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Timetable (RMAW04)

Random Matrix Theory and Arithmetic Aspects of Quantum Chaos

Monday 28th June 2004 to Friday 2nd July 2004

Monday 28th June 2004
09:30 to 10:25 Quantum vesus classical fluctuations on the modular surface

In joint work with W.Luo we have computed the variance of the fluctuations of a quantum observable on the modular surface. It corresponds to the classical fluctuations of the observable after insertion of a subtle arithmetic correction factor.

11:00 to 11:40 S De Bievre ([Lille])
Long time propagation of coherent states under perturbed cat map dynamics

I will describe recent work with J.M. Bouclet (Lille) on the propagation of coherent states up to times logarithmic in hbar under quantized perturbed cat maps. We show that, for long enough times, the quantum evolution equidistributes the coherent states throughout phase space. The proof requires a good control on the error term in the Egorov theorem on the one hand and on the classical rate of mixing on the other. This generalizes to perturbed cat maps a result obtained previously with F. Bonechi.

11:50 to 12:30 Random matrix theory and entanglement in quantum spin chains

We compute the entropy of entanglement in the ground states of a general class of quantum spin-chain Hamiltonians --- those that are related to quadratic forms of Fermi operators --- between the first $N$ spins and the rest of the system in the limit of infinite total chain length. We show that the entropy can be expressed in terms of averages over the classical compact groups and establish an explicit correspondence between the symmetries of a given Hamiltonian and those characterizing the Haar measure of the associated group. These averages are either Toeplitz determinants or determinants of combinations of Toeplitz and Hankel matrices. Recent generalizations of the Fisher-Hartwig conjecture are used to compute the leading order asymptotics of the entropy as $N\rightarrow\infty$. This is shown to grow logarithmically with $N$. The constant of proportionality is determined explicitly, as is the next (constant) term in the asymptotic expansion. The logarithmic growth of the entropy was previously predicted on the basis of numerical computations and conformal-field-theoretic calculations. In these calculations the constant of proportionality was determined in terms of the central charge of the Virasoro algebra. Our results therefore lead to an explicit formula for this charge. We also show that the entropy is related to solutions of ordinary differential equations of Painlev\'e type. In some cases these solutions can be evaluated to all orders using recurrence relations.

14:30 to 15:25 Evolution and constrains on scarring for (perturbed) cat maps

We consider quantized cat maps on the 2-dimensional torus, as well as their nonlinear perturbations. We first analyze the evolution up to the Ehrenfest time of states localized around a periodic point, showing a transition to equidistribution. Using this transition, we obtain constraints on the localization properties of eigenstates around periodic orbits. The analysis is much simpler in the unperturbed case, where one uses the algebraic properties of the map. Besides, the constraints we obtain are known to be sharp only for the unperturbed case.

16:00 to 16:40 C Hughes ([AIM])
On the number of lattice points in a thin annulus

We count the number of integer lattice points in an annulus of inner-radius $t$ and outer-radius $t+\rho$. If $\rho \to 0$ sufficiently slowly then the distribution of this counting function as $t\to\infty$ weakly converges to the normal distribution.

16:50 to 17:30 N Anantharaman ([Lyon])
The ``Quantum unique ergodicity" problem for anosov geodesic flows: an approach by entropy

Let $M$ be compact, negatively curved Riemannian manifold, and let $(\psi_n)$ be an orthonormal basis of eigenfunctions of the Laplacian on $M$. The Quantum Unique Ergodicity problem concerns the behaviour of the sequence of probability measures $|\psi_n(x)|^2 dx$ on $M$, or, more precisely, of their "microlocal" lifts to the tangent bundle $TM$. The limits of convergent subsequences must be invariant probability measures of the geodesic flow (sometimes called "quantum invariant measures"), and it is known that a very large subsequence converges to the Liouville measure on the unit tangent bundle. A conjecture of Rudnick and Sarnak says that this should actually be the only possible limit. E. Lindenstrauss proved the conjecture recently, in the case when $M$ is an arithmetic surface (of constant negative curvature) and $(\psi_n)$ is a common basis of eigenfunctions for the Laplacian and the Hecke operators. However, very little is known in the non-arithmetic case.

In the general case of an Anosov geodesic flow, I present an attempt to bound from below the metric entropy of "quantum invariant measures". I actually prove the following: if the $L^p$ norms of the $\psi_n$s do not grow too fast with $n$, then the corresponding quantum invariant measures cannot be entirely carried on a set of zero topological entropy.

