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Sep 23, 2013
09/13

by
Renjie Feng; Steve Zelditch

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We study the asymptotic distribution of critical values of random holomorphic `polynomials' s_n on a Kaehler manifold M as the degree n tends to infinity. By `polynomial' of degree n we mean a holomorphic section of the nth power of a positive Hermitian holomorphic line bundle $(L, h). In the special case M = CP^m and L = O(1), and h is the Fubini-Study metric, the random polynomials are the SU(m + 1) polynomials. By a critical value we mean the norm ||s_n||_h of s_n at a non-zero critical...

Source: http://arxiv.org/abs/1212.4762v1

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Sep 21, 2013
09/13

by
Bernard Shiffman; Steve Zelditch

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The Newton polytope $P_f$ of a polynomial $f$ is well known to have a strong impact on its zeros, as in the Kouchnirenko-Bernstein theorem on the number of simultaneous zeros of $m$ polynomials with given Newton polytopes. In this article, we show that $P_f$ also has a strong impact on the distribution of zeros of one or several polynomials. We equip the space of (holomorphic) polynomials of degree $\leq N$ in $m$ complex variables with its usual $SU(m+1)$-invariant Gaussian measure and then...

Source: http://arxiv.org/abs/math/0203074v2

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Sep 18, 2013
09/13

by
John Toth; Steve Zelditch

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The standard eigenfunctions $\phi_{\lambda} = e^{i < \lambda, x >}$ on flat tori $\R^n / L$ have $L^{\infty}$-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that $L^2$-normalized eigenfunctions have uniformly bounded $L^{\infty}$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with completely integrable geodesic flows.

Source: http://arxiv.org/abs/math-ph/0002038v2

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Sep 19, 2013
09/13

by
Nalini Anantharaman; Steve Zelditch

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We construct an explicit intertwining operator $\lcal$ between the Schr\"odinger group $e^{it \frac\Lap2} $ and the geodesic flow $g^t$ on certain Hilbert spaces of symbols on the cotangent bundle $T^* \X$ of a compact hyperbolic surface $\X = \Gamma \backslash \D$. Thus, the quantization Op(\lcal^{-1} a) satisfies an exact Egorov theorem. The construction of $\lcal$ is based on a complete set of Patterson-Sullivan distributions.

Source: http://arxiv.org/abs/1010.0867v1

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Sep 23, 2013
09/13

by
Renjie Feng; Steve Zelditch

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We prove that the expected value and median of the supremum of $L^2$ normalized random holomorphic fields of degree $n$ on $m$-dimensional K\"ahler manifolds are asymptotically of order $\sqrt{m\log n}$. This improves the prior result of Shiffman-Zelditch (arXiv:math/0303335) that the upper bound of the media is of order $\sqrt{\log n}$ The estimates are based on the entropy methods of Dudley and Sudakov combined with a precise analysis of the relevant pseudo-metric and its covering...

Source: http://arxiv.org/abs/1303.4096v1

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Sep 18, 2013
09/13

by
Bernard Shiffman; Steve Zelditch

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We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers $L^N$ of a positive holomorphic Hermitian line bundle $L$ over a compact complex manifold $M$. Our first result concerns `random' sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases $\{S^N_j\}$ of $H^0(M, L^N)$, we show that for almost every sequence $\{S^N_j\}$, the associated sequence of zero currents $1/N Z_{S^N_j}$ tends to the...

Source: http://arxiv.org/abs/math/9803052v1

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Sep 17, 2013
09/13

by
Tatsuya Tate; Steve Zelditch

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We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers $V_{\lambda}^{\otimes N}$ of an irreducible representation $V_{\lambda}$ of a compact connected Lie group $G$. The weights are allowed to depend on $N$, and we obtain several regimes of pointwise asymptotics, ranging from a central limit region to a large deviations region. We use a complex steepest descent method that applies to general asymptotic counting problems for lattice paths...

