pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

transient behavior of the m g 1 workload process in this paper we describe the time dependent moments of the workload process in the m g 1 queue the kth moment as a function of time can be characterized in terms of a differential equation involving lower moment functions and the time dependent server occupation probability for general initial conditions we show that the first two moment functions can be represented as the difference of two nondecreasing functions one of which is the moment function starting at zero the two nondecreasing components can be regarded as probability cumulative distribution function cdf s after appropriate normalization the normalized moment functions starting empty are called moment cdf s the other normalized components are called moment difference cdf s we establish relations among these cdf s using stationary excess relations we apply these relations to calculate moments and derivatives at the origin of these cdf s we also obtain results for the covariance function of the stationary workload process it is inte

chaotic scattering theory of transport and reaction rate coefficients the chaotic scattering theory is here extended to obtain escape rate expressions for the transport coefficients appropriate for a simple classical fluid or for a chemically reacting system this theory allows various transport coefficients such as the coefficients of viscosity thermal conductivity etc to be expressed in terms of the positive lyapunov exponents and kolmogorov sinai entropy of a set of phase space trajectories that take place on an appropriate fractal repeller this work generalizes the previous results of gaspard and nicolis for the coefficient of diffusion of a particle moving in a fixed array of scatterers

poisson law for the number of lattice points in a random strip with finite area let a smooth curve be given by a functionr f in polar coordinate system in the plane and letr be a uniformly distributed random variable on the interval a 1 l a 2 l with somea 2 a 1 0 and a largel 0 ya g sinai has conjectured that given some real numbersc 2 c 1 the number of lattice points in the domain between the curves left r frac c 1 r right and left r frac c 2 r right is asymptotically poisson distributed for good functionsf we cannot prove this conjecture but we show that if a probability measure with some nice properties is given on the space of smooth functions then almost all functions with respect to this measure satisfy sinai s conjecture this is an improvement of an earlier result of sinai 9 and actually the proof also contains many ideas of that paper

lyapunov exponent of random walkers on a bond disordered lattice abstract the chaotic properties of a random walker in a quenched random environment are studied analytically following the work of gaspard et al on lorentz gases for systems with closed periodic or open absorbing boundaries the model of interest describes random walkers hopping on a disordered lattice on which the hopping probabilities across bonds are quenched random variables for closed systems an exact expression for the lyapunov exponent is derived which not only depends on the composition but also on the number of clusters of certain type for open systems escape rates and lyapunov exponents are calculated from a mean field approximation the theoretical predictions are compared with the results of extensive computer simulations based on the thermodynamic formalism for large systems the agreement is excellent

chaotic scattering and diffusion in the lorentz gas a chaotic scattering theory of diffusion in the lorentz gas is presented the scattering process is considered on disk scatterers of increasing sizes in this way chaotic and fractal properties of the scattering process are related to diffusion a formula is obtained that gives the diffusion coefficient in terms of the lyapunov exponent and the hausdorff codimension of the fractal repeller of orbits trapped in the scatterer numerical results are presented that support our theoretical results

the pair correlation function of fractional parts of polynomials we investigate the pair correlation function of the sequence of fractional parts of n d n 1 2 n where d 2 is an integer and an irrational we conjecture that for badly approximable the normalized spacings between elements of this sequence have poisson statistics as n

semiclassical form factor for chaotic systems with spin we study the properties of the two point spectral form factor for classically chaotic systems with spin in the semiclassical limit with a suitable semiclassical trace formula as our principal tool to this end we introduce a regularized form factor and discuss the limit in which the so called diagonal approximation can be recovered the incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics provided this skew product is mixing we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory

