We study the monic polynomials Pn(x; t), orthogonal with respect to a symmetric perturbed Gaussian weight function w(x)=w(x;t)≔e−x21+tx2λ,x∈R, t>0,λ∈R. This problem is related single-user multiple-input multiple-output systems in information theory. It shown that recurrence coefficient βn(t) particular Painlevé V transcendent, and sub-leading p(n, t) of (Pn(x; = xn + t)xn−2 ⋯) satisfies ...

We study the Emptiness Formation Probability (EFP) for the spin 1/2 XXZ spin chain. EFP P (n) detects a formation of ferromagnetic string of the length n in the ground state. It is expected that EFP decays in a Gaussian way for large strings P (n) ∼ n−γC−n2. Here, we propose the explicit expressions for the rate of Gaussian decay C as well as for the exponent γ. In order to confirm the validity...

We study a vertex operator algebra (VOA) V related to the M(3, p) Virasoro minimal series. This VOA reduces in the simplest case p = 4 to the level two integrable vacuum module of ŝl2. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of ŝl2. Our main concern is the subspace W generated by this action from the highest weight vector of V . Using the Fo...

Schlesinger transformations are discrete monodromy preserving symmetry transformations of a meromorphic connection which shift by integers the eigenvalues of its residues. We study Schlesinger transformations for twisted slN -valued connections on the torus. A universal construction is presented which gives the elementary two-point transformations in terms of Belavin’s elliptic quantum R-matrix...

The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann–Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solut...

We study a vertex operator algebra (VOA) V related to the M(3, p) Virasoro minimal series. This VOA reduces in the simplest case p = 4 to the level two integrable vacuum module of ŝl2. On V there is an action of a commutative current a(z), which is an analog of the current e(z) of ŝl2. Our main concern is the subspace W generated by this action from the highest weight vector of V . Using the Fo...

The rational solutions of the Painlevé-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann–Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solut...

Scaling level-spacing distribution functions in the “bulk of the spectrum” in random matrix models of N × N hermitian matrices and then going to the limit N → ∞, leads to the Fredholm determinant of the sine kernel sinπ(x− y)/π(x− y). Similarly a scaling limit at the “edge of the spectrum” leads to the Airy kernel [Ai(x)Ai′(y)−Ai′(x)Ai(y)] /(x − y). In this paper we derive analogues for this Ai...

Following the work with Jimbo and Miwa[1], we introduce a certain degeneration of the elliptic algebra Aq,p ( ŝl2 ) and its boson realization. We investigate its rational limit. The limit is the central extension of the Yangian double DY (sl2) at level one. We give a new boson realization of it. Based on these algebras, we reformulate the Smirnov’s form factor bootstrap approach to the sine-Gor...