The Hankel determinant associated with a singularly perturbed Laguerre unitary ensemble

Abstract

We are concerned with the probability that all the eigenvalues of a unitary ensemble with the weight function w(x;t) = x α e −x− t x , x ∈ [0, ∞), α > −1, t ≥ 0, are greater than s. This probability is expressed as the quotient of D n (s,t) and its value at s = 0, where D n (s,t) denotes the determinant of the n dimensional Hankel matrices generated by the moments of w(x;t) on x ∈ [s, ∞). In this paper we focus specifically on the Hankel determinant D n (s,t) and its properties.Based on the ladder operators adapted to the monic polynomials orthogonal with respect to w(x;t), and from the associated supplementary conditions and a sum-rule, we show that the log-derivative of the Hankel determinant, viewed as a function of s and t, satisfies a second order sixth degree partial differential equation, where n appears as a parameter. In order to go to the thermodynamic limit, of infinitely large matrices, we envisage a scenario where n → ∞, s → 0, and t → 0 such that S := 4ns and T := (2n + 1 + α)t are finite. After such a double scaling, the large finite n equation reduces to a second order second degree equation, in the variables S and T , from which we derive the asymptotic expansion of the scaled Hankel determinant in three cases of S and T : S → ∞ with T fixed, S → 0 with T > 0 fixed, and T → ∞ with S > 0 fixed. The constant term in the asymptotic expansion is shown to satisfy a difference equation and one of its solutions is the Tracy-Widom constant.

Full counting statistics for interacting trapped fermions

“…Similarly for the periodic model one can always choose a 2 = 0 by translation, which leads to (117). Finally for the hyperbolic solutions one can always reduce to either a 1 = a 2 , which leads again to (125), or to a 2 = 0 (whenever a 1 > a 2 ), which leads to (134), or to a 1 = 0 (whenever a 1 < a 2 ), which leads to (140). One can check that T 3 is indeed a constant for each of these models, again via non trivial trigonometric identities.…”

“…This allows to state that e.g. the model(125) is equivalent to the model with λ(x) = e px . Although there is then a relative factor of 4 in their respective values for T 3 , this is compensated by the change in T 2 from the change v → ṽ see discussion below(125).…”

Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systems

“…Analogs of Riccati equations for , and , , and coupled PDEs satisfied by , Combining the expressions involving the recurrence coefficients together, namely, (12), (13), (17), and (18), with the aid of the difference equations (10) and (11), we arrive at the following four first-order PDEs for , and , . Lemma 1.…”

“…For problems involving two such variables, a second‐order PDE can be deduced. The two variables could be, for example, two deformation variables, 16 two interval variables, 3 two jump variables, 5 one interval variable together with one perturbation variable 17 . As the dimension of the unitary ensemble tends to and under suitable double scaling, based on the aforementioned Painlevé equations or PDEs, we can have a rough understanding of the asymptotic expansion of the interested quantities.…”

The smallest eigenvalue distribution of the Jacobi unitary ensembles

“…For finite n analysis, the ladder operators adapted to monic orthogonal polynomials are usually used. See, for instance, previous studies 20–23 …”