Tuesday 29th June 2004
09:30 to 10:25 Semiclassical evidence for universal spectral correlations in quantum chaos

Almost all quantum systems which are chaotic in their classical limit exhibit universality when statistical distributions of energy levels are evaluated. Spectral correlations are found to agree with those between eigenvalues of random matrices. We report on recent progress in semiclassical methods that provide a theoretical basis for the connection between quantum chaos and random matrix theory.

11:00 to 11:40 R Schubert ([Bristol])
Propagation of wavepackets for large times

We study the semiclassical propagation of a class of wavepackets for large times on manifolds of negative curvature. The time evolution is generated by the Laplace-Beltrami operator and the wavepackets considered are Lagrangian states. The principal result is that these wavepackets become weakly equidistributed in the joint limit $\hbar\to 0$ and $t\to\infty$ with $t<<|\ln \hbar|$. The main ingredient in the proof is hyperbolicity and mixing of the geodesic flow.

11:50 to 12:30 On the distribution of matrix elements for the quantum cat map

For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. We will study the fluctuations of the matrix elements for the desymmetrized quantum cat map and present a conjecture for the distribution of the normalized matrix elements, namely that their distribution is that of a certain weighted sum of traces of independent matrices in SU(2). This is in contrast to generic chaotic systems where the distribution is expected to be Gaussian. We will show that the second and fourth moment of the distribution agree with the conjecture, and also present some numerical evidence.

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14:30 to 15:25 Quantum chaos on locally symmetric spaces

I will discuss a proof of QUE, w.r.t. the basis of Hecke eigenforms, for certain locally symmetric spaces of rank > 1.

16:00 to 16:40 On the remiainder in weyl's law for heisenberg manifolds

We examine the error term in Weyl's law for the distribution of Laplace eigenvalues for Heisenberg manifolds equipped with a left-invariant metric. We formulate conjectures on the optimal size and evidence for the conjectures. We explain the analogies with the classical Dirichlet divisor problem in analytic number theory. This program has been developped jointly with D. Chung, M. Khosravi and J. Toth.

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16:50 to 17:30 S Müller ([Essen])
Semiclassical foundation of universality in quantum chaos

We sketch the semiclassical core of a proof of the so-called Bohigas-Giannoni-Schmit conjecture: A dynamical system with full classical chaos has a quantum energy spectrum with universal fluctuations on the scale of the mean level spacing. We show how in the semiclassical limit all system specific properties fade away, leaving only ergodicity, hyperbolicity, and combinatorics as agents determining the contributions of pairs of classical periodic orbits to the quantum spectral form factor. The small-time form factor is thus reproduced semiclassically. Bridges between classical orbits and (the non-linear sigma model of) quantum field theory are built by revealing the contributing orbit pairs as topologically equivalent to Feynman diagrams.

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Wednesday 30th June 2004
09:30 to 10:25 Classical dynamics of billiards in rational polygons

This talk is a survey of some recent developments in the field.

11:00 to 11:40 The triangle map: a model for quantum chaos

We intend to discuss some recent results concerning classical and semiclassical properites of a particular weakly chaotic discrete dynamical system.

11:50 to 12:30 Eigenfunction statistics for star graphs

I will review some recent results, obtained in collaborations with Gregory Berkolaiko, Jens Marklof and Brian Winn, relating to the title of the talk.

Thursday 1st July 2004
09:30 to 10:25 A central limit theorem for the spectrum of the modular domain

We study the fluctuations in the discrete spectrum of the hyperbolic Laplacian for the modular domain using smooth counting functions. We show that in a certain regime, these have Gaussian fluctuations.

11:00 to 11:40 Subconvexity of L-functions and the uniqueness principle

We consider the triple L-function $L(1/2,f\times g\times \phi_i)$ for fixed Maass forms f and g as the eigenvalue of $\phi_i$ goes to infinity.

We deduce a subconvexity bound for this L-function from the uniqueness principle in representation theory and from simple geometric properties of the corresponding invariant functional. Joint with J. Bernstein.

11:50 to 12:30 On distribution of zeros of Heine-Stieltjes polynomials

We introduce Heine-Stieltjes polynomials, describe classical results of Heine, Stieltjes, Van Vleck and Shah on the distribution of their zeros as well as more recent asymptotic results, both in the semiclassical and thermodynamic regime.

