Subleading Coefficient Research Articles

Hankel determinant and orthogonal polynomials for a perturbed Gaussian weight: From finite n to large n asymptotics

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, with t>0,λ∈R. This problem is related to single-user multiple-input multiple-output systems in information theory. It is shown that the recurrence coefficient βn(t) is related to a particular Painlevé V transcendent, and the sub-leading coefficient p(n, t) of Pn(x; t) (Pn(x; t) = xn + p(n, t)xn−2 + ⋯) satisfies the Jimbo–Miwa–Okamoto σ-form of the Painlevé V equation. Furthermore, we derive the second-order difference equations satisfied by βn(t) and p(n, t), respectively. This enables us to obtain the large n full asymptotic expansions for βn(t) and p(n, t) with the aid of Dyson’s Coulomb fluid approach in the one-cut case [i.e., λt ≤ 1 (t > 0)]. We also consider the Hankel determinant Dn(t), generated by the perturbed Gaussian weight. It is found that Φn(t), a quantity allied to the logarithmic derivative of Dn(t) via Φn(t)=2t2ddtlnDn(t)−2nλt, can be expressed in terms of βn(t) and p(n, t). Based on this result, we obtain the large n asymptotic expansion of Φn(t) and then that of the Hankel determinant Dn(t) in the one-cut case.

From tree- to loop-simplicity in affine Toda theories I: Landau singularities and their subleading coefficients

Various features of the even order poles appearing in the S-matrices of simply-laced affine Toda field theories are analysed in some detail. In particular, the coefficients of first- and second-order singularities appearing in the Laurent expansion of the S-matrix around a general 2Nth order pole are derived in a universal way using perturbation theory at one loop. We show how to cut loop diagrams contributing to the pole into particular products of tree-level graphs that depend on the on-shell geometry of the loop; in this way, we recover the coefficients of the Laurent expansion around the pole exploiting tree-level integrability properties of the theory. The analysis is independent of the particular simply-laced theory considered, and all the results agree with those obtained in the conjectured bootstrapped S-matrices of the ADE series of theories.

A note on the asymptotics of the Hankel determinant associated with time-dependent Jacobi polynomials

In 2010, Basor, Chen and Ehrhardt [J. Phys. A 43 (2010), 015204 (25 pp)] studied the (monic) time-dependent Jacobi polynomials. They proved that the diagonal recurrence coefficient α n ( t ) \alpha _n(t) satisfies a particular Painlevé V equation and the sub-leading coefficient p ( n , t ) \mathrm {p}(n,t) satisfies the Jimbo-Miwa-Okamoto σ \sigma -form of Painlevé V under suitable transformations. In the end they made some conjectures about the large n n asymptotics of the associated Hankel determinant. In this note we will derive the large n n asymptotics of the Hankel determinant, together with the recurrence coefficients, the sub-leading coefficient and the square of L 2 L^2 -norm of the time-dependent Jacobi polynomials. We will also show the relation between our problem and Heun’s differential equations.

Semi-classical Jacobi polynomials, Hankel determinants and asymptotics

We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient $\beta_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials. This enables us to obtain the large $n$ asymptotics of $\beta_n(t)$ and $\mathrm{p}(n,t)$ based on the result of Kuijlaars et al. [Adv. Math. \textbf{188} (2004) 337-398]. In addition, we show the second-order differential equation satisfied by the orthogonal polynomials, with all the coefficients expressed in terms of $\beta_n(t)$. From the $t$ evolution of the auxiliary quantities, we prove that $\beta_n(t)$ satisfies a second-order differential equation and $R_n(t)=2n+1+2\alpha-2t(\beta_n(t)+\beta_{n+1}(t))$ satisfies a particular Painlev\'{e} V equation under a simple transformation. Furthermore, we show that the logarithmic derivative of the associated Hankel determinant satisfies both the second-order differential and difference equations. The large $n$ asymptotics of the Hankel determinant is derived from its integral representation in terms of $\beta_n(t)$ and $\mathrm{p}(n,t)$.

Nonlinear difference equations arising from the generalized Stieltjes‐Wigert and q‐Laguerre weights

In this paper, we investigate the generalized Stieltjes‐Wigert and q‐Laguerre polynomials. We derive the second‐ and the third‐order nonlinear difference equations for the subleading coefficients of these polynomials and use them to find a few terms of the formal expansions in powers of qn/2. We also show how the recurrence coefficients in the three‐term recurrence relation for these polynomials can be computed efficiently by using the nonlinear difference equations for the subleading coefficient. Moreover, we obtain systems of difference equations with one of the equations being q‐discrete Painlevé III or V equations and analyze them by a singularity confinement. We also discuss certain generalized weights.

Nonlinear difference equations for the generalized little q-Laguerre polynomials

In this paper, a study is made on polynomials orthogonal with respect to the generalized little q-Laguerre weight, defined by , . Here , and . This weight is supported on the exponential lattice . Let the subleading coefficient of of the monic polynomials be From the q-analogue of the ladder operators, the associated supplementary conditions and a ‘sum rule’, we deduce a system of difference equations satisfied by This system is used to obtain the first few terms in a formal asymptotic expansion of . We express the recurrence coefficients in terms of this subleading coefficient and show that the first few terms in the formal expansions in powers of agree with the first few terms for the corresponding expansions of the recurrence coefficients in the classical case. Moreover, we find certain non-linear difference equations for the recurrence coefficients of the monic polynomials, auxiliary functions in the ladder operators and for . We also observe the phenomenon of singularity confinement, related to that observed in the q-discrete Painlevé equations. Furthermore, we give a generalization of the weight function, characterized by for on the exponential lattice. In this situation we find another system of difference equations satisfied by and study its behaviour for large n. The paper ends with the discussion on a deformation of the generalized little q-Laguerre weight.

