arXiv subjects

Steve Zelditch

Publications and source records attributed to Steve Zelditch.

At least 127 records · Page 7Linked to original sources

arXiv2003

Asymptotics of polynomials and eigenfunctions

We review some recent results on asymptotic properties of polynomials of large degree, of general holomorphic sections of high powers of positive line bundles over Kahler manifolds, and of Laplace eigenfunctions of large eigenvalue on compact Riemannian manifolds. We describe statistical patterns in the zeros, critical points and L^p norms of random polynomials and holomorphic sections, and the influence of the Newton polytope on these patterns. For eigenfunctions, we discuss L^p norms and mass concentration of individual eigenfunctions and their relation to dynamics of the geodesic flow.

math.CA↗

arXiv2003

Distribution laws for integrable eigenfunctions

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 tails of the eigenfunctions on large length scales. These are not universal but depend on the global geometry of the toric variety and in particular on the details of the exponential decay of the eigenfunctions away from the classically allowed set.

math.CV↗

arXiv2003

Random polynomials with prescribed Newton polytope

We show that the Newton polytope of a polynomial has a strong impact on the distribution of its mass and zeros. The basic theme is that Newton polytopes determine allowed and forbidden regions for these distributions. We equip the space of (holomorphic) polynomials of degree $\leq p$ in $m$ complex variables with its usual $SU(m + 1)$-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope $P$. We then determine the asymptotics of the conditional expectation $E_{|N P}(Z^k)$ of simultaneous zeros of $k$ polynomials with Newton polytope $NP$ as $N \to \infty$. When $P = Σ$, the unit simplex, then it is obvious and well-known that the expected zero distribution $E_{|NΣ}(Z^k)$ is uniform relative to the Fubini-Study form. For a convex polytope $P\subset pΣ$, we show that there is an allowed region on which $N^{-k}E_{|N P}(Z^k)$ is asymptotically uniform as the scaling factor $N\to\infty$. However, the zeros have an exotic distribution in the complementary forbidden region, and when $k=m$, the expected percentage of simultaneous zeros in the forbidden region approaches 0 as $N\to\infty$, yielding a quantitative version of the Kouchnirenko-Bernstein theorem.

math.AG↗

arXiv2003

Macdonald's identities and the large N limit of $YM_2$ on the cylinder

We give a rigorous calculation of the large N limit of the partition function of SU(N) gauge theory on a 2D cylinder in the case where one boundary holomony is a so-called special element of type $ρ$. By MacDonald's identity, the partition function factors in this case as a product over positive roots and it is straightforward to calculate the large N asymptotics of the free energy. We obtain the unexpected result that the free energy in these cases is asymptotic to N times a functional of the limit densities of eigenvalues of the boundary holonomies. This appears to contradict the predictions of Gross-Matysin and Kazakov-Wynter that the free energy should have a limit governed by the complex Burgers equation.

hep-th↗

arXiv2003

Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers

We give asymptotic formulas for the multiplicities of weights and irreducible summands in high-tensor powers $V_λ^{\otimes N}$ of an irreducible representation $V_λ$ 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 with steps in a convex polytope.

math.RT↗

arXiv2003

Inverse spectral problem for analytic plane domains I: Balian-Bloch trace formula

We give a rigorous version of the classical Balian-Bloch trace formula, a semiclassical expansion around a periodic reflecting ray of the (regularized) resolvent of the Dirichlet Laplacian on a bounded smooth plane domain. It is equivalent to the Poisson relation (or wave trace formula) between spectrum and closed geodesics. We view it primarily as a computational device for explicitly calculating wave trace invariants. Its effectiveness will be illustrated in subsquent articles in the series in which concrete inverse spectral results are proved.

math.SP↗

arXiv2002

L^p norms of eigenfunctions in the completely integrable case

The eigenfunctions e^{i λx} 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 λ^{p-2/4p - ε} for every ε>0 as λ\to \infty.

math.SP↗

arXiv2002

Equilibrium distribution of zeros of random polynomials

We consider ensembles of random polynomials of the form $p(z)=\sum_{j = 1}^N a_j P_j$ where $\{a_j\}$ are independent complex normal random variables and where $\{P_j\}$ are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain $Ω\subset C$ relative to an analytic weight $ρ(z) |dz|$. In the simplest case where $Ω$ is the unit disk and $ρ=1$, so that $P_j(z) = z^j$, it is known that the average distribution of zeros is the uniform measure on $S^1$. We show that for any analytic $(Ω, ρ)$, the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on $\partialΩ$ as $N\to\infty$. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of $(Ω, ρ)$.

math.CV↗

arXiv2002

Random polynomials with prescribed Newton polytope, I

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 consider the conditional measures $γ_{|NP}$ induced on the subspace of polynomials whose Newton polytope $P_f\subset NP$. When $P=Σ$, the unit simplex, then it is obvious and well-known that the expected distribution of zeros $Z_{f_1,...,f_k}$ (regarded as a current) of $k$ polynomials $f_1,...,f_k$ of degree $N$ is uniform relative to the Fubini-Study form. Our main results concern the conditional expectation $E_{|NP} (Z_{f_1,...,f_k})$ of zeros of $k$ polynomials with Newton polytope $NP\subset NpΣ$ (where $p=°P$); these results are asymptotic as the scaling factor $N\to\infty$. We show that $E_{|NP} (Z_{f_1,...,f_k})$ is asymptotically uniform on the inverse image $A_P$ of the open scaled polytope $p^{-1}P^\circ$ via the moment map $μ:CP^m\toΣ$ for projective space. However, the zeros have an exotic distribution outside of $A_P$ and when $k=m$ (the case of the Kouchnirenko-Bernstein theorem) the percentage of zeros outside $A_P$ approaches 0 as $N\to\infty$.

