Strongly Correlated Extreme Fluctuations

Our Advanced Study Group (ASG) focuses on understanding the impact of long-range correlations on spectral density and eigenvalue statistics in non-equilibrium complex systems. Extreme fluctuations, though rare, can significantly influence these systems, making the study of Extreme Value Theory (EVT) crucial. EVT traditionally deals with independent and identically distributed (i.i.d.) random variables, but real-world applications often involve correlated variables, posing a challenge to classical EVT principles.

Objectives

1. Spectral Properties of Random Matrices: We will investigate the spectral properties of random matrices with strongly correlated elements, focusing on the statistics of their extreme eigenvalues. This includes studying the distribution of the largest eigenvalue in random matrices and understanding the novel universality classes that emerge in strongly correlated systems.

2. Characterizing the Role of Fluctuations in the Emergent Multifractality of Quantum Systems: In the context of the Integer Quantum Hall Effect (IQHE), critical quantum wave functions at the transitions between different quantized Hall plateaus exhibit significant fluctuations and multifractal properties. These wave functions display complex spatial patterns characterized by a spectrum of exponents, indicating multifractal behavior. They form intricate, self-similar structures that play a pivotal role in the phase transition phenomena inherent to the IQHE. We will discuss how such multifractality is related to the correlated fluctuations in the system.

3. Revising Classical EVT: The group aims to discuss the necessity of revising classical EVT to account for the influence of correlated structures. We will explore the relationship between EVT and the parent distributions of correlated data, seeking to develop a comprehensive theory for strongly correlated extremes.

4. Real-World Applications: We will examine recent observations and applications of EVT in various fields such as biophysics, finance, climate science, and disordered systems. This includes understanding the impact of extreme events on biological populations, financial markets, and power-grid stability.

5. Machine Learning and AI: The potential of modern techniques like machine learning (ML) and artificial intelligence (AI) in predicting extreme events will be a key focus. We will explore how these technologies can enhance the predictability of extreme events in complex, interacting systems by identifying underlying correlated structures from extreme records.

Importance and Challenges

• Extreme Events in Natural and Man-Made Systems: Natural disasters like floods, earthquakes, and extreme weather events, as well as fluctuations in power grids, highlight the practical importance of studying extreme values. Understanding and predicting these events can help mitigate their devastating impacts.

• Biological and Socio-Economic Systems: Extreme events can lead to significant biological responses, such as species extinction, and socio-economic disruptions, including market crashes. Addressing these challenges requires extending EVT to account for the strong correlations observed in these systems.

• Theoretical and Practical Advances: Despite progress, there remains a lack of comprehensive theories for EVT in strongly correlated variables. The ASG will facilitate discussions on theoretical advancements and practical applications, aiming to bridge this gap and provide new insights into the behavior of complex systems under extreme conditions.

Our ASG brings together experts from statistical physics, mathematical physics, biophysics, finance, and climate science to explore these interconnected disciplines. By combining theoretical foundations with real-world applications, we aim to advance our understanding of extreme events and their impacts on strongly correlated systems.