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The distribution function for the first eigenvalue spacing in the Laguerreunitary ensemble of finite rank random matrices is found in terms of aPainlev\'e V system, and the solution of its associated linear isomonodromicsystem. In particular it is characterised by the polynomial solutions to theisomonodromic equations which are also orthogonal with respect to a deformationof the Laguerre weight. In the scaling to the hard edge regime we find ananalogous situation where a certain Painlev\'e \IIId system and its associatedlinear isomonodromic system characterise the scaled distribution. We undertakeextensive analytical studies of this system and use this knowledge toaccurately compute the distribution and its moments for various values of theparameter $ a $. In particular choosing $ a=\pm 1/2 $ allows the firsteigenvalue spacing distribution for random real orthogonal matrices to becomputed.

The Distribution of the first Eigenvalue Spacing at the Hard Edge of the Laguerre Unitary Ensemble