dynamical localization of trajectories on a bond disordered lattice the chaotic dynamics of a random walker in a quenched environment is studied via the thermodynamic formalism of ruelle sinai and bowen in which chaotic properties are expressed in terms of a free energy type function of an inverse temperature like parameter localization phenomena in this system are elucidated both analytically and numerically the infinite system limit of the ruelle pressure at 1 and 1 is shown to be controlled by rare configurations of the bond disorder and this is related respectively to the extreme configurations associated to the minimum and maximum lyapunov exponent in finite systems these extreme values and the corresponding configurations are obtained numerically from monte carlo simulations based on the thermodynamic formalism

distribution of energy levels of quantum free particle on the liouville surface and trace formulae we consider the weyl asymptotic formula left e n leqq r 2 right frac area q 4 pi r 2 n r for eigenvalues of the laplace beltrami operator on a two dimensional torusq with a liouville metric which is in a sense the most general case of an integrable metric we prove that if the surfaceq is non degenerate then the remainder termn r has the formn r r1 2 r where r is an almost periodic function of the besicovitch classb1 and the fourier amplitudes and the fourier frequencies of r can be expressed via lengths of closed geodesics onq and other simple geometric characteristics of these geodesics we prove then that if the surfaceq is generic then the limit distribution of r has a densityp t which is an entire function oft possessing an asymptotics on a real line logp t c t4 ast an explicit expression for the fourier transform ofp t via fourier amplitudes of r is also given we obtain the analogue of the guillemin duistermaat trace formula for the liouville surfaces and discuss its accuracy

localized eigenfunctions in seba billiards we describe some new families of quasimodes for the laplacian perturbed by the addition of a potential formally described by a dirac delta function as an application we find under some additional hypotheses on the spectrum subsequences of eigenfunctions of seba billiards that localize around a pair of unperturbed eigenfunctions

the energy level spacing for two harmonic oscillators with golden mean ratio of frequencies the limit distribution of energy level spacing is studied for the system of two harmonic oscillators with generic ratio of frequencies it is proved that for any fixed generic ratio no limit distribution exists but for random with any absolutely continuous distributionp d on 0 1 a universal random limit distribution of the energy level spacing exists some properties of the random limit distribution are discussed

dynamic transient and stationary behavior of the m gi 1 queue via martingales l auteur associe une martingale exponentielle a la chaine de markov du nombre de clients d une file d attente m gi 1 il demontre a l aide de la theorie de renouvellement que cette martingale fournit un cadre probabiliste unifie pour deduire differentes fonctions generatrices usuelles pour les files d attente m gi 1

spectra of sol manifolds arithmetic and quantum monodromy the spectral problem of three dimensional manifolds m 3 a admitting sol geometry in thurston s sense is investigated topologically m 3 a are torus bundles over a circle with a unimodular hyperbolic gluing map a the eigenfunctions of the corresponding laplace beltrami operators are described in terms of modified mathieu functions it is shown that the multiplicities of the eigenvalues are the same for generic values of the parameters in the metric and are directly related to the number of representations of an integer by a given indefinite binary quadratic form as a result the spectral statistics is shown to disagree with the berry tabor conjecture the topological nature of the monodromy for both classical and quantum systems on sol manifolds is demonstrated

the uniform central limit theorem for theta sums sommes gaussiennes sommes de kloosterman theoreme de limite centrale uniforme pour des series theta theoreme de limite centrale uniformes pour des sommes theta moments sur des intervalles arbitraires

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

dynamical chaos in the lorentz lattice gas this paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics in particular to a study of nonequilibrium phenomena in lorentz lattice gases with stochastic collision rules using simple arguments based upon discussions in the mathematical literature we show that such lattice gases belong to the category of dynamical systems with positive lyapunov exponents this is accomplished by showing how such systems can be expressed in terms of continuous phase space variables expressions for the lyapunov exponent of a one dimensional lorentz lattice gas with periodic boundaries are derived other quantities of interest for the theory of irreversible processes are discussed

