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Aspects of a Phase Transition in High-Dimensional Random Geometry
Axel Prüser1, Imre Kondor2,3,4, Andreas Engel1
1Institute of Physics, Carl von Ossietzky University of Oldenburg, D-26111 Oldenburg, Germany.
Entropy (Basel, Switzerland)
|July 2, 2021
Summary
This study analyzes a geometric phase transition in high-dimensional random systems. This transition impacts financial risk measures like Expected Shortfall, portfolio optimization, and ecological resource competition.
Area of Science:
- Random matrix theory
- Statistical physics
- High-dimensional geometry
Background:
- Phase transitions are critical phenomena observed in various physical and mathematical systems.
- High-dimensional random geometry presents complex challenges in analysis and application.
- Understanding these transitions can unify seemingly disparate scientific problems.
Purpose of the Study:
- To analyze a phase transition in high-dimensional random geometry.
- To connect this transition to diverse applications including finance, machine learning, and ecology.
- To provide a unified framework for understanding these phenomena.
Main Methods:
- Analysis of a minimax problem representing extremal financial risk measures.
- Investigation of the feasibility of random geometric configurations.
- Application of concepts from statistical physics and random matrix theory.
Main Results:
- Identified a geometric phase transition in high-dimensional random systems.
- Demonstrated the transition's relevance to Expected Shortfall calculations and portfolio optimization.
- Showcased the transition's role in perceptron storage capacity and ecological competition.
Conclusions:
- The geometric phase transition offers a unifying principle across multiple scientific domains.
- This framework facilitates a deeper understanding of complex systems.
- Potential for new insights and solutions in finance, computer science, and ecology.
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