Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

Our interest is in the cumulative probabilities Pr(L(t) \le l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) \le l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.Comment: LaTeX, 19 page

Applications and generalizations of Fisher-Hartwig asymptotics

Fisher-Hartwig asymptotics refers to the large $n$ form of a class of Toeplitz determinants with singular generating functions. This class of Toeplitz determinants occurs in the study of the spin-spin correlations for the two-dimensional Ising model, and the ground state density matrix of the impenetrable Bose gas, amongst other problems in mathematical physics. We give a new application of the original Fisher-Hartwig formula to the asymptotic decay of the Ising correlations above $T_c$, while the study of the Bose gas density matrix leads us to generalize the Fisher-Hartwig formula to the asymptotic form of random matrix averages over the classical groups and the Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our generalizations is that they extend to Hankel determinants the Fisher-Hartwig asymptotic form known for Toeplitz determinants.Comment: 25 page

Particles in a magnetic field and plasma analogies: doubly periodic boundary conditions

The $N$-particle free fermion state for quantum particles in the plane subject to a perpendicular magnetic field, and with doubly periodic boundary conditions, is written in a product form. The absolute value of this is used to formulate an exactly solvable one-component plasma model, and further motivates the formulation of an exactly solvable two-species Coulomb gas. The large $N$ expansion of the free energy of both these models exhibits the same O(1) term. On the basis of a relationship to the Gaussian free field, this term is predicted to be universal for conductive Coulomb systems in doubly periodic boundary conditions.Comment: 12 page

Pfaffian Expressions for Random Matrix Correlation Functions

It is well known that Pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices. Moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of Pfaffians. In this article, we review the formulations and applications of Pfaffian formulas. For that purpose, we first present the general Pfaffian expressions in terms of the corresponding skew orthogonal polynomials. Then we clarify the relation to Eynard and Mehta's determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials. Applications of Pfaffian formulas to random matrix theory and other fields are also mentioned.Comment: 28 page

Random walks and random fixed-point free involutions

A bijection is given between fixed point free involutions of $\{1,2,...,2N\}$ with maximum decreasing subsequence size $2p$ and two classes of vicious (non-intersecting) random walker configurations confined to the half line lattice points $l \ge 1$. In one class of walker configurations the maximum displacement of the right most walker is $p$. Because the scaled distribution of the maximum decreasing subsequence size is known to be in the soft edge GOE (random real symmetric matrices) universality class, the same holds true for the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page

Finite N Fluctuation Formulas for Random Matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic $\sum_{j=1}^N (x_j - )$ is computed exactly and shown to satisfy a central limit theorem as $N \to \infty$. For the circular random matrix ensemble the p.d.f.'s for the linear statistics ${1 \over 2} \sum_{j=1}^N (\theta_j - \pi)$ and $- \sum_{j=1}^N \log 2|\sin \theta_j/2|$ are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem as $N \to \infty$.Comment: LaTeX 2.09, 11 pages + 3 eps figs (needs epsf.sty

A monomial basis for the Virasoro minimal series M(p,p') : the case 1<p'/p<2

Quadratic relations of the intertwiners are given explicitly in two cases of chiral conformal field theory, and monomial bases of the representation spaces are constructed by using the Fourier components of the intertwiners. The two cases are the (p,p')-minimal series for the Virasoro algebra where 1<p'/p<2, and the level k integrable highest weight modules for the affine Lie algebra \hat{sl}_2.Comment: Latex, 29 page

Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages

In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval $(0,s)$ at the hard edge contains $k$ eigenvalues, was evaluated in terms of a Painlev\'e V transcendent in $\sigma$-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small $s$ behaviour of the Painlev\'e V equation in $\sigma$-form known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlev\'e \IIId transcendent in $\sigma$-form. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function

{\bf τ\tau-Function Evaluation of Gap Probabilities in Orthogonal and Symplectic Matrix Ensembles}

It has recently been emphasized that all known exact evaluations of gap probabilities for classical unitary matrix ensembles are in fact $\tau$-functions for certain Painlev\'e systems. We show that all exact evaluations of gap probabilities for classical orthogonal matrix ensembles, either known or derivable from the existing literature, are likewise $\tau$-functions for certain Painlev\'e systems. In the case of symplectic matrix ensembles all exact evaluations, either known or derivable from the existing literature, are identified as the mean of two $\tau$-functions, both of which correspond to Hamiltonians satisfying the same differential equation, differing only in the boundary condition. Furthermore the product of these two $\tau$-functions gives the gap probability in the corresponding unitary symmetry case, while one of those $\tau$-functions is the gap probability in the corresponding orthogonal symmetry case.Comment: AMS-Late

Universal Behavior of Correlations between Eigenvalues of Random Matrices

The universal connected correlations proposed recently between eigenvalues of unitary random matrices is examined numerically. We perform an ensemble average by the Monte Carlo sampling. Although density of eigenvalues and a bare correlation of the eigenvalues are not universal, the connected correlation shows a universal behavior after smoothing.Comment: ISSP-September-199