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14:30 to 15:25 Complex zeros of real ergodic eigenfunctions

A well-known problem in geometry of eigenfunctions of Laplacians on Riemannian manifolds is to determine how the nodal hypersurface (zero set) is asymptotically distributed as the eigenvalue tends to infinity. The random wave model predicts that the normalized measure of integration over the nodal hypersurface tends to the volume measure on the manifold. My talk is a preliminary report on the distribution of complex zeros of analytic continuations of eigenfunctions of real analytic Riemannian manifolds with ergodic geodesic flow. We describe how the complex nodal hypersurfaces are distributed in the cotangent bundle. The (perhaps surprising) result is that the complex zeros concentrate around the real ones.

16:00 to 16:40 Energy asymptotics for gaudin spin chains

I will discuss recent work (joint with M. Min-Oo) on partition function asymptotics for the integrable Gaudin spin chains in various thermodynamic regimes.

Friday 2nd July 2004
09:30 to 10:25 F Steiner ([Ulm])
The cosmic microwave background and the shape of the Universe

The anisotropy of the cosmic microwave background (CMB) is analysed in nearly flat hyperbolic universes possessing a non-trivial topology with a fundamental cell which is stretched out into an infinitely long horn. It is shown that the horned topology does not lead to a flat spot in the CMB sky maps in the direction of the horn as predicted by Levin, Barrow, Bunn, and Silk.Two particular topologies are discussed in detail: the Sokolov-Starobinsky model having an infinite spatial volume, and the Picard orbifold which has a finite volume. It is demonstrated that the recent observations of the "Wilkinson Microwave Anisotropy Probe" (WMAP) hint that our Universe may be shaped like the Picard space.

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11:00 to 11:40 Zeros of the derivative of a selberg zeta function

In this talk, we will study the distribution of non-trivial zeros of Selberg zeta functions on cofinite hyperbolic surfaces, in particular obtain the asymptotic formula for the zero density with bounded height, which is similar to the well-known Weyl law. Then we will relate the distribution of the zeros to the issue of bounding the multiplicity of Laplacian eigenvalues.

11:50 to 12:30 A Strombergsson ([Uppsala])
Numerical computations with the trace formula and the selberg eigenvalue conjecture

I will report on some numerical computations with the trace formula on congruence subgroups of the modular group. In particular I will discuss how to check numerically the validity of the Selberg eigenvalue conjecture for specific congruence subgroups.

This is joint work with Andrew Booker.

14:30 to 15:25 Granular bosonization (or Fyodorov meets SUSY)

Random matrix methods can be roughly divided into two categories: methods which rely on having the joint probability density of the eigenvalues in closed form, and others which don't. Supersymmetry methods belong to the latter category.

The supersymmetry method used in the theory of disordered metals goes back to Schaefer and Wegner (1980). Quite recently, Fyodorov has proposed a related but different method, which computes averages of inverse characteristic polynomials for granular systems or random matrices with a hierarchical structure.

In this talk Fyodorov's method is reviewed, and it is shown how to generalize it to the case of characteristic polynomials (where it amounts to a form of bosonization) and to the case of ratios of such polynomials (the supersymmetric variant). Some applications to granular systems are presented.

16:00 to 16:40 The double Riemann zeta function

A double zeta function is defined as a function having zeros at sum of zeros of an original zeta function. The aim of this talk is to construct the double Riemann zeta function, and express it as a double Euler product which is a product over pairs of prime numbers.

16:50 to 17:30 A Gamburd ([Stanford])
Expander graphs, random matrices and quantum chaos

A basic problem in the theory of expander graphs, formulated by Lubotzky and Weiss, is to what extent being an expander family for a family of Cayley graphs is a property of the groups alone, independent of the choice of generators. While recently Alon, Lubotzky and Wigderson constructed an example demonstrating that expansion is not in general a group property, the problem is open for "natural" families of groups. In particular for SL(2, p) numerical experiments indicate that it might be an expander family for "generic" choices of generators (Independence Conjecture).

A basic conjecture in Quantum Chaos, formulated by Bohigas, Giannoni, and Shmit, asserts that the eigenvalues of a quantized chaotic Hamiltonian behave like the spectrum of a typical member of the appropriate ensemble of random matrices. Both conjectures can be viewed as asserting that a deterministically constructed spectrum "generically" behaves like the spectrum of a large random matrix: "in the bulk" (Quantum Chaos Conjecture) and at the "edge of the spectrum" (Independence Conjecture). After explaining this approach in the context of the spectra of random walks on groups, we review some recent related results and numerical experiments.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons