Painleve II in random matrix theory and related fields

We review some occurrences of Painlev\'e II transcendents in the study of two-dimensional Yang-Mills theory, fluctuation formulas for growth models, and as distribution functions within random matrix theory. We first discuss settings in which the parameter $\alpha$ in the Painlev\'e equation is zero, and the boundary condition is that of the Hasting-MacLeod solution. As well as expressions involving the Painlev\'e transcendent itself, one encounters the sigma form of the Painlev\'e II equation, and Lax pair equations in which the Painlev\'e transcendent occurs as coefficients. We then consider settings which give rise to general $\alpha$ Painlev\'e II transcendents. In a particular random matrix setting, new results for the corresponding boundary conditions in the cases $\alpha = \pm 1/2$, 1 and 2 are presented.Comment: 25 pages, prepared for the special issue on Painleve equations in the journal Constructive Approximatio

Discrete Painlev\'e equations and random matrix averages

The $\tau$-function theory of Painlev\'e systems is used to derive recurrences in the rank $n$ of certain random matrix averages over U(n). These recurrences involve auxilary quantities which satisfy discrete Painlev\'e equations. The random matrix averages include cases which can be interpreted as eigenvalue distributions at the hard edge and in the bulk of matrix ensembles with unitary symmetry. The recurrences are illustrated by computing the value of a sequence of these distributions as $n$ varies, and demonstrating convergence to the value of the appropriate limiting distribution.Comment: 25 page

Moments of the Gaussian β\beta Ensembles and the large-NN expansion of the densities

The loop equation formalism is used to compute the $1/N$ expansion of the resolvent for the Gaussian $\beta$ ensemble up to and including the term at $O(N^{-6})$. This allows the moments of the eigenvalue density to be computed up to and including the 12-th power and the smoothed density to be expanded up to and including the term at $O(N^{-6})$. The latter contain non-integrable singularities at the endpoints of the support --- we show how to nonetheless make sense of the average of a sufficiently smooth linear statistic. At the special couplings $\beta = 1$, $2$ and $4$ there are characterisations of both the resolvent and the moments which allows for the corresponding expansions to be extended, in some recursive form at least, to arbitrary order. In this regard we give fifth order linear differential equations for the density and resolvent at $\beta = 1$ and $4$, which complements the known third order linear differential equations for these quantities at $\beta = 2$.Comment: 30 pages, final version has been trimmed and additional moments have been computed, which are recorded in the Appendi

Application of the τ\tau-function theory of Painlev\'e equations to random matrices: \PVI, the JUE, CyUE, cJUE and scaled limits

Okamoto has obtained a sequence of $\tau$-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 multi-dimensional 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. 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 $\tau$-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.Comment: AMSLate

On the Variance of the Index for the Gaussian Unitary Ensemble

We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a $\tau$-function of the fourth Painlev\'e equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation for the variance.Comment: Added references and authors dat

Discrete Painlev\'e equations, Orthogonal Polynomials on the Unit Circle and N-recurrences for averages over U(N) -- \PIIIa and \PV τ\tau-functions

In this work we show that the $N\times N$ Toeplitz determinants with the symbols $z^{\mu}\exp(-{1/2}\sqrt{t}(z+1/z))$ and $(1+z)^{\mu}(1+1/z)^{\nu}\exp(tz)$ -- known $\tau$-functions for the \PIIIa and \PV systems -- are characterised by nonlinear recurrences for the reflection coefficients of the corresponding orthogonal polynomial system on the unit circle. It is shown that these recurrences are entirely equivalent to the discrete Painlev\'e equations associated with the degenerations of the rational surfaces $D^{(1)}_{6} \to E^{(1)}_{7}$ (discrete Painlev\'e {\rm II}) and $D^{(1)}_{5} \to E^{(1)}_{6}$ (discrete Painlev\'e {\rm IV}) respectively through the algebraic methodology based upon of the affine Weyl group symmetry of the Painlev\'e system, originally due to Okamoto. In addition it is shown that the difference equations derived by methods based upon the Toeplitz lattice and Virasoro constraints, when reduced in order by exact summation, are equivalent to our recurrences. Expressions in terms of generalised hypergeometric functions ${{}^{\vphantom{(1)}}_0}F^{(1)}_1, {{}^{\vphantom{(1)}}_1}F^{(1)}_1$ are given for the reflection coefficients respectively.Comment: AMS-Latex2e with AMS macro

