spectral statistics for unitary transfer matrices of binary graphs quantum graphs have recently been introduced as model systems to study the spectral statistics of linear wave problems with chaotic classical limits it is proposed here to generalize this approach by considering arbitrary directed graphs with unitary transfer matrices an exponentially increasing contribution to the form factor is identified when performing a diagonal summation over periodic orbit degeneracy classes a special class of graphs so called binary graphs is studied in more detail for these the conditions for periodic orbit pairs to be correlated including correlations due to the unitarity of the transfer matrix can be given explicitly using combinatorial techniques it is possible to perform the summation over correlated periodic orbit pair contributions to the form factor for some low dimensional cases gradual convergence towards random matrix results is observed when increasing the number of vertices of the binary graphs

quantum graphs a simple model for chaotic scattering we connect quantum graphs with infinite leads and turn them into scattering systems we show that they display all the features which characterize quantum scattering systems with an underlying classical chaotic dynamics typical poles delay time and conductance distributions ericson fluctuations and when considered statistically the ensemble of scattering matrices reproduces quite well the predictions of the appropriately defined random matrix ensembles the underlying classical dynamics can be defined and it provides important parameters which are needed for the quantum theory in particular we derive exact expressions for the scattering matrix and an exact trace formula for the density of resonances in terms of classical orbits analogous to the semiclassical theory of chaotic scattering we use this in order to investigate the origin of the connection between random matrix theory and the underlying classical chaotic dynamics being an exact theory and due to its relative simplicity it offers new insights into this problem which is at the forefront of the research in chaotic scattering and related fields

fractal weyl laws in discrete models of chaotic scattering we analyse simple models of quantum chaotic scattering namely quantized open baker s maps we numerically compute the density of quantum resonances in the semiclassical regime this density satisfies a fractal weyl law where the exponent is governed by the fractal dimension of the set of trapped trajectories this type of behaviour is also expected in the physically more relevant case of hamiltonian chaotic scattering within a simplified model we are able to rigorously prove this weyl law and compute quantities related to the coherent transport through the system namely the conductance and shot noise the latter is close to the prediction of random matrix theory

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

regular quantum graphs we introduce the concept of regular quantum graphs and construct connected quantum graphs with discrete symmetries the method is based on a decomposition of the quantum propagator in terms of permutation matrices which control the way incoming and outgoing channels at vertex scattering processes are connected symmetry properties of the quantum graph as well as its spectral statistics depend on the particular choice of permutation matrices also called connectivity matrices and can now be controlled easily the method may find applications in the study of quantum random walks and may also prove to be useful in analysing universality in spectral statistics

towards a semiclassical theory of the quantum baker s map abstract we develop semiclassical 0 treatments of the quantum baker s map paying special attention to a the discrete and finite nature of its hilbert space and b the asymptotic analysis of the true quantum objects themselves new techniques are described at the theoretical level exact symbolic decomposition of the propagator and at the numerical level color graphics display of operators in the kirkwood representation both of which are of general interest for other quantized chaotic maps periodic orbit theory is tested numerically and we find that log corrections need to be included before an accurate calculation of the spectrum from periodic orbits can be achieved however the asymptotic analysis of the traces of the propagator reveals unexpected departures from standard semiclassical theory those anomalies which we compute explicitly in a few cases are traced ultimately to the discontinuity of the map and to the compactness of the phase space the analysis is far from complete but the results point towards the necessity of careful assessment of semiclassical folklore when applied to chaotic maps

experimental versus numerical eigenvalues of a bunimovich stadium billiard a comparison we compare the statistical properties of eigenvalue sequences for a gamma 1 bunimovich stadium billiard the eigenvalues have been obtained in two ways one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator real system and the other set is calculated numerically ideal system we show influence of mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system we also discuss the influence of a family of marginally stable orbits the bouncing ball orbits in two microwave stadium billiards with different geometrical dimensions

