lntegrability and geometry in random plane partitions

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We study integrable systems lying with geometorical structures that appear and play important roles in the recent progress of mathematical physics, in particular, gauge theories and string theories. Exact solutions for supersymmetric gauge theories and exact amplitudes of topological strings show that random partitions and random plane partitions reflect combinatorial and geometrical structures of these theories. We investigate the integrable structure(1-Toda hierarchy and its dispertion-less limit) of random plane partitions and its thermodynamic limit by utilizing quantum torus symmetry, giving the generalized string equation in a partial form. We also derive the semiclassical generalized string equation which describes the thermodynamic limit of random partitions.