Non-perturbative effects in complex systems: A study through the theory of random matrices and orthogonal polynomials

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The paradigm of complex systems is that an extermely simple model (for example, a collection of a large number of identical particles, called Fermions, which only communicate through an extremely simple rule: no two particles are allowed to occupy the same point in space) can exhibit highly organised behaviour as n, the number of constituents, becomes large.A random matrix is an array of numbers arranged in an n columns by n rows grid with the numbers determined from say, the throw of a die, which introducesthe element of unpredictability. It turns out that for a particular, but ubiqiutous familyof random matrices, certain real numbers (eigenvalues) that are fundamental to its description are in one-to-one correspondence with the n Fermions mentioned above.To understand the collective behaviour of this system, we apply a small external probeand observe the response of the system. It turns out that if the probe is asmall and in some sense smooth perturbation, thesystem will tend to remain in the original unperturbed state.However, even a weak but non-smooth external probe can produce responses qualitatively distinct from what can be normally expected. In such situations, the usual approach of expansion in terms of small parameters --- the perturbative treatment --- breaks down completely. New methods to deal with non-perturbative effects will be developed in the proposed research to explain such behaviour.