the Hamiltonian. To simplify the analysis we use two approximating models including either step-wise or pulse-type time-dependence ...

Added: September 14, 2026

Coherence restoring in communication line via controlled interaction with environment

Fel’dman E. B., Lazarev I., Pechen A. N. et al., Quantum Information Processing 2026 Vol. 25 Article 121

We consider the state-restoring protocol based on the controlled interaction of a linear chain with environment through incoherent control by the specially adjusted step-wise time-dependent Lindblad operators. We show that the best restoring result (maximal scale factors in the restored state) corresponds to the symmetrical Lindblad equation. (0, 1)-excitation dynamics is considered numerically, and restoring ...

Added: September 14, 2026

Macroscopic entanglement distribution with atomic ensembles

Li S., Hu J., Lazarev I. et al., Annals of Physics 2026 Vol. 494 Article 170618

The distribution of entanglement is a crucial task for quantum communication towards realizing a globe-spanning quantum internet. Recently, a protocol for deterministic long-distance distribution of macroscopic entanglement over a network of ensembles of qubits was introduced (Pyrkov et al., 2025). It was shown that this protocol allows for the propagation of macroscopic amounts of entanglement ...

Added: September 14, 2026

Quantum Repeater Protocol for Deterministic Distribution of Macroscopic Entanglement, Advanced Quantum Technologies

Pyrkov A., Lazarev I., Byrnes T., Advanced Quantum Technologies 2025 Article 2400524

Distributing long-distance entanglement is a fundamental goal that is necessary for a variety of tasks such as quantum communication, distributed quantum computing, and quantum metrology. Currently quantum repeater schemes typically aim to distribute one ebit at a time, the equivalent of one Bell pair's worth of entanglement. Here, a scheme is presented to distribute a ...

Added: September 14, 2026

Hybrid quantum-classical unsupervised data clustering based on the self-organizing feature map

I. D. Lazarev, Narozniak M., Byrnes T. et al., Physical Review A: Atomic, Molecular, and Optical physics 2025 Vol. 111 No. 012416 Article 012416

Unsupervised machine learning is one of the main techniques employed in artificial intelligence. We introduce an algorithm for quantum-assisted unsupervised data clustering using the self-organizing feature map, a type of artificial neural network. The complexity of our algorithm scales as 𝑂⁡(𝐿⁢𝑁), in comparison to the classical case which scales as 𝑂⁡(𝐿⁢𝑀⁢𝑁), where 𝑁 is the ...

Added: September 14, 2026

Mixing of real and quantum gases

Борщевский А. Я., Physical Chemistry Chemical Physics 2025 Vol. 27 No. 9 P. 4793–4803

Through the consistent application of quantum statistical thermodynamics, functions of isothermal mixing of real rarefied gases and quantum gases in states of varying degrees of degeneracy are obtained. The reasons for the emergence and resolution of the “Gibbs paradox” and other “paradoxes” are analyzed. Numerical calculations of entropy and pressure jumps were performed using the ...

Added: September 14, 2026

Автоматизированное формирование журналов событий на основе неструктурированных Интернет-источников для задач анализа процессов

Воронова К. Д., Lyadova L. N., Proceedings of the Institute for System Programming of the RAS 2026 Vol. 38 No. 4 P. 153–170

Title: Automated Event Logs Generation Based on Unstructured Internet Sources for Process Analysis Tasks Abstract. This paper presents an approach to automated structuring event-related information extracted from unstructured textual Internet sources for process mining tasks. In many practical cases, information on events associated with various processes is not presented in the form of ready-made event logs, but is ...

Added: September 14, 2026

Intelligent Decision Technologies. Proceedings of the 17th KES-IDT 2025 Conference

Springer, 2026.

This book gathers selected papers from the KES-IDT 2025 conference, held in Solin, Croatia, on June 25–27, 2025. The book presents and discusses the latest research results and generates new ideas in the field of intelligent decision-making. The range of topics discussed is classification, prediction, data analysis, big data, data science, decision support, knowledge engineering, ...

Added: September 13, 2026

Интегральная оценка сценариев кибератак на АСУ ТП с применением цифрового двойника и киберполигона

Lebedev O. B., Митяков Е. С., Саенко И. Б. et al., Информация и космос 2026 № 1 С. 53–58

The article presents a methodology for the integrated assessment of cyberattacks on industrial control systems (ICS) using a combination of the Ampire cyber range and a digital twin. The approach is based on calculating five criteria—severity, attacker capabilities, vulnerability, complexity, and defense effectiveness—that form an integrated threat index. Testing across three scenarios confirmed the index's ...

