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Inter-relations between random matrix ensembles with different symmetry typesprovide inter-relations between generating functions for the gap probabilitesat the spectrum edge. Combining these in the scaled limit with the exactevaluation of the gap probabilities for certain superimposed ensembles withorthogonal symmetry allows for the exact evaluation of the gap probabilities atthe hard and soft spectrum edges in the cases of orthogonal and symplecticsymmetry. These exact evaluations are given in terms of Painlev\'etranscendents, and in terms of Fredholm determinants.

Hard and soft edge spacing distributions for random matrix ensembles with orthogonal and symplectic symmetry