Application of theτ-function theory of Painlevé equations to random matrices:P_VI, the JUE, CyUE, cJUE and scaled limits

Abstract

Okamoto has obtained a sequence of τ-functions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter a an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the τ-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter a = 0; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

“…Such an analog was defined as the circular Jacobi ensemble in [12] and [14]. If δ ∈ R, we recover the cJUE as in Witte and Forrester [35].…”

“…Using the union bound and the Markov inequality, we may write 14) and then use explicit distributions. Indeed, we showed in Theorem 3.3 that the vector…”

Random Matrix Theory and Entanglement in Quantum Spin Chains

“…As immediate consequence of (5.15) the remaining normalized eigenvectors of matrix (5.11) are 20) where k = 2πl/M. The corresponding eigenvalues can be obtained by direct substitution:…”

“…When the average is over the other compact groups, the determinant (3.12) can still be interpreted as gap probability generating function for the respective group and its logarithm is still a solution of a differential equation related to the Painlevé VI equation. However, although recurrence formulae analogous to (9.9) exist, at each step the values of (σ 1 , σ 2 ) that label the integral (5.4) change and in general do not even identify one of the classical compact groups (see, e.g., [20]). …”

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

“…We show that our P VI transcendent evaluation for the finite system scales to the infinite system result (1.6). As well as being a special case of the class of integrals related to P VI systems in [11], the multidimensional integral formula for ρ N (x; y) on a circle is also a special case of a class of integrals over the unitary group shown to satisfy integrable recurrence relations in [1]. We will show that these recurrences can alternatively be derived from orthogonal polynomial theory [33].…”