Source: http://arxiv.org/abs/math/0305251v1

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Sep 19, 2013
09/13

by
Jian Song; Steve Zelditch

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The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly approximated by geodesics in the finite dimensional spaces of Bergman metrics. We prove a stronger C^2-approximation in the special case of toric (i.e. S^1-invariant) metrics on CP^1.

Source: http://arxiv.org/abs/math/0703517v1

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Jul 19, 2013
07/13

by
John Toth; Steve Zelditch

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Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $\phi_j |_H$ to $H$ of $\Delta$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic on $H$. We prove two kinds of results: First (i) for any smooth hypersurface $H$, the Cauchy data $(\phi_j|H, \partial \phi_j|H)$ is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly (ii) we give conditions on $H$ so that...

Source: http://arxiv.org/abs/1005.1636v2

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Jun 30, 2018
06/18

by
Junehyuk Jung; Steve Zelditch

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It is an open problem in general to prove that there exists a sequence of $\Delta_g$-eigenfunctions $\phi_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(\phi_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(\phi_{j_k}) \to \infty$ along a subsequence of eigenvalues of density $1$ if the $(M, g)$ is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related...

Topics: Mathematics, Spectral Theory, Analysis of PDEs

Source: http://arxiv.org/abs/1401.4520

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Sep 21, 2013
09/13

by
Christopher D. Sogge; Steve Zelditch

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Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the general results of Burq, G\'erard and Tzvetkov \cite{burq}. By earlier results of Bourgain \cite{bourgainef} and the first author \cite{Sokakeya}, they are equivalent to improvements of the general $L^p$-estimates in \cite{soggeest} for $n=2$ and $2

Source: http://arxiv.org/abs/1108.2726v2

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Sep 19, 2013
09/13

by
Yanir A. Rubinstein; Steve Zelditch

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We continue our study of the Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA). In the prequel a quantum mechanical approach for solving the HCMA was developed, and was shown to coincide with the well-known Legendre transform approach in the case of the HRMA. In this article---that uses tools of convex analysis and can be read independently---we prove that the candidate solution produced by these methods ceases to solve the HRMA, even in a weak sense, as...

Source: http://arxiv.org/abs/1010.2442v2

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Sep 19, 2013
09/13

by
Christopher D. Sogge; Steve Zelditch

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Let $\ncal_{\phi_{\lambda}}$ be the nodal hypersurface of a $\Delta$-eigenfunction $\phi_{\lambda}$ of eigenvalue $\lambda^2$ on a smooth Riemannian manifold. We prove the following lower bound for its surface measure: $\hcal^{n-1}(\ncal_{\phi_{\lambda}}) \geq C \lambda^{\frac74-\frac{3n}4} $. The best prior lower bound appears to be $e^{- C \lambda}$.

Source: http://arxiv.org/abs/1009.3573v3

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6.0

Jun 30, 2018
06/18

by
Christopher D. Sogge; Steve Zelditch

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In this note we show how improved $L^p$-estimates for certain types of quasi-modes are naturally equaivalent to improved operator norms of spectral projection operators associated to shrinking spectral intervals of the appropriate scale. Using this, one can see that recent estimates that were stated for eigenfunctions also hold for the appropriate types of quasi-modes.

Topics: Mathematics, Analysis of PDEs, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1401.0345

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Sep 20, 2013
09/13

by
Yanir A. Rubinstein; Steve Zelditch

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We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real Monge-Ampere equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the C^3 lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and...

Source: http://arxiv.org/abs/1205.4793v1

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Sep 24, 2013
09/13

by
Christopher D. Sogge; Steve Zelditch

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On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not...

Source: http://arxiv.org/abs/math/0103172v2

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Sep 18, 2013
09/13

by
John A. Toth; Steve Zelditch

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This is a partly expository, partly new paper on sup norm estimates of eigenfunctions. The focus is on the quantum completely integrable case. We give a new proof of the main result of our paper ``Riemannian manifolds with uniformly bounded eigenfunctions' (Duke Math J. 111 (2002), 97-132), based on the analysis of quasi-modes and Birkhoff normal forms. We also discuss related issues of resonant normal forms and tunnelling between tori.

