Browsing by Author "Roeder, Roland"

Now showing 1 - 10 of 11
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    The Dynamics of Twisted Tent Maps
    (2013-07-12) Chamblee, Stephen Joseph; Misiurewicz, Michał, 1948-; Roeder, Roland; Geller, William; Eremenko, Alexandre; Mukhin, Evgeny
    This paper is a study of the dynamics of a new family of maps from the complex plane to itself, which we call twisted tent maps. A twisted tent map is a complex generalization of a real tent map. The action of this map can be visualized as the complex scaling of the plane followed by folding the plane once. Most of the time, scaling by a complex number will \twist" the plane, hence the name. The "folding" both breaks analyticity (and even smoothness) and leads to interesting dynamics ranging from easily understood and highly geometric behavior to chaotic behavior and fractals.
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    Gaudin models associated to classical Lie algebras
    (2020-08) Lu, Kang; Mukhin, Evgeny; Its, Alexander; Roeder, Roland; Tarasov, Vitaly
    We study the Gaudin model associated to Lie algebras of classical types. First, we derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe Ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic. We also show that except for the type D, the joint spectrum of Gaudin Hamiltonians in such tensor products is simple. Second, we define a new stratification of the Grassmannian of N planes. We introduce a new subvariety of Grassmannian, called self-dual Grassmannian, using the connections between self-dual spaces and Gaudin model associated to Lie algebras of types B and C. Then we obtain a stratification of self-dual Grassmannian.
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    Lee–Yang zeros for the DHL and 2D rational dynamics, I. Foliation of the physical cylinder
    (Elsevier, 2017-05) Bleher, Pavel; Lyubich, Mikhail; Roeder, Roland; Department of Mathematical Sciences, School of Science
    In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal–Kadanoff Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the renormalization transformation). We prove that R is partially hyperbolic on an invariant cylinder C. The Lee–Yang zeros are organized in a transverse measure for the central-stable foliation of R|C. Their distribution is absolutely continuous. Its density is C∞ (and non-vanishing) below the critical temperature. Above the critical temperature, it is C∞ on a open dense subset, but it vanishes on the complementary set of positive measure.
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    Lee–Yang–Fisher Zeros for the DHL and 2D Rational Dynamics, II. Global Pluripotential Interpretation
    (Springer, 2019) Bleher, Pavel; Lyubich, Mikhail; Roeder, Roland; Mathematical Sciences, School of Science
    In a classical work of the 1950s, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to zero. Limiting distributions of Lee–Yang and of Fisher zeros are physically important as they control phase transitions in the model. One can also consider the zeros of the partition function simultaneously in both complex magnetic field and complex temperature. They form an algebraic curve called the Lee–Yang–Fisher (LYF) zeros. In this paper, we continue studying their limiting distribution for the Diamond Hierarchical Lattice (DHL). In this case, it can be described in terms of the dynamics of an explicit rational function R in two variables (the Migdal–Kadanoff renormalization transformation). We study properties of the Fatou and Julia sets of this transformation and then we prove that the LYF zeros are equidistributed with respect to a dynamical (1, 1)-current in the projective space. The free energy of the lattice gets interpreted as the pluripotential of this current. We also prove a more general equidistribution theorem which applies to rational mappings having indeterminate points, including the Migdal–Kadanoff renormalization transformation of various other hierarchical lattices.
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    Level spacing statistics for the multi-dimensional quantum harmonic oscillator: Algebraic case
    (AIP, 2022-01) Haynes, Alan; Roeder, Roland; Mathematical Sciences, School of Science
    We study the statistical properties of the spacings between neighboring energy levels for the multi-dimensional quantum harmonic oscillator that occur in a window [E, E + ΔE) of fixed width ΔE as E tends to infinity. This regime provides a notable exception to the Berry–Tabor conjecture from quantum chaos, and, for that reason, it was studied extensively by Berry and Tabor in their seminal paper from 1977. We focus entirely on the case that the (ratios of) frequencies ω1, ω2, …, ωd together with 1 form a basis for an algebraic number field Φ of degree d + 1, allowing us to use tools from algebraic number theory. This special case was studied by Dyson, Bleher, Bleher–Homma–Ji–Roeder–Shen, and others. Under a suitable rescaling, we prove that the distribution of spacings behaves asymptotically quasiperiodically in log E. We also prove that the distribution of ratios of neighboring spacings behaves asymptotically quasiperiodically in log E. The same holds for the distribution of finite words in the finite alphabet of rescaled spacings. Mathematically, our work is a higher dimensional version of the Steinhaus conjecture (three gap theorem) involving the fractional parts of a linear form in more than one variable, and it is of independent interest from this perspective.
