Catastrophic events, though rare, do occur and when they occur, they have devastating effects. The study of the critical dynamics in complex systems is always interesting yet challenging. First, we present a brief overview of the random matrix theory and correlated Wishart ensemble. Then, we choose financial market as an example of a complex system, and do the analysis of the S&P 500 (USA) stock market based on the evolution of cross-correlation structure patterns. We identify â€œmarket statesâ€ as clusters of similar correlation structures, which occur more frequently than by pure chance (randomness). Power mapping method from the random matrix theory is used to suppress the noise on correlation patterns, and an adaptation of the intra-cluster distance method is used to obtain the optimum number of market states, and also identify the â€œprecursorsâ€ to the crashes. The dynamics of the transitions between the states are interesting. Further, the resulting â€œemerging spectrumâ€ of eigenvalues near zero, have intriguing properties: (i) the shape of the emerging spectrum reflects the market instability, (ii) the smallest eigenvalue is able to statistically distinguish the nature of a market crash or crisis. We finally investigate whether the smallest eigenvalue is able to predict a high market correlation, which is a signature of a crash.
Anirban Chakraborti is a Professor at the Jawaharlal Nehru University, New Delhi. He was working as an Associate Professor at the Ã‰cole Centrale Paris, France, during 2009-14. He obtained a Ph.D. in Physics from the Saha Institute of Nuclear Physics, India and then the Habilitation in Physics from UniversitÃ© Pierre et Marie Curie (Paris VI), France. He has the experience of working as a scientist in many reputed universities and educational institutions in India, USA, Europe and Japan. He was awarded the prestigious Young Scientist medal of the Indian National Science Academy in 2009. He has published several books and research articles from internationally renowned publishers.