Source: http://arxiv.org/abs/math/0202074v1

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Sep 21, 2013
09/13

by
Yanir A. Rubinstein; Steve Zelditch

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The Cauchy problem for the homogeneous (real and complex) Monge-Ampere equation (HRMA/HCMA) arises from the initial value problem for geodesics in the space of Kahler metrics. It is an ill-posed problem. We conjecture that, in its lifespan, the solution can be obtained by Toeplitz quantizing the Hamiltonian flow defined by the Cauchy data, analytically continuing the quantization, and then taking a kind of logarithmic classical limit. In this article, we prove that in the case of torus...

Source: http://arxiv.org/abs/1008.3577v3

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Sep 18, 2013
09/13

by
Yanir A. Rubinstein; Steve Zelditch

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We generalize the results of Song-Zelditch on geodesics in spaces of Kahler metrics on toric varieties to harmonic maps of any compact Riemannian manifold with boundary into the space of Kahler metrics on a toric variety. We show that the harmonic map equation can always be solved and that such maps may be approximated in the C^2 topology by harmonic maps into the spaces of Bergman metrics. In particular, WZW maps, or equivalently solutions of a homogeneous Monge-Ampere equation on the product...

Source: http://arxiv.org/abs/0803.1249v1

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Sep 21, 2013
09/13

by
Christopher D. Sogge; Steve Zelditch

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Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with Laplacian, $\Delta_g$. If $e_\lambda$ are the associated eigenfunctions of $\sqrt{-\Delta_g}$ so that $-\Delta_g e_\lambda = \lambda^2 e_\lambda$, then it has been known for some time \cite{soggeest} that $\|e_\lambda\|_{L^4(M)}\lesssim \lambda^{1/8}$, assuming that $e_\lambda$ is normalized to have $L^2$-norm one. This result is sharp in the sense that it cannot be improved on the standard sphere because of highest...

Source: http://arxiv.org/abs/1011.0215v1

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Sep 19, 2013
09/13

by
John A. Toth; Steve Zelditch

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The eigenfunctions e^{i \lambda x} of the Laplacian on a flat torus have uniformly bounded L^p norms. In this article, we prove that for every other quantum integrable Laplacian, the L^p norms of the joint eigenfunctions must blow up at a rate \gg \lambda^{p-2/4p - \epsilon} for every \epsilon >0 as \lambda \to \infty.

Source: http://arxiv.org/abs/math/0208045v1

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183

Sep 23, 2013
09/13

by
John A. Toth; Steve Zelditch

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We consider the zeros on the boundary $\partial \Omega$ of a Neumann eigenfunction $\phi_{\lambda}$ of a real analytic plane domain $\Omega$. We prove that the number of its boundary zeros is $O (\lambda)$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(\lambda)$. It follows that the number of nodal lines of $\phi_{\lambda}$ (components of the nodal set) which touch the...

Source: http://arxiv.org/abs/0710.0101v4

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61

Sep 23, 2013
09/13

by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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We study the limit as $N\to\infty$ of the correlations between simultaneous zeros of random sections of the powers $L^N$ of a positive holomorphic line bundle $L$ over a compact complex manifold $M$, when distances are rescaled so that the average density of zeros is independent of $N$. We show that the limit correlation is independent of the line bundle and depends only on the dimension of $M$ and the codimension of the zero sets. We also provide some explicit formulas for pair correlations....

Source: http://arxiv.org/abs/math-ph/9904020v1

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Sep 22, 2013
09/13

by
Dmitry Jakobson; Alexander Strohmaier; Steve Zelditch

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On a compact K\"ahler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after...

