Random matrices
Gregory Schehr
Since the pioneering work of Wishart in statistics followed by Wigner and others in nuclear physics, random matrix theory (RMT) has found a huge number of applications ranging from statistical physics of disordered systems all the way to number theory and finance. During the last two decades, the study of the largest eigenvalue of large random matrices has attracted a special attention. In particular, its limiting law, the Tracy-Widom distribution, has emerged in a wide variety of fundamental problems in statistical mechanics, e.g., in the so-called Kardar-Parisi-Zhang universality class, which describes stochastic growth processes or a directed polymer in disordered media in 1+1 dimensions.
During the last few years, our group has unveiled deep connections between non-interacting trapped fermions and RMT, paving the way to observe experimentally the Tracy-Widom distribution in simple one-dimensional cold atom experiments. More recent developments have explored the relations between interacting trapped fermions and RMT. Indeed, for a certain choice of interactions and trapping potential in one-dimension, the ground-state of such systems is related to the so-called Gaussian beta-ensembles, well known in RMT.
