385 research outputs found

    Combinatorics of Classical Unitary Invariant Ensembles and Integrable Systems

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    The first part of this thesis is devoted to the combinatorics, geometry, and effective computation of correlators of unitary invariant ensembles of random hermitian matrices with classical potentials. The main results are the subject of the publications [7, 8] with my supervisors T.~Grava and G.~Ruzza, and are summarized as follows. We provide generating functions for correlators of general Hermitian matrix models; formulae of this sort have already appeared in the literature [1, 5], we rederive them here with different methods which lend themselves to further generalizations. Such formulae are not recursive in the genus and hence particularly effective. Moreover, these formulae express the correlators of classical unitary ensembles as linear combinations of products of discrete hypergeometric polynomials; this generalizes relations to discrete orthogonal polynomials for the one-point correlators \langle \tr M^k \rangle of the classical ensembles recently discovered by Cunden et al. [3]. Hence, we turn our attention on the combinatorial interpretation of correlators for the Laguerre and Jacobi ensembles. We prove that the coefficients in the topological expansion of Jacobi correlators are multiparametric single Hurwitz numbers involving combinations of triple monotone Hurwitz numbers. Via a simple limit, this reproduces formulae of [2] on the Laguerre ensemble. This completes the combinatorial interpretation of correlators of unitary ensembles with classical potential. Combining results of Dubrovin et al. [4], and of Norbury [10] connecting integrable systems with enumerative geometry, we obtain ELSV-like formulae linking the multiparametric single Hurwitz numbers of LUE and JUE respectively to cubic Hodge integrals and Θ\Theta-GW invariants. In the second part of the thesis we analyse various integrable dynamical systems from a probabilistic point of view. Specifically, we study the spectrum of their random Lax Matrix equipped with the associated Gibbs Measure, in the spirit of [9, 11]. This is the content of the preprint [6], in collaboration with T.~Grava, G.~Gubbiotti and G.~Mazzuca. We explicitly compute the density of states for the exponential Toda lattice and the Volterra lattice showing they are connected to the Laguerre ÎČ\beta-ensemble at high temperatures and the ÎČ\beta-antisymmetric Gaussian ensemble at high temperatures respectively. For generalizations of these system we derive numerically their density of states and compute their ground states. [1] M. Bertola, B. Dubrovin, and D. Yang, Correlation functions of the KdV hierarchy and applications to intersection numbers over Mg,n, Phys. D, 327 (2016), pp. 30–57. [2] F. D. Cunden, A. Dahlqvist, and N. O’Connell, Integer moments of complex Wishart matrices and Hurwitz numbers, Ann. Inst. Henri Poincar ́e D, 8 (2021), pp. 243–268. [3] F. D. Cunden, F. Mezzadri, N. O’Connell, and N. Simm, Moments of random matrices and hypergeometric orthogonal polynomials, Comm. Math. Phys, 369 (2019), pp. 1091–1145. [4] B. Dubrovin, S. Q. Liu, D. Yang, and Y. Zhang, Hodge-GUE correspondence and the discrete KdV equation, Comm. Math. Phys, 379 (2020), pp. 461–490. [5] B. Eynard, T. Kimura, and S. Ribault, Random matrices, arXiv preprint arXiv:1510.04430, (2015). [6] M. Gisonni, T. Grava, G. Gubbiotti, and G. Mazzuca, Discrete integrable systems and random Lax matrices, arXiv preprint arXiv:2206.15371, (2022). [7] M. Gisonni, T. Grava, and G. Ruzza, Laguerre ensemble: Correlators, Hurwitz numbers and Hodge integrals, Ann. Henri Poincar ́e, 21 (2020), pp. 3285–3339. [8] M. Gisonni, T. Grava, and G. Ruzza, Jacobi ensemble, Hurwitz numbers and Wilson polynomials, Lett. Math. Phys., 111 (2021), pp. 1–38. [9] T. Grava and G. Mazzuca, Generalized gibbs ensemble of the Ablowitz-Ladik lattice, Circular ÎČ-ensemble and double confluent Heun equation, arXiv preprint: 2107.02303, (2021). [10] P. Norbury, Gromov-Witten invariants of P1 coupled to a KdV tau function, Adv. Math., 399 (2022), p. 108227. [11] H. Spohn, Generalized Gibbs ensembles of the classical Toda chain, J. Stat. Phys., 180 (2020), pp. 4–2