The sub-leading coefficient of the L-function of an elliptic curve

We show that there is a relation between the leading term at s=1 of an L-function of an elliptic curve defined over an number field and the term that follows.

Scaling dimensions from the mirror thermodynamic Bethe ansatz

The mirror thermodynamic Bethe ansatz equations proposed by Arutyunov, Suzuki and the author are solved numerically up to 't Hooft's coupling λ ≈ 2340 for several two-particle states dual to SYM operators from the sector. The data obtained for states with mode numbers n = 1, 2, 3, 4 are used to propose a general charge J-dependent formula for the first nonvanishing subleading coefficient in the strong-coupling expansion of scaling dimensions. In addition, we find that the first critical and subcritical values of the coupling for the J = 4, n = 1 operator are at λ ≈ 133 and λ ≈ 190, respectively.

Structure of large spin expansion of anomalous dimensions at strong coupling

The anomalous dimensions of planar N = 4 SYM theory operators like tr ( Φ D + S Φ ) expanded in large spin S have the asymptotics γ = f ln S + f c + 1 S ( f 11 ln S + f 10 ) + ⋯ , where f (the universal scaling function or cusp anomaly), f c and f m n are given by power series in the 't Hooft coupling λ. The subleading coefficients appear to be related by the so-called functional relation and parity (reciprocity) property of the function expressing γ in terms of the conformal spin of the collinear group. Here we study the structure of such large spin expansion at strong coupling via AdS/CFT, i.e. by using the dual description in terms of folded spinning string in AdS 5 . The large spin expansion of the classical string energy happens to have exactly the same structure as that of γ in the perturbative gauge theory. Moreover, the functional relation and the reciprocity constraints on the coefficients are also satisfied. We compute the leading string 1-loop corrections to the coefficients f c , f 11 , f 10 and verify the functional/reciprocity relations at subleading 1 λ order. This provides a strong indication that these relations hold not only in weak coupling (gauge-theory) but also in strong coupling (string-theory) perturbative expansions.

Strong-coupling expansion of cusp anomaly and gluon amplitudes from quantum open strings in [formula omitted

An important “observable” of planar N = 4 SYM theory is the scaling function f ( λ ) that appears in the anomalous dimension of large spin twist 2 operators and also in the cusp anomaly of light-like Wilson loop. The non-trivial relation between the anomalous dimension and the Wilson loop interpretations of f ( λ ) is well-understood on the perturbative gauge theory side of the AdS/CFT duality. In the first part of this paper we present the dual string-theory counterpart of this relation, i.e., the equivalence between the closed-string and the open-string origins of f ( λ ) . We argue that the coefficient of the log S term in the energy of the closed string with large spin S in AdS 5 should be equal to the coefficient in the logarithm of expectation value of the null cusp Wilson loop, to all orders in λ − 1 / 2 expansion. The reason is that the corresponding minimal surfaces happen to be related by a conformal transformation (and an analytic continuation). As a check, we explicitly compute the leading 1-loop string sigma model correction to the cusp Wilson loop, reproducing the same subleading coefficient in f ( λ ) as found earlier in the spinning closed string case. The same function f ( λ ) appears also in the resummed form of the 4-gluon amplitude as discussed at weak coupling by Bern, Dixon and Smirnov and recently found at the leading order at strong coupling by Alday and Maldacena (AM). Here we attempt to extend the latter approach to a subleading order in λ − 1 / 2 by computing the IR singular part of the 1-loop string correction to the corresponding T-dual Wilson loop. We discuss explicitly the 1-cusp case and comment on apparent problems with the dimensional regularization proposal of AM when directly applied order by order in strong coupling (string inverse tension) expansion.

Spinning strings in AdS5× S5 and integrable systems

We show that solitonic solutions of the classical string action on the AdS 5× S 5 background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. We derive equations which determine the energy of these solitons as a function of spins. In the limit of large spins J, the first subleading 1/ J coefficient in the expansion of the string energy is expected to be non-renormalized to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous dimensions of the corresponding composite operators in N=4 super-YM theory. We obtain a closed system of equations that determines this subleading coefficient and, therefore, the 1-loop anomalous dimensions of the dual SYM operators. We expect that an equivalent system of equations should follow from the thermodynamic limit of the algebraic Bethe ansatz for the SO(6) spin chain derived from SYM theory. We also identify a particular string solution whose classical energy exactly reproduces the one-loop anomalous dimension of a certain set of SYM operators with two independent R charges J 1, J 2.

An improved theoretical prediction for the 2-jet rate in e+e− annihilation

We show that an Ansatz to resum all leading and next-to-leading logarithms in the theoretical prediction for the 2-jet rate in e + e − → hadrons, where jets are defined with the k t -algorithm, is consistent with a full O( α s 2) calculation done by Monte Carlo integration. From the asymptotic behaviour of the full O( α s 2) calculation we extract the subleading coefficient G 21 and the constant C 2.