Source: http://arxiv.org/abs/0805.2376v1

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Sep 22, 2013
09/13

by
Bernard Shiffman; Steve Zelditch; Scott Zrebiec

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We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a holomorphic section of the N-th power of a positive line bundle on a compact Kaehler manifold. In particular, we show that for all $\delta>0$, the probability that this volume differs by more than $\delta N$ from its average value is less than $\exp(-C_{\delta,U}N^{m+1})$, for some constant $C_{\delta,U}>0$. As a consequence, the...

Source: http://arxiv.org/abs/0805.2598v2

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Sep 20, 2013
09/13

by
Hans Christianson; John Toth; Steve Zelditch

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We prove a quantum ergodic restriction theorem for the Cauchy data of a sequence of quantum ergodic eigenfunctions on a hypersurface $H$ of a Riemannian manifold $(M, g)$. The technique of proof is to use a Rellich type identity to relate quantum ergodicity of Cauchy data on $H$ to quantum ergodicity of eigenfunctions on the global manifold $M$. This has the interesting consequence that if the eigenfunctions are quantum unique ergodic on the global manifold $M$, then the Cauchy data is...

Source: http://arxiv.org/abs/1205.0286v2

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Jul 20, 2013
07/13

by
Frank Ferrari; Semyon Klevtsov; Steve Zelditch

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We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a background-independent measure on the space of metrics can be easily constructed from first principles. Our framework suggests the relevance of a new gravitational effective action and we show that it occurs when coupling the massive scalar field to two-dimensional...

Source: http://arxiv.org/abs/1107.4022v2

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Jul 20, 2013
07/13

by
Frank Ferrari; Semyon Klevtsov; Steve Zelditch

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The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a K\"ahler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of K\"ahler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all K\"ahler metrics. Several examples are considered.

Source: http://arxiv.org/abs/1107.4575v2

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Sep 22, 2013
09/13

by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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In our previous work [math-ph/9904020], we proved that the correlation functions for simultaneous zeros of random generalized polynomials have universal scaling limits and we gave explicit formulas for pair correlations in codimensions 1 and 2. The purpose of this paper is to compute these universal limits in all dimensions and codimensions. First, we use a supersymmetry method to express the n-point correlations as Berezin integrals. Then we use the Wick method to give a closed formula for the...

Source: http://arxiv.org/abs/math-ph/0011016v2

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Sep 23, 2013
09/13

by
Frank Ferrari; Semyon Klevtsov; Steve Zelditch

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The Mabuchi energy is an interesting geometric functional on the space of K\"ahler metrics that plays a crucial r\^ole in the study of the geometry of K\"ahler manifolds. We show that this functional, as well as other related geometric actions, contribute to the effective gravitational action when a massive scalar field is coupled to gravity in two dimensions in a small mass expansion. This yields new theories of two-dimensional quantum gravity generalizing the standard Liouville...

Source: http://arxiv.org/abs/1112.1352v2

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Sep 18, 2013
09/13

by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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This note is concerned with the scaling limit as N approaches infinity of n-point correlations between zeros of random holomorphic polynomials of degree N in m variables. More generally we study correlations between zeros of holomorphic sections of powers L^N of any positive holomorphic line bundle L over a compact Kahler manifold. Distances are rescaled so that the average density of zeros is independent of N. Our main result is that the scaling limits of the correlation functions and, more...

Source: http://arxiv.org/abs/math-ph/9903012v1

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3.0

Jun 30, 2018
06/18

by
Xiaonan Ma; George Marinescu; Steve Zelditch

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We consider a general Hermitian holomorphic line bundle $L$ on a compact complex manifold $M$ and let ${\Box}^q_p$ be the Kodaira Laplacian on $(0,q)$ forms with values in $L^p$. The main result is a complete asymptotic expansion for the semi-classically scaled heat kernel $\exp(-u{\Box}^q_p/p)(x,x)$ along the diagonal. It is a generalization of the Bergman/Szeg\"o kernel asymptotics in the case of a positive line bundle, but no positivity is assumed. We give two proofs, one based on the...

