Universal scaling of the tail of the density of eigenvalues in random matrix models

Large random matrices have eigenvalue density distributions limited to a finite support. Near the endpoint of the support, when the size N of the matrices is large, one can study a scaling region of size N^-μ in which the cross-over from a non-zero density to a vanishing density takes place. This cross-over is shown to be universal (for random hermitian matrices with a unitary-invariant probability distribution), in the sense that it depends only on the order of multicriticality of the problem. For a multicritical point of odd order k, the large-N density vanishes as |λ-λ_c|^{k- 1 2} near λ_c. The cross-over function of the scaling variables (λ_c-λ)N^μ and (g_c-g)N^ν (where g is the coupling constant characterizing the potential or equivalently the cosmological constant of 2D quantum gravity) is related to the resolvent of the Schrödinger operator in which the potential is the scaling function which satisfies the string equation. The exponents are found to be μ= 2 (2k+1) and ν= 2k (2k+1).