Application of the τ\tau-function theory of Painlev\'e equations to random matrices: \PV, \PIII, the LUE, JUE and CUE

With $$ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is $\tilde{E}_N(I;a,\mu) :=$ for $I = (0,s)$ and $I = (s,\infty)$, where $\chi_I^{(l)} = 1$ for $\lambda_l \in I$ and $\chi_I^{(l)} = 0$ otherwise. Using Okamoto's development of the theory of the Painlev\'e V equation, it is shown that $\tilde{E}_N(I;a,\mu)$ is a $\tau$-function associated with the Hamiltonian therein, and so can be characterised as the solution of a certain second order second degree differential equation, or in terms of the solution of certain difference equations. The cases $\mu = 0$ and $\mu = 2$ are of particular interest as they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue. In the case $I = (s,\infty)$, $\tilde{E}_N(I;a,\mu)$ is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard edge and soft edge scaled limits of $\tilde{E}_N(I;a,\mu)$. In particular, in the hard edge scaled limit it is shown that the limiting quantity $E^{\rm hard}((0,s);a,\mu)$ can be evaluated as a $\tau$-function associated with the Hamiltonian in Okamoto's theory of the Painlev\'e III equation.Comment: AMSLatex with CIMS macro

Discrete Painlev\'e equations for a class of \PVI τ\tau-functions given as U(N) averages

In a recent work difference equations (Laguerre-Freud equations) for the bi-orthogonal polynomials and related quantities corresponding to the weight on the unit circle $w(z)=\prod^m_{j=1}(z-z_j(t))^{\rho_j}$ were derived.Here it is shown that in the case $m=3$ these difference equations, when applied to the calculation of the underlying U(N) average, reduce to a coupled system identifiable with that obtained by Adler and van Moerbeke using methods of the Toeplitz lattice and Virasoro constraints. Moreover it is shown that this coupled system can be reduced to yield the discrete fifth Painlev\'e equation \dPV as it occurs in the theory of the sixth Painlev\'e system. Methods based on affine Weyl group symmetries of B\"acklund transformations have previously yielded the \dPV equation but with different parameters for the same problem. We find the explicit mapping between the two forms. Applications of our results are made to give recurrences for the gap probabilities and moments in the circular unitary ensemble of random matrices, and to the diagonal spin-spin correlation function of the square lattice Ising model.Comment: Companion work to CA/041239

Exact Wigner surmise type evaluation of the spacing distribution in the bulk of the scaled random matrix ensembles

Random matrix ensembles with orthogonal and unitary symmetry correspond to the cases of real symmetric and Hermitian random matrices respectively. We show that the probability density function for the corresponding spacings between consecutive eigenvalues can be written exactly in the Wigner surmise type form $a(s) e^{-b(s)}$ for $a$ simply related to a Painlev\'e transcendent and $b$ its anti-derivative. A formula consisting of the sum of two such terms is given for the symplectic case (Hermitian matrices with real quaternion elements).Comment: 6 pages, Latex2

Singular Values of Products of Ginibre Random Matrices

The squared singular values of the product of $M$ complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a hard edge scaling to a form specified in terms of certain Meijer G-functions, or equivalently hypergeometric functions ${}_0 F_M$, also referred to as hyper-Bessel functions. In the case $M=1$ it is well known that the corresponding gap probability for no squared singular values in $(0,s)$ can be evaluated in terms of a solution of a particular sigma form of the Painlev\'e III' system. One approach to this result is a formalism due to Tracy and Widom, involving the reduction of a certain integrable system. Strahov has generalised this formalism to general $M \ge 1$, but has not exhibited its reduction. After detailing the necessary working in the case $M=1$, we consider the problem of reducing the 12 coupled differential equations in the case $M=2$ to a single differential equation for the resolvent. An explicit 4-th order nonlinear is found for general hard edge parameters. For a particular choice of parameters, evidence is given that this simplifies to a much simpler third order nonlinear equation. The small and large $s$ asymptotics of the 4-th order equation are discussed, as is a possible relationship of the $M=2$ systems to so-called 4-dimensional Painlev\'e-type equations.Comment: 33 pages, 4 figures, 1 tabl