spectral statistics on zoll surfaces the subject of this article is a family of partly conjectural spectral invariants of a laplace operator with a discrete spectrum j known as its k level correlation functions k and level spacings distribution the 2 level or pair correlation function 2 for instance measures the asymptotic number of spacings i j between normalized eigenvalues i as the eigenvalues tend to infinity the normalization is so that the mean level spacing i 1 i is one these invariants have been studied by many physicists including wigner dyson and berry and are conjectured to have certain universal forms depending on the qualitative behaviour of the corresponding classical geodesic flow

distribution of the error term for the number of lattice points inside a shifted circle define begin array 20 c n x r left n in z d x n leqslant r right x in 0 1 d end array 1 1 and f x r r frac d 1 2 n x r r d text vol d b 1 1 2 where b 1 is the unit ball in r d thus n x r is the number of lattice points in z d in the r ball centered at x and f x r is the normalized difference with the volume of the ball we are interested in the problem of the distribution of f x r as a function of r for given x

values of indefinite quadratic forms at integral points and flows on spaces of lattices this mostly expository paper centers on recently proved conjectures in two areas a a conjecture of a oppenheim on the values of real indefinite quadratic forms at integral points b conjectures of dani raghunathan and margulis on closures of orbits in spaces of lattices such as sl r sl z at first sight a belongs to analytic number theory and b belongs to ergodic and lie theory and they seem to be quite unrelated they are discussed together here because of a very interesting connection between the two pointed out by m s raghunathan namely a special case of b yields a proof of a the first main goal of this talk is to describe the oppenheim conjecture and various refinements and to derive them from one statement about closures of orbits in the space of unimodular lattices in r3 see proposition 2 in 2 3 in 3 and 4 we put this statement in context and describe more general conjectures and results on orbit closures and invariant probability measures on quotients of lie groups by discrete subgroups 5 gives some brief comments on the proofs and further developments 6 7 and 8 are devoted to the so called s arithmetic setting where we consider products of real and p adic groups 6 is concerned with a generalized oppenheim conjecture 7 with a generalization of the orbit closure theorem proved by m ratner r8 and 8 with applications to quadratic forms since the subject matter of that last section has not been so far discussed elsewhere we take this opportunity to present proofs obtained jointly with g prasad finally 9 gives the proof of a lemma on symmetric simple lie algebras a special case of which is used in 8 i am glad to thank m ratner and g prasad for a number of remarks on and corrections to a preliminary version of this paper thanks to which many typos and inaccuracies have been eliminated

level repulsion in the spectrum of two dimensional harmonic oscillators the asymptotic energy limit of the density p s of spacings between adjacent levels of the two dimensional harmonic oscillator tdho spectrum is studied it is shown that in any integer segment m m 1 containing approximately m levels of the tdho spectrum m n p s has the form wi s si where i takes on at most three values for large m p s displays strong level repulsion for irrational but it does not settle on a stationary form nor does its average over m this is in marked contrast with the behaviour of generic integrable systems for which the poisson statistics p s exp s is known to apply

the generic spacing distribution of the two dimensional harmonic oscillator the level spacing distribution of the two dimensional harmonic oscillator is studied by choosing a suitable average a stable distribution is obtained this is then shown to have a fixed form for the generic frequency ratio a delta function is also found for a dense class of ratio with hausdorff dimension thus the distribution is unstable under perturbation of the ratio all of this differs from the universal poissonian statistics associated with integrable systems 1

energy level statistics of model quantum systems universality and scaling in a lattice point problem we investigate the statistics of the numbern r s of lattice pointsn z 2 in an annular domain r w r w a ra wherer w 0 herea is a fixed convex set with smooth boundary andw is chosen so that the area of r w iss the statistics comes fromr being taken as random with a smooth density in some interval c 1 t c 2 t c 2 c 1 0 we find that in the limitt the variance and distribution of n n r s s depend strongly on hows grows witht there is a saturation regimes t ast in which the fluctuations in n coming from the two boundaries of are independent then there is a scaling regime s t z 0 z in which the distribution depends onz in an almost periodic way going to a gaussian asz 0 the variance in this limit approachesz for generic a but can be larger for degenerate cases the former behavior is what one would expect from the poisson limit of a distribution for annuli of finite area

two point spectral correlations for the square billiard we investigate the two point correlations in the quantum spectrum of the square billiard this system is unusual in that the degeneracy of the energy levels increases in the semiclassical limit in such a way that the average level separation is not given by the inverse of the mean density of states hence for example the standard level spacings distribution does not tend to the poissonian limit expected for integrable systems in this paper we calculate the leading order asymptotic form of a degeneracy weighted two point correlation function using a combination of probabilistic techniques and classical number theory the result exhibits number theoretical fluctuations about a mean which is a sum of two terms one having the usual constant poissonian form and the second representing a small correction which decays as the inverse of the correlation distance this is confirmed by numerical computations

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

thermodynamic formalism and localization in lorentz gases and hopping models the thermodynamic formalism expresses chaotic properties of dynamical systems in terms of the ruelle pressure the inverse temperature like variable allows one to scan the structure of the probability distributin in the dynamic phase space this formalism is applied here to a lorentz lattice gas where a particle moving on a lattice of sizeld collides with fixed scatterers placed at random locations here we give rigorous arguments that the ruelle pressure in the limit of infinite systems has two branches joining with a slope discontinuity at 1 the low and high branches correspond to localization of trajectories on respectively the most chaotic highest density region and the most deterministic lowest density region i e is completely controlled by rare fluctuations in the distribution of scatterers on the lattice and it dose not carry and information on the global structure of the static disorder as approaches unity from either side a localization delocalization transition leads to a state where trajectories are extended and carry information on transprot properties at finitel the narrow region around 1 where the trajectories are extended scales as inl 2 where depends on the sign of 1 ifd 1 and as l inl 1 ifd 1 this result appears to be general for diffusive systems with static disorder such as random walks in random environments or for the continuous lorentz gas other models of random walks on disordered lattices showing the same phenomenon are discussed

statistical properties of spectral fluctuations for a quantum system with infinitely many components extending the idea formulated in makino phys rev e 67 066205 2003 that is based on the berry robnik approach m v berry and m robnik j phys a 17 2413 1984 we investigate the statistical properties of a two point spectral correlation for a classically integrable quantum system the eigenenergy sequence of this system is regarded as a superposition of infinitely many independent components in the semiclassical limit we derive the level number variance lnv in the limit of infinitely many components and discuss its deviations from poisson statistics the slope of the limiting lnv is found to be larger than that of poisson statistics when the individual components have a certain accumulation this property agrees with the result from the semiclassical periodic orbit theory that is applied to a system with degenerate torus actions d biswas m azam and s v lawande phys rev a 43 5694 1991

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

periodic orbits contribution to the 2 point correlation form factor for pseudo integrable systems using heuristic arguments based on the trace formulas we relate the 2 point correlation form factor k2 at small values of with sums over classical periodic orbits for typical examples of pseudo integrable systems the later sums have been explicitly calculated for the following models i plane billiards in the form of right triangles with one angle n and ii rectangular billiards with the aharonov bohm flux line in the first model using the properties of the veech structure it is shown that k2 0 n e n 3 n 2 where e n 0 for odd n e n 2 for even n not divisible by 3 and e n 6 for even n divisible by 3 for completeness we also recall informally the main features of the veech construction in the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle when these ratios are non commensurable irrational numbers k2 0 1 3 where is the fractional part of the flux through the rectangle when and it is symmetric with respect to the line when the comparison of these results with numerical calculations of the form factor is discussed in detail the above values of k2 0 differ from all known examples of spectral statistics thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo integrable systems

degeneracy moments for the square billiard we calculate the semiclassical asymptotics of the moments of a function that characterizes the accidental energy level degeneracies of the square billiard the results quantify the deviations from generic poisson statistics in this spectrum