Asymptotic forms for hard and soft edge general β conditional gap probabilities
Introduction
Fundamental to Dysonʼs [13] pioneering studies of random matrices is the log-gas picture. The essential point is that the joint eigenvalue probability density function for various random matrix ensembles can be computed exactly, and is seen to be identical to the Boltzmann factor for certain classical particle systems interacting via a repulsive logarithmic pair potential. These so-called log-gas systems are in equilibrium at particular values of the dimensionless inverse temperature β, restricted to the value in the case of the matrix ensemble exhibiting an orthogonal symmetry, for a unitary symmetry and for a unitary symplectic symmetry. In the application of random matrix ensembles to quantum mechanics these cases correspond to a time reversal symmetry T with , no time reversal symmetry, and a time reversal symmetry T with , respectively.
The significance of the log-gas picture is that it permits the use of macroscopic arguments to compute certain probabilistic quantities in the bulk scaling limit (where the number of eigenvalues goes to infinity with their bulk density fixed to the value ρ). Specifically, the particle system is approximated as a continuum fluid confined to a one-dimensional domain but obeying the laws of two-dimensional electrostatics. As an example, consider , the probability that there are no eigenvalues in the interval for the bulk scaling limit of a random matrix ensemble indexed by β. Dysonʼs first application of the log-gas analogy was to derive the asymptotic form Although it was remarked [13, above Eq. (92)] that the variational principle underlying (1.1) may contain errors of order , other arguments were presented that suggested the log-gas prediction of the explicit term may in fact be correct at this order.
A noteworthy feature is that (1.1) is valid for all , and thus in particular applies uniformly to all three random matrix symmetries. In fact, the general form of (1.1) is relevant to contemporary random matrix theory [23]. Thus there are random tridiagonal matrices [11] and random Hessenberg unitary matrices [24] that β-generalize the eigenvalue probability density functions for the Gaussian and circular ensembles from classical random matrix theory. A remarkable development relating to these constructions has been the characterization of in terms of a stochastic differential equation [25], [33]. This was subsequently used [34] to give a rigorous derivation of (1.1), and to furthermore establish that the explicit form of the term is not (1.2) but rather
Generalizing is the probability of there being exactly n eigenvalues in the interval for the bulk scaling limit of a random matrix ensemble indexed by β. Many years after his pioneering work [13], Dyson [14] and independently Fogler and Shklovskii [18] used the log-gas picture — now applied directly to the infinite bulk state (in [13] use was made of a scaling limit of a large deviation formula in the finite system; see Section 5) — to derive the generalization of (1.1) In fact, in the cases and 4 this same formula, except requiring n to be fixed, had been obtained earlier [2] using an analysis based on explicit Painlevé/Fredholm determinant expression for .
The primary aim of the present paper is to extend Dysonʼs [14] log-gas study of the bulk conditioned gap probability to the analogous quantities and corresponding to there being exactly n eigenvalues in the intervals and about the soft and hard spectrum edges respectively, for the regime , . The soft edge refers to the neighbourhood of the largest or smallest eigenvalue, in the situation that the corresponding eigenvalue density exhibits a square root profile, which from our formalism extending [14] is taken to be (semi-)infinite in extent. The hard edge is the neighbourhood of the smallest eigenvalue for matrices with non-negative eigenvalues, when the spectral density exhibits an inverse square root singularity profile, which again is to be taken of (semi-)infinite extent.
As examples of explicit realizations of these edge states in terms of limits of finite N matrix ensembles, let denote the eigenvalue probability density proportional to The choice defines the Gaussian β-ensemble, and the choice () defines the Laguerre β-ensemble. We know from [19] that the largest eigenvalues are, respectively, to leading order in N at and , and that [16], [28] upon the scalings [19] the correlations have the same well defined . The corresponding scaled density admits an explicit β-dimensional integral formula in the case of β even [10], and this exhibits the square root singularity characterizing the soft edge state by way of the asymptotic formula The standard realization of the hard edge is the Laguerre β-ensemble, upon the scaling [15], [19] For , c a positive integer, and β even the corresponding scaled density admits an explicit β-dimensional integral formula [20]. Asymptotic analysis of this integral shows thus exhibiting the characterizing density profile at the hard edge.
The relevance of the spectrum edge to problems in theoretical and mathematical physics was not apparent in the era of Dysonʼs first works. In fact, at the soft edge there is a somewhat hidden, but nonetheless fundamental, relationship to growth problems of the KPZ class (see e.g. the recent review [17]). And in the case of the hard edge the statistical properties of the smallest eigenvalue can be related to data from lattice QCD simulations (see e.g. [32]).
A priori the log-gas formalism can be applied to the study of the asymptotic form of gap probabilities in the cases the gap contains no eigenvalues, or in the cases there are n eigenvalues with . In the former case the asymptotic form of the logarithm of the gap probability for general β at both the hard and soft edges is known from other considerations [20], [5], [4], [29], up to and including terms . This information is important for two reasons. First, it tells us the infinite log-gas formalism at both the hard and soft edge is in error in the second order (free energy/entropy) contribution. But fortunately, guided by the bulk case, an ansatz can be made to correct this error. Second, it allows the asymptotic expansions obtained for to be very naturally extended to hold (as a conjecture) uniformly in n, up to and including terms .
These latter expansions, which are the main result of the paper, read and As a check, using results from [22], asymptotic duality equations — relating gap probabilities with given to gap probabilities at and with modified n and length scale — are identified for both and , and (1.7), (1.8) are shown to exhibit the required inter-relations.
Section snippets
Formalism for a general background potential
The finite log-gas system consists of N mobile charges, of strength +1 say, and a smeared out neutralizing background charge density . We are interested in the limit, in the circumstance that is supported on some semi-infinite or infinite domain I. A physical argument tells us that an equilibrium state will exhibit local charge neutrality, and so , where denotes the partial density (otherwise the charge imbalance would create a non-zero electric field, which
Direct calculation
In the log-gas formalism the hard edge is specified by the background density and, furthermore, by an electrostatic coupling between a fixed charge of strength at the origin, and the charge density, where (see [23]). This additional term accounts for the parameter a in the Laguerre weight , which in the hard edge scaling is independent of N.
Comparing (3.1) with (2.2) shows Conditioning so that exactly n eigenvalues
The case
In our log-gas formalism, we are specifying the soft edge by In the realization of the soft edge as the scaled neighbourhood of the largest eigenvalue of the Gaussian and Laguerre β-ensembles as discussed in the Introduction, this corresponds to shifting the origin to the location of the largest eigenvalue, and then changing the sense of direction by the mapping . Thus now corresponds to the region of the eigenvalue support. With the system conditioned so that exactly n
Concluding remarks
An infinite log-gas formalism, due in the bulk to Dyson [14] and independently Fogler and Shklovskii [18], has been applied to the computation of the conditioned soft and hard edge gap probabilities and . For this purpose the hard and soft edges are characterized by their asymptotic densities (1.6), (1.5), which are taken to be the exact profiles of the background densities in the log-gas. The hypothesis (2.1) asserts that the conditioned gap probability is given
Acknowledgements
This work was supported by the Australian Research Council. The assistance in the preparation of this manuscript by Anthony Mays is acknowledged, as are the considered remarks of the referee.
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