Added: September 13, 2026

Децентрализованное управление группой автономных подвижных объектов при формировании траектории движения

Lebedev O. B., Лебедев Б. К., Бесхмельнов М. И., Известия ЮФУ. Технические науки 2024 № 2(242) С. 177–190

This article examines algorithms for generating motion trajectories for unmanned aerial vehicles (UAVs) during search-and-rescue and emergency response operations. It describes methods and algorithms for controlling the movement of a UAV group in various formations—such as line, rank, column, and during turns. Control is achieved through alternative collective adaptation algorithms based on principles of collective ...

Added: September 13, 2026

Управление передвижением группы БПЛА с соблюдением геометрической структуры строя на основе альтернативной коллективной адаптации

Lebedev O. B., Котов Д. В., Известия ЮФУ. Технические науки 2024 № 1(237) С. 155–167

The primary approach to solving motion planning and control problems involves the use of intelligent technologies. These technologies are applied to define and adjust control objectives and action programs, as well as to formulate control algorithms under conditions of uncertainty—stemming from various factors—across actuator elements and the motion control, planning, and behavioral subsystems. This paper ...

Added: September 13, 2026

Гибридный алгоритм оптимизации на основе объединения моделей роевого интеллекта: роя хромосом и пчел

Lebedev O. B., Лебедев Б. К., Ганжур М. А., Информатизация и связь 2023 № 2 С. 80–84

A hybrid algorithm has been developed to solve the VLSI component placement problem by combining procedures based on bee colony behavior and a chromosome swarm. The optimization criterion is based on estimates of the number of nets crossing specific lines within the routing area. A modified chromosome swarm paradigm is described; unlike the canonical method, ...

Added: September 13, 2026

МОДЕЛИРОВАНИЕ ПОСТРОЕНИЯ ТРАЕКТОРИИ ПОДВИЖНОГО ОБЪЕКТА НА ОСНОВЕ МЕТОДОВ МНОГОКРИТЕРИАЛЬНОЙ ОПТИМИЗАЦИИ

Lebedev O. B., Лебедев Б. К., Бесхмельнов М. И., Вестник Тверского государственного технического университета 2025 № 3(27) С. 103–119

The article presents a detailed analysis of a hybrid path-planning algorithm designed for navigation in a two-dimensional space, accounting for partial uncertainty regarding terrain information. The algorithm operates in stages and is divided into two levels. At the first level, the object's local visibility zone is established, and the corresponding region on the map is ...

Added: September 12, 2026

ГИБРИДНЫЙ МЕТОД ПЛАНИРОВАНИЯ КОНФИГУРАЦИИ МАРШРУТА НА КАРТЕ МЕСТНОСТИ В УСЛОВИЯХ ЧАСТИЧНОЙ НЕОПРЕДЕЛЕННОСТИ

Lebedev O. B., Лебедев Б. К., Бесхмельнов М. И., Известия ЮФУ. Технические науки 2025 Т. 2(244) С. 68–82

This paper describes a hybrid situational trajectory planning algorithm for 2D space operating under conditions of partial uncertainty. Based on the integration of wave-propagation and ant colony optimization algorithms, it enables the real-time generation of minimum-length trajectories while simultaneously optimizing other path quality criteria. The processes of generating a trajectory segment and moving the object ...

Added: September 12, 2026

OPTIMIZATION BY THE SWARM OF TRANSFORMING CHROMOSOMES METHOD

Lebedev O. B., Beskhielnov M. I., OPTIMIZATION BY THE SWARM OF TRANSFORMING CHROMOSOMES METHOD 2024 Vol. 34 No. 3 P. 652–658

A new strategy for finding effective solutions based on modification and hybridization of the canonical representations of the particle swarm algorithm is proposed. A paradigm of a swarm of transforming chromosomes is described, making it possible to represent solutions in the affine space of chromosomes with integer parameter values. The mechanisms of chromosome transformation in ...

Added: September 12, 2026

On phase-lock area parquet in a special slow-fast limit of model of Josephson junction.

Glutsyuk A., / Series arXiv "math". 2026.

B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, dθdτ=1ω(cosθ+B+Acosτ), which is known as ...

Added: September 8, 2026

Infinitely many graph manifolds with unique geometrical piece that admit arbitrarily many Anosov flows

Pochinka O., Shmukler V., / Series math.RT "arXiv:1808.06395 [math.RT]". 2026.

Anosov flows have a long and rich history, firstly motivated by the study of geodesic flows in negative curvature surface by Anosov and Sinai. Not every closed manifold admits an Anosov flow for well-known reasons: the fundamental group of a 3-manifold M admitting an Anosov flow must have exponential growth, and M must be universally covered by R3. Nevertheless, there ...

Added: August 31, 2026