Topics: Complex Variables, Mathematics, Differential Geometry

Source: http://arxiv.org/abs/1406.0201

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Sep 18, 2013
09/13

by
Pavel Bleher; Bernard Shiffman; Steve Zelditch

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This article is concerned with random holomorphic polynomials and their generalizations to algebraic and symplectic geometry. A natural algebro-geometric generalization studied in our prior work involves random holomorphic sections $H^0(M,L^N)$ of the powers of any positive line bundle $L \to M$ over any complex manifold. Our main interest is in the statistics of zeros of $k$ independent sections (generalized polynomials) of degree $N$ as $N\to\infty$. We fix a point $P$ and focus on the ball...

Source: http://arxiv.org/abs/math-ph/0002039v2

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Sep 20, 2013
09/13

by
Jens Marklof; Stephen O'Keefe; Steve Zelditch

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For a general class of unitary quantum maps, whose underlying classical phase space is divided into several invariant domains of positive measure, we establish analogues of Weyl's law for the distribution of eigenphases. If the map has one ergodic component, and is periodic on the remaining domains, we prove the Schnirelman-Zelditch-Colin de Verdiere Theorem on the equidistribution of eigenfunctions with respect to the ergodic component of the classical map (quantum ergodicity). We apply our...

Source: http://arxiv.org/abs/nlin/0404038v2

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2.0

Jun 29, 2018
06/18

by
Boris Hanin; Steve Zelditch; Peng Zhou

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We study the scaling asymptotics of the eigenspace projection kernels $\Pi_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 \Delta + |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $\Pi_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets...

Topics: Probability, Spectral Theory, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1602.06848

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Sep 18, 2013
09/13

by
Bernard Shiffman; Tatsuya Tate; Steve Zelditch

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We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the...

Source: http://arxiv.org/abs/math/0306189v1

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Sep 24, 2013
09/13

by
Frank Ferrari; Semyon Klevtsov; Steve Zelditch

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Recently, the authors have proposed a new approach to the theory of random metrics, making an explicit link between probability measures on the space of metrics on a Kahler manifold and random matrix models. We consider simple examples of such models and compute the one and two-point functions of the metric. These geometric correlation functions correspond to new interesting types of matrix model correlators. We study a large class of examples and provide in particular a detailed study of the...

Source: http://arxiv.org/abs/1112.4382v2

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Jul 19, 2013
07/13

by
Bernard Shiffman; Steve Zelditch; Qi Zhong

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We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros, but we show that it has quite a different small distance behavior. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension one. To prove this, we...

Source: http://arxiv.org/abs/1005.4166v1

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Sep 21, 2013
09/13

by
Michael R. Douglas; Bernard Shiffman; Steve Zelditch

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Supersymmetric vacua (`universes') of string/M theory may be identified with certain critical points of a holomorphic section (the `superpotential') of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed. The present paper initiates the study of the statistics of critical points $\nabla s = 0$ of Gaussian random holomorphic sections with respect to a connection $\nabla$. Even the...

Source: http://arxiv.org/abs/math/0402326v2

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Sep 20, 2013
09/13

by
Michael R. Douglas; Bernard Shiffman; Steve Zelditch

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A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold $X$ with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas and Denef-Douglas are given, together with van der Corput style remainder estimates. We also...

Source: http://arxiv.org/abs/math-ph/0506015v4

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Jul 20, 2013
07/13

by
Michael R. Douglas; Bernard Shiffman; Steve Zelditch

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Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive Hermitian line bundle $(L, h)\to M$ over a compact K\"ahler manifold: the expected distribution of critical points of a Gaussian random holomorphic section $s \in H^0(M, L)$ with respect to the Chern connection $\nabla_h$. It is a measure on $M$ whose total mass is the average number $\mathcal{N}^{crit}_h$ of critical points of a random holomorphic section. We are interested in...

Source: http://arxiv.org/abs/math/0406089v3

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Sep 22, 2013
09/13

by
Christopher D. Sogge; John A. Toth; Steve Zelditch

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On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${\phi_{\lambda}}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not...

Source: http://arxiv.org/abs/0908.0688v1