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    On Random Polynomials Spanned by OPUC
    (2020-12) Aljubran, Hanan; Yattselev, Maxim; Bleher, Pavel; Mukhin, Evgeny; Roeder, Roland
    We consider the behavior of zeros of random polynomials of the from \begin{equation*} P_{n,m}(z) := \eta_0\varphi_m^{(m)}(z) + \eta_1 \varphi_{m+1}^{(m)}(z) + \cdots + \eta_n \varphi_{n+m}^{(m)}(z) \end{equation*} as \( n\to\infty \), where \( m \) is a non-negative integer (most of the work deal with the case \( m =0 \) ), \( \{\eta_n\}_{n=0}^\infty \) is a sequence of i.i.d. Gaussian random variables, and \( \{\varphi_n(z)\}_{n=0}^\infty \) is a sequence of orthonormal polynomials on the unit circle \( \mathbb T \) for some Borel measure \( \mu \) on \( \mathbb T \) with infinitely many points in its support. Most of the work is done by manipulating the density function for the expected number of zeros of a random polynomial, which we call the intensity function.
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    On the Gaudin and XXX models associated to Lie superalgebras
    (2020-08) Huang, Chenliang; Mukhin, Evgeny; Bleher, Pavel; Roeder, Roland; Tarasov, Vitaly
    We describe a reproduction procedure which, given a solution of the gl(m|n) Gaudin Bethe ansatz equation associated to a tensor product of polynomial modules, produces a family P of other solutions called the population. To a population we associate a rational pseudodifferential operator R and a superspace W of rational functions. We show that if at least one module is typical then the population P is canonically identified with the set of minimal factorizations of R and with the space of full superflags in W. We conjecture that the singular eigenvectors (up to rescaling) of all gl(m|n) Gaudin Hamiltonians are in a bijective correspondence with certain superspaces of rational functions. We establish a duality of the non-periodic Gaudin model associated with superalgebra gl(m|n) and the non-periodic Gaudin model associated with algebra gl(k). The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an (m+n) by (m+n) matrix in the case of gl(m|n) and of a column determinant of a k by k matrix in the case of gl(k). We obtain our results by proving Capelli type identities for both cases and comparing the results. We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian Y(gl(m|n)). To a solution we associate a rational difference operator D and a superspace of rational functions W. We show that the set of complete factorizations of D is in canonical bijection with the variety of superflags in W and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
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    Quantum Toroidal Superalgebras
    (2020-05) Pereira Bezerra, Luan; Mukhin, Evgeny; Ramras, Daniel; Roeder, Roland; Tarasov, Vitaly
    We introduce the quantum toroidal superalgebra E(m|n) associated with the Lie superalgebra gl(m|n) and initiate its study. For each choice of parity "s" of gl(m|n), a corresponding quantum toroidal superalgebra E(s) is defined. To show that all such superalgebras are isomorphic, an action of the toroidal braid group is constructed. The superalgebra E(s) contains two distinguished subalgebras, both isomorphic to the quantum affine superalgebra Uq sl̂(m|n) with parity "s", called vertical and horizontal subalgebras. We show the existence of Miki automorphism of E(s), which exchanges the vertical and horizontal subalgebras. If m and n are different and "s" is standard, we give a construction of level 1 E(m|n)-modules through vertex operators. We also construct an evaluation map from E(m|n)(q1,q2,q3) to the quantum affine algebra Uq gl̂(m|n) at level c=q3^(m-n)/2.
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    Ricci Curvature of Finsler Metrics by Warped Product
    (2020-05) Marcal, Patricia; Shen, Zhongmin; Buse, Olguta; Ramras, Daniel; Roeder, Roland
    In the present work, we consider a class of Finsler metrics using the warped product notion introduced by B. Chen, Z. Shen and L. Zhao (2018), with another “warping”, one that is consistent with the form of metrics modeling static spacetimes and simplified by spherical symmetry over spatial coordinates, which emerged from the Schwarzschild metric in isotropic coordinates. We will give the PDE characterization for the proposed metrics to be Ricci-flat and construct explicit examples. Whenever possible, we describe both positive-definite solutions and solutions with Lorentz signature. For the latter, the 4-dimensional metrics may also be studied as Finsler spacetimes.
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    Superstable manifolds of invariant circles
    (2013-12-10) Kaschner, Scott R.; Roeder, Roland; Bleher, Pavel, 1947-; Misiurewicz, Michał, 1948-; Buzzard, Gregory; Mukhin, Evgeny
    Let f:X\rightarrow X be a dominant meromorphic self-map, where X is a compact, connected complex manifold of dimension n > 1. Suppose there is an embedded copy of \mathbb P^1 that is invariant under f, with f holomorphic and transversally superattracting with degree a in some neighborhood. Suppose also that f restricted to this line is given by z\rightarrow z^b, with resulting invariant circle S. We prove that if a ≥ b, then the local stable manifold W^s_loc(S) is real analytic. In fact, we state and prove a suitable localized version that can be useful in wider contexts. We then show that the condition a ≥ b cannot be relaxed without adding additional hypotheses by resenting two examples with a < b for which W^s_loc(S) is not real analytic in the neighborhood of any point.