Linen Boxes and Slices: Raoul De Keyser and American Modernism in Belgium in the 1960s and 1970s

Before his international breakthrough shortly before the turn of the century, Belgian painter Raoul De Keyser (1930–2012) had a long career that reaches back to the 1960s, when he was associated with Roger Raveel and the so-called Nieuwe Visie (New Vision in Dutch), Belgium’s variation on postwar figurative painting that also entails Anglo-Saxon Pop Art and French nouveau réalisme. Dealing with De Keyser’s works of the 1960s and 1970s, this article discusses the reception of American late-modernist art currents such as Color-Field Painting, Hard Edge, Pop Art, and Minimal Art in Belgium. Drawing on contemporaneous reflections (by, among others, poet and critic Roland Jooris) as well as on recently resurfaced materials from the artist’s personal archives, this essay focuses on the ways innovations associated with these American trends were appropriated by De Keyser, particularly in the production of his so-called Linen Boxes and Slices. Made between 1967 and 1971, Linen Boxes and Slices are paintings that evolved into three-dimensional objects, free-standing on the floor or leaning against the wall. Apart from situating these constructions in De Keyser’s oeuvre, this article interprets Linen Boxes and Slices as particular variations on Pop Art’s fascination for consumer items and on Minimalism’s interest in the spatial and material aspects of “specific objects”.

Applications in random matrix theory of a PIII’ τ-function sequence from Okamoto’s Hamiltonian formulation

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular [Formula: see text] Bessel functions. Earlier studies have identified the same determinant, but with the [Formula: see text] Bessel functions replaced by [Formula: see text] Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a [Formula: see text]-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III[Formula: see text], and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same [Formula: see text]-form Painlevé III[Formula: see text] differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the [Formula: see text] limit, both with the Laguerre parameter [Formula: see text] fixed, and with [Formula: see text] (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

Critical edge behavior in the singularly perturbed Pollaczek–Jacobi type unitary ensemble

We investigate the orthogonal polynomials associated with a singularly perturbed Pollaczek–Jacobi type weight [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Based on our observation, we find that this weight includes the symmetric constraint [Formula: see text]. Our main results obtained here include two aspects: (1) Strong asymptotics: we deduce strong asymptotics of monic orthogonal polynomials with respect to the above weight function in different regions in the complex plane when the polynomial degree [Formula: see text] goes to infinity. Because of the effect of [Formula: see text] for varying [Formula: see text], the asymptotic behavior in a neighborhood of point [Formula: see text] is described in terms of the Airy function as [Formula: see text], but the Bessel function as [Formula: see text]. Due to symmetry, the similar local asymptotic behavior near the singular point [Formula: see text] can be derived. (2) Limiting eigenvalue correlation kernels: We calculate the limit of the eigenvalue correlation kernel of the corresponding unitary random matrix ensemble in the bulk of the spectrum described by the sine kernel, and at both sides of hard edge, expressed as a Painlevé III kernel. Our analysis is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.

Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib–Borodin Ensembles

We consider the limiting process that arises at the hard edge of Muttalib–Borodin ensembles. This point process depends on$$\theta > 0$$θ>0and has a kernel built out of Wright’s generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form$$\begin{aligned} {\mathbb {P}}(\text{ gap } \text{ on } [0,s]) = C \exp \left( -a s^{2\rho } + b s^{\rho } + c \ln s \right) (1 + o(1)) \qquad \text{ as } s \rightarrow + \infty , \end{aligned}$$P(gapon[0,s])=Cexp-as2ρ+bsρ+clns(1+o(1))ass→+∞,where the constants$$\rho $$ρ,a, andbhave been derived explicitly via a differential identity insand the analysis of a Riemann–Hilbert problem. Their method can be used to evaluatec(with more efforts), but does not allow for the evaluation ofC. In this work, we obtain expressions for the constantscandCby employing a differential identity in$$\theta $$θ. When$$\theta $$θis rational, we find thatCcan be expressed in terms of Barnes’G-function. We also show that the asymptotic formula can be extended to all orders ins.

A degenerate Gaussian weight connected with Painlevé equations and Heun equations

In this paper, we study the recurrence coefficients of a deformed Hermite polynomials orthogonal with respect to the weight [Formula: see text] where [Formula: see text] and [Formula: see text]. It is an extension of Chen and Feigin [J. Phys. A., Math. Gen. 39 (2006) 12381–12393]. By using the ladder operator technique, we show that the recurrence coefficients satisfy a particular Painlevé IV equation and the logarithmic derivative of the associated Hankel determinant satisfies the Jimbo–Miwa–Okamoto [Formula: see text] form of the Painlevé IV. Furthermore, the asymptotics of the recurrence coefficients and the Hankel determinant are obtained at the hard-edge limit and can be expressed in terms of the solutions to the Painlevé XXXIV and the [Formula: see text]-form of the Painlevé II equation at the soft-edge limit, respectively. In addition, for the special case [Formula: see text], we obtain the asymptotics of the Hankel determinant at the hard-edge limit via semi-classical Laguerre polynomials with respect to the weight [Formula: see text], which reproduced the result in Charlier and Deano, [Integr. Geom. Methods Appl. 14(2018) 018 (p. 43)].