math.AG↗

arXiv2002

Norms of Modes and Quasi-modes revisited

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.

math.AP↗

arXiv2001

Riemannian manifolds with maximal eigenfunction growth

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{ϕ_λ\}$ satisfy $||ϕ_λ||_{\infty} \leq C λ^{\frac{n-1}{2}}$ where $-Δϕ_λ = λ^2 ϕ_λ.$ 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 $\R^n/Γ$. We say that $S^n$, but not $\R^n/Γ$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the $(M, g)$ with maximal eigenfunction growth. Our main result is that such an $(M, g)$ must have a point $x$ where the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in $S^*_x M$. We show that if $(M, g)$ is real analytic, this puts topological restrictions on $M$, e.g. only $M = S^2$ (topologically) in dimension 2 can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any $M$ fail to have maximal eigenfunction growth. In addition, we construct an example of $(M, g)$ for which ${\mathcal L}_x$ has positive measure for an open set of $x$ but which does not have maximal eigenfunction growth, thus disproving a naive converse to the main result.

math.AP↗

arXiv2001

Riemannian Manifolds With Uniformly Bounded Eigenfunctions

The standard eigenfunctions $ϕ_λ = e^{i < λ, x >}$ 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.

math-ph↗

arXiv2000

Correlations between zeros and supersymmetry

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 limit pair correlation function for the point case in all dimensions.

math-ph↗

arXiv2000

Universality and scaling of zeros on symplectic manifolds

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 of radius $1/\sqrt{N}$ about $P$. Under a microscope magnifying the ball by the factor $\sqrt{N}$, the statistics of the configurations of simultaneous zeros of random $k$-tuples of sections tends to a universal limit independent of $P,M,L$. We review this result and generalize it further to the case of pre-quantum line bundles over almost-complex symplectic manifolds $(M,J,ω)$. Following [SZ2], we replace $H^0(M,L^N)$ in the complex case with the `asymptotically holomorphic' sections defined by Boutet de Monvel-Guillemin and (from another point of view) by Donaldson and Auroux. Using a generalization to an $m$-dimensional setting of the Kac-Rice formula for zero correlations together with the results of [SZ2], we prove that the scaling limits of the correlation functions for zeros of random $k$-tuples of asymptotically holomorphic sections belong to the same universality class as in the complex case.

math-ph↗

arXiv2000

From Random Polynomials to Symplectic Geometry

We review some recent results on random polynomials and their generalizations in complex and symplectic geometry. The main theme is the universality of statistics of zeros and critical points of (generalized) polynomials of degree $N$ on length scales of order $\frac{D}{\sqrt{N}}$ (complex case), resp. $\frac{D}{N}$ (real case).

math-ph↗

arXiv2000

Inverse Spectral Problem for Surfaces of Revolution

This paper concerns the inverse spectral problem for analytic simple surfaces of revolution. By `simple' is meant that there is precisely one critical distance from the axis of revolution. Such surfaces have completely integrable geodesic flows with global action-angle variables and possess global quantum Birkhoff normal forms (Colin de Verdiere). We prove that isospectral surfaces within this class are isometric. The first main step is to show that the normal form at meridian geodesics is a spectral invariant. The second main step is to show that the metric is determined from this normal form.

math-ph↗

arXiv2000

Wave invariants at elliptic closed geodesics

This paper concerns spectral invariants of the Laplacian on a compact Riemannian manifold (M,g) known as wave invariants. If U(t) denotes the wave group of (M,g), then the trace Tr U(t) is singular when t = 0 or when ti is the length of a closed geodesic. It has a special type of singularity expansion at each length and the coefficients are known as the wave invariants. Our main purpose is to calculate the wave invariants explicitly in terms of curvature, Jacobi fields etc. when the closed geodesic is non-degenerate elliptic. We do this by putting the Laplacian into quantum Birkhoff normal form at the closed geodesic. Such a normal form was previously introduced by V. Guillemin. We give a new algorithm for calculating it, and for expressing wave invariants in terms of normal form invariants.

math.SP↗

arXiv2000

Wave Invariants for Non-degenerate Closed Geodesics

This paper generalizes the methods and results of our article xxx.lanl.gov math.SP/0002036 from elliptic to general non-degenerate closed geodesics. The main purpose is to introduce a quantum Birkhoff normal form of the Laplacian at a general non-degenerate closed geodesic in the sense of V.Guillemin. Guillemin proved that the coefficients of the normal form at an elliptic closed geodesic could be determined from the wave invariants of this geodesic. We give a new proof, and extend its range to any non-degenerate closed geodesic.

math.SP↗