Statistics of Extremes and Records in Random Sequences

Rare events such as earthquakes, tsunamis, and floods fortunately do not occur every day, but when they do, their effects are devastating. Such rare events are particularly important in understanding and characterizing global warming and climate changes. In addition to natural catastrophes, rare events such as big financial crashes also play a significant role in the economy. In the absence of predictive models, the best way forward is to analyse the statistics of these extreme events and draw conclusions about the probability of their occurrences.

Extreme value statistics (EVS) and the statistics of records in a random sequence are examples of a truly interdisciplinary topic, spanning from statistics and mathematics on one side to physics of disordered systems on the other. They have tremendous importance and practical applications in a wide variety of fields, such as climate science, finance, spin-glasses, and random matrices.

Statistics and mathematical literature have explored the subject of the classical theory of EVS. However, more recently, EVS started to play a very important role in statistical physics, in particular in disordered systems. This has led to a plethora of activities, both in the statistical physics and in the mathematics communities over the last few decades. This book develops the theory of rare events, both for the classical uncorrelated as well as for correlated sequences, in terms of simple models and examples.

Statistics of Extremes and Records in Random Sequences is a pedagogical book with examples illustrating the basic tools and techniques that are essential to a student starting to work in this interesting and rapidly developing field.

Satya N. Majumdar, Director of Research, CNRS, LPTMS, Université Paris-Saclay,Grégory Schehr, Senior researcher, CNRS, Sorbonne Université

Satya Majumdar has been the CNRS Research Director at LPTMS (Université Paris-Saclay) since 2004. He is also an adjunct professor at the Tata Institute in Bombay, the Weizmann Institute in Rehovot, Israel, the Higgs Centre of the University of Edinburgh, and the Raman Research Institute in Bangalore, India. He has won several awards including the Paul Langevin Medal of the French Physical Society (2005), the European Physical Society Prize for Statistical and Nonlinear Physics (2019), the CNRS Silver Medal (2019), and the Gay-Lussac Humboldt Prize awarded by the Alexander von Humboldt Foundation (2019).

Grégory Schehr has been a CNRS Research Director at LPTHE (Sorbonne Université) since 2019. Since 2023 he has also been "Professeur Chargé de Cours" at Ecole Polytechnique. In 2010 he was awarded the CNRS Bronze Medal, and in 2022 the Aniuta Winter-Klein Prize of the French Academy of Science.

1. Introduction2. The two principal models and some basic tools3. First-passage probability4. Extreme statistics5. Time of the maximum and the minimum6. Order statistics7. Records8. Extremes in other correlated systems9. Conclusion and perspectivesReferences