Topic: Investigating the Asymptotic Behavior and Central Limit Theorems for Random Walks on Non-Euclidean Geometries and Their Applications in Graph Embeddings

Project Abstract

This work investigates the asymptotic behavior and central limit phenomena for random walks defined on non-Euclidean geometries, with a focus on hyperbolic, spherical, and more general curved manifolds, and explores their implications for graph embeddings and network analysis. We develop a unified probabilistic and geometric framework to characterize long-term displacement, hitting times, and distributional limits of random walks whose transition mechanisms are constrained by curvature, local connectivity, and metric measures intrinsic to the underlying space. By leveraging tools from stochastic processes, Riemannian geometry, and harmonic analysis, we derive conditions under which scaling limits converge to stable, Gaussian, or mixture distributions, elucidating how curvature, volume growth, and spectral properties influence the rate of convergence and the form of the limiting process. A central objective is to extend classical central limit theorems to non-Euclidean settings, identifying the correct normalization sequences and identifying anisotropic effects induced by curvature that yield nontrivial covariance structures. We also investigate concentration phenomena and large deviations for functionals of the random walk, such as the empirical distribution of visited states and the mean position in geodesic coordinates, establishing asymptotic equivalences and perturbative expansions around symmetric reference spaces. The methodological core combines heat kernel analysis, sub-Gaussian estimates, and coupling techniques with spectral theory of Laplace-type operators on manifolds, providing explicit error bounds for finite-time approximations and practical criteria for model selection in simulations. In parallel, we translate these asymptotic results into graph embedding algorithms. By interpreting non-Euclidean random walks as processes on graphs with curvature-aware transition probabilities, we develop embedding schemes that preserve intrinsic geometric features such as hyperbolic growth patterns and spherical symmetry. Our embeddings demonstrate improved fidelity for networks with hierarchical or clustered structures, enabling more accurate link prediction, motif discovery, and visualization in high-dimensional data. A series of numerical experiments illustrate how different curvatures influence diffusion speed, spectral gaps, and embedding distortions, while theoretical results offer guidance on choosing step sizes and normalization to achieve stable convergence. The study contributes to a deeper understanding of how geometry governs stochastic dynamics and informs practical embedding techniques for complex networks, with potential applications in neuroscience, social networks, and data science where non-Euclidean geometries naturally arise. We conclude with a discussion of limitations, potential generalizations to manifolds with variable curvature and singularities, and directions for future research at the intersection of probability, geometry, and machine learning.

Project Overview

What This Project Is About

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A plain-language overview of the topic and what the project investigates.

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The Problem It Addresses

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What problem or gap this project tackles and why it matters to the field or society.

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Objectives of the Project

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1. Understand how random walks behave on non-Euclidean shapes.
2. Explore central limit behavior in these settings with simple examples.
3. Connect findings to practical ideas in graph embeddings (placing graphs in a space).
4. Develop intuition about asymptotic (long-term) trends without heavy math.
5. Identify limitations and potential real-world applications.

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What You Will Do Step by Step

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1. Study basic concepts of random walks and non-Euclidean geometries in plain terms.
2. Review simple literature examples to see how similar problems are approached.
3. Formulate small, manageable models of walks on curved spaces or graphs.
4. Analyze long-run behavior with approachable, non-technical explanations.
5. Illustrate ideas with diagrams and basic simulations or visualizations.
6. Discuss how these ideas could improve graph layouts used in data analysis.

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Expected Outcome

Clear, student-friendly understanding of how randomness behaves on non-flat spaces and how that informs graph embedding techniques, along with simple visual demonstrations and a short written summary of findings.