lyndon word decompositions and pseudo orbits on q nary graphs abstract a foundational result in the theory of lyndon words words that are strictly earlier in lexicographic order than their cyclic permutations is the chen fox lyndon theorem which states that every word has a unique non increasing decomposition into lyndon words this article extends this factorization theorem obtaining the proportion of these decompositions that are strictly decreasing this result is then used to count primitive pseudo orbits sets of primitive periodic orbits on q nary graphs as an application we obtain a diagonal approximation to the variance of the characteristic polynomial coefficients of q nary quantum graphs

unitary stochastic matrix ensembles and spectral statistics we propose to study unitary matrix ensembles defined in terms of unitary stochastic transition matrices associated with markov processes on graphs we argue that the spectral statistics of such an ensemble after ensemble averaging depends crucially on the spectral gap between the leading and subleading eigenvalue of the underlying transition matrix it is conjectured that unitary stochastic ensembles follow one of the three standard ensembles of random matrix theory in the limit of infinite matrix size n if the spectral gap of the corresponding transition matrices closes slower than 1 n the hypothesis is tested by considering several model systems ranging from binary graphs to uniformly and non uniformly connected star graphs and diffusive networks in arbitrary dimensions

experimental investigation of wigner s reaction matrix for irregular graphs with absorption we use tetrahedral microwave networks consisting of coaxial cables and attenuators connected by t joints to make an experimental study of wigner s reaction k matrix for irregular graphs in the presence of absorption from measurements of the scattering matrix s for each realization of the microwave network we obtain distributions of the imaginary and real parts of k our experimental results are in good agreement with theoretical predictions

random stochastic matrices from classical compact lie groups and symmetric spaces we consider random stochastic matrices m with elements given by mij uij 2 with u being uniformly distributed on one of the classical compact lie groups or some of the associated symmetric spaces we observe numerically that for large dimensions the spectral statistics of m discarding the perron frobenius eigenvalue 1 are similar to those of the gaussian orthogonal ensemble for symmetric matrices and to those of the real ginibre ensemble for nonsymmetric matrices we compute some spectral statistics using weingarten functions and establish connections with some difficult enumerative problems involving permutations

experimental and numerical investigation of the reflection coefficient and the distributions of wigner s reaction matrix for irregular graphs with absorption we present the results of an experimental and numerical study of the distribution of the reflection coefficient p r and the distributions of the imaginary p v and the real p u parts of the wigner reaction k matrix for irregular fully connected hexagon networks graphs in the presence of strong absorption in the experiment we used microwave networks which were built of coaxial cables and attenuators connected by joints in the numerical calculations experimental networks were described by quantum fully connected hexagon graphs the presence of absorption introduced by attenuators was modeled by optical potentials the distribution of the reflection coefficient p r and the distributions of the reaction k matrix were obtained from measurements and numerical calculations of the scattering matrix s of the networks and graphs respectively we show that the experimental and numerical results are in good agreement with the exact analytic ones obtained within the framework of random matrix theory

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

universal quantum computation with unlabelled qubits we show that an nth root of the walsh hadamard transform obtained from the hadamard gate and a cyclic permutation of the qubits together with two diagonal matrices namely a local qubit flip for a fixed but arbitrary qubit and a non local phase flip for a fixed but arbitrary coefficient can do universal quantum computation on n qubits a quantum computation making use of n qubits and based on these operations is then a word of variable length but whose letters are always taken from an alphabet of cardinality three therefore in contrast with other universal sets no choice of qubit lines is needed for the application of the operations described here a quantum algorithm based on this set can be interpreted as a discrete diffusion of a quantum particle on a de bruijn graph corrected on the fly by auxiliary modifications of the phases associated with the arcs

spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices to a unitary matrix u we associate a doubly stochastic matrix m by taking the squared modulus of each element of u to study the connection between onset of quantum chaos on graphs and ergodicity of the underlying markov chain specified by m we study the limiting distribution of the spectral gap of m when u is taken from the circular unitary ensemble and the dimension n of u is taken to infinity we prove that the limiting distribution is degenerate the gap tends to its maximal value 1 the shape of the gap distribution for finite n is also discussed

experimental simulation of quantum graphs by microwave networks we present the results of experimental and theