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Item Domain Wall Six-Vertex Model with Half-Turn Symmetry(Springer, 2018-02) Bleher, Pavel; Liechty, Karl; Mathematical Sciences, School of ScienceWe obtain asymptotic formulas for the partition function of the six-vertex model with domain wall boundary conditions and half-turn symmetry in each of the phase regions. The proof is based on the Izergin–Korepin–Kuperberg determinantal formula for the partition function and its reduction to orthogonal polynomials, and on an asymptotic analysis of the orthogonal polynomials under consideration in the framework of the Riemann–Hilbert approach.Item Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge(Springer, 2016-10) Bogatskiy, A.; Claeys, T.; Its, Alexander R.; Department of Mathematical Sciences, School of ScienceWe compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz–Segur family of solutions to the Painlevé II equation. Our results complement the ones in [33]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlevé II transcendents, and we also prove a new result on the poles of the Ablowitz–Segur solutions to the Painlevé II equation. We also highlight applications of our results in random matrix theory.Item Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials(Springer, 2015-12) Aptekarev, Alexander L.; Yattselev, Maxim L.; Department of Mathematical Sciences, School of ScienceLet f be a germ of an analytic function at infinity that can be analytically continued along any path in the complex plane deprived of a finite set of points, f∈A(C¯∖A)f∈A(C¯∖A), #A<∞#A<∞. J. Nuttall has put forward the important relation between the maximal domain of f where the function has a single-valued branch and the domain of convergence of the diagonal Padé approximants for f. The Padé approximants, which are rational functions and thus single-valued, approximate a holomorphic branch of f in the domain of their convergence. At the same time most of their poles tend to the boundary of the domain of convergence and the support of their limiting distribution models the system of cuts that makes the function f single-valued. Nuttall has conjectured (and proved for many important special cases) that this system of cuts has minimal logarithmic capacity among all other systems converting the function f to a single-valued branch. Thus the domain of convergence corresponds to the maximal (in the sense of minimal boundary) domain of single-valued holomorphy for the analytic function f∈A(C¯∖A)f∈A(C¯∖A). The complete proof of Nuttall’s conjecture (even in a more general setting where the set A has logarithmic capacity 0) was obtained by H. Stahl. In this work, we derive strong asymptotics for the denominators of the diagonal Padé approximants for this problem in a rather general setting. We assume that A is a finite set of branch points of f which have the algebro-logarithmic character and which are placed in a generic position. The last restriction means that we exclude from our consideration some degenerated “constellations” of the branch points.Item Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality(Elsevier, 2022-02-12) Denisov, Sergey A.; Yattselev, Maxim L.; Mathematical Sciences, School of ScienceWe study Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials. Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each subspace, we find generators and the generalized eigenfunctions written in terms of the orthogonal polynomials. The spectrum and its spectral type are studied for large classes of orthogonality measures.Item Symmetric contours and convergent interpolation(Elsevier, 2018-01) Yattselev, Maxim L.; Mathematical Sciences, School of ScienceThe essence of Stahl–Gonchar–Rakhmanov theory of symmetric contours as applied to the multipoint Padé approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ, if there exists a contour that is “symmetric” with respect to the interpolation scheme, does not separate the plane, and in the complement of which the germ has a single-valued continuation with non-identically zero jump across the contour, then the interpolants converge to that continuation in logarithmic capacity in the complement of the contour. The existence of such a contour is not guaranteed. In this work we do construct a class of pairs interpolation scheme/symmetric contour with the help of hyperelliptic Riemann surfaces (following the ideas of Nuttall and Singh, 1977; Baratchart and Yattselev, 2009). We consider rational interpolants with free poles of Cauchy transforms of non-vanishing complex densities on such contours under mild smoothness assumptions on the density. We utilize ∂̄-extension of the Riemann–Hilbert technique to obtain formulae of strong asymptotics for the error of interpolation.