Principal Investigator: Mark Adler
The project aims at investigating various connections between random matrix theory, various statistical processes,integrable mechanics and the Virasoro algebra Specifically,one project is to study non-colliding Brownian motion a la Dyson,on the line and on the circle,including the Dyson elliptic Brownian motion;the connection with matrix models in a chain is particularly relevant to finding PDE's for the joint probabilities of the motion.Using scaling limits arising in the context of random matrices and permutations,the Dyson motions above tend to novel limiting random processes.The motion of the outmost particle tends to the so-called Airy process.Finding stochastic differential equations for these processes hinges on intricate statistical questions about the universality laws appearing in random matrix theory.Another project is to find large time asymptotics,like asymptotic covariances,for the Dyson processes and the limiting processes as well.The investigators believe that each of the universality laws in random matrix theory connects with an integrable system and the algebra of Virasoro constraints.The problem is to find these sysems and to extract interesting information about the distribution functions and their differential equations.Finally,the Dyson circular motion has an interesting realization in terms of the "Stochastic Loewner equation",providing a conformal map realization of this motion.Its Ito stochastic differential equation is related-in a mysterious way-to the Virasoro algebra,which also naturally comes up in questions of non-colliding random walks and the Fokker-Planck equations.These connections will be investigated.
The mathematical physics above has applications in the statistiical analysis that comes up in many practical problems involving a small number of sources ,each generating lots of data,like antennas receiving information,analyzing ecological data from a small number of sources ,each generating lots of data,etc.The point being that in many practical problems large rectangular arrays of data come up,which are big in one direction,but not the other.The statistical processes that come up also seem to come up in lots of growth models that should be relevant in industrial processes.In addition the universality laws that arise in the random matrix theory arise quite naturally in quantum mechanics,in studying large atoms and so may prove useful in understanding chemical reactions in physical chemistry and hence in manufacturing drugs through simulation experiments one day.