    Restrictions and extensions of semibounded operators

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    We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure

    Design methods for microwave filters and multiplexers

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    This thesis is concerned with developing synthesis and design procedures for microwave filters and multiplexers. The core of this thesis presents the following topics. 1) New classes of lumped lowpass prototype filters satisfying generalized Chebyshev characteristics have been investigated. Exact synthesis procedures are given using a relatively new technique termed the alternating pole synthesis technique to solve the accuracy problem. The properties of these filters and their practical advantages have been discussed. Tables of element values for commonly used specifications are included. 2) A new design procedure has been developed for bandpass channel multiplexers connected at a common junction. This procedure is for multiplexers having any number of Chebyshev channel filters, with arbitrary degrees, bandwidths and inter-channel spacings. The procedure has been modified to allow the design of multi-octave bandwidth combline channel filter multiplexers. It is shown that this procedure gives very good results for a wide variety of specifications, as demonstrated by the computer analysis of several multiplexers examples and by the experimental results. 3) A compact exact synthesis method is presented for a lumped bandpass prototype filter up to degree 30 and satisfies a generalized Chebyshev response. This prototype has been particularly utilized in designing microwave broadband combline filters. 4) Different forms of realization have been discussed and used in design and construction of different devices. This includes a new technique to realize TEM networks in coaxial structure form having equal diameter coupled circular cylindrical rods between parallel ground planes. Other forms of realization have been discussed ranging from equal diameter posts, direct coupled cavity waveguide filters to microwave integrated circuits using suspended substrate stripline structure. The experimental results are also given. In addition, the fundamentals of lumped circuits and distributed circuits have been briefly reviewed. The approximation problem was also discussed

    Quantum mechanics of layers with a finite number of point perturbations

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    We study spectral and scattering properties of a spinless quantum particle confined to an infinite planar layer with hard walls containing a finite number of point perturbations. A solvable character of the model follows from the explicit form of the Hamiltonian resolvent obtained by means of Krein's formula. We prove the existence of bound states, demonstrate their properties, and find the on-shell scattering operator. Furthermore, we analyze the situation when the system is put into a homogeneous magnetic field perpendicular to the layer; in that case the point interactions generate eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian essential spectrum.Comment: LateX 2e, 48 pages, with 3 ps and 3 eps figure

    Edge fluctuations for random normal matrix ensembles

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    A famous result going back to Eric Kostlan states that the moduli of the eigenvalues of random normal matrices with radial potential are independent yet non identically distributed. This phenomenon is at the heart of the asymptotic analysis of the edge, and leads in particular to the Gumbel fluctuation of the spectral radius when the potential is quadratic. In the present work, we show that a wide variety of laws of fluctuation are possible, beyond the already known cases, including for instance Gumbel and exponential laws at unusual speeds. We study the convergence in law of the spectral radius as well as the limiting point process at the edge. Our work can also be seen as the asymptotic analysis of the edge of two-dimensional determinantal Coulomb gases and the identification of the limiting kernels.Comment: 43 pages, improved version with more general theorem

    Passivity and Maximum Quality Factor Assessment in Lossy 2-port Transfer Functions

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    International audienceLossy transfer functions are appealing in the design of filters and electric networks, as they can be exactly implemented by physical passive components. However, lossy techniques relax most of the constraints governing the design and thus offer many degrees of freedom but with unclear effects on realizability. This work describes first an analytical method to check whether a given 2-port matrix transfer function is passive. Moreover, for comparison purposes, a technique to assess the maximum allowed predistortion is proposed, related to the highest required quality factor
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