Longrange Voter Model on the Real Line and Related Topics

• Applicant: Professor Dr. Achim Klenke
• Subject Area: Mathematics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 561077737

This project investigates the long-range voter model on the real line. The focus is on the alpha-stable voter model, which displays a more complex behaviors than the nearest-neighbor case. Key objectives include understanding the properties of the interface and behavior of the support, martingale problem characterization of the model, and studying coalescing alpha-stable processes. Interface Properties: In the alpha-stable voter model, the interface (the boundary between different “opinions”) is more intricate than for the nearest neighbor case. The upper bound on the Hausdorff dimension of this interface has been established, but determining if this bound is sharp and finding a lower bound remains open. This project aims to resolve these questions and also to check existence of isolated interface points, potentially occurring at random times. Support Properties: Prior work has shown that in the alpha-stable model with bounded initial support for opinion 1, the support remains bounded at fixed times. However, it is unknown if there are exceptional times when this support becomes unbounded. We conjecture that such times exist and give a precise conjecture for their Hausdorff dimension. This project seeks to verify these conjectures, which would deepen understanding of the support’s behavior over time. Martingale Problem Characterization: Currently, the alpha-stable voter model is understood mainly through duality. This project aims to derive a martingale problem characterization of the model. Success in this area could provide insights into colony lifetimes and even related models like symbiotic branching processes. Coalescing alpha-Stable Processes: As the dual to the alpha-stable voter model, coalescing alpha-stable processes exhibit a “coming down from infinity” property. One of the project aims if to determine the rate of coming down from infinity for this model. We also plan to construct a space-time duality for the alpha-stable voter model, similar to the Brownian web’s duality with the nearest-neighbor model. This duality could offer a deeper understanding of both the alpha-stable voter model and its dual. Overall, this research seeks to address significant questions around long-range voter models, which are substantially more complex than nearest-neighbor models. By making progress in the above problems, this project aims to shed light on fundamental open questions related to interacting population models on a continuum space and to the behavior of coalescing processes, contributing to current research in probability.

• DFG Programme: Research Grants
• International Connection: Israel
• Partner Organisation: The Israel Science Foundation
• Cooperation Partner: Professor Leonid Mytnik, Ph.D.