Critical edge behavior in unitary random matrix ensembles and the thirty fourth Painleve transcendent

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We describe a new universality class for unitary invariant random matrix ensembles. It arises in the double scaling limit of ensembles of random n × n Hermitian matrices Zₙ,ₙ⁻¹ | M|²α e-N V(M) dM with α> -1/2, where the factor | M|²α induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we compute the limiting eigenvalue correlation kernel in the double scaling limit as n, N → ∞ such that n²/³(n/N-1) = O(1). We use the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight |x|²α e-NV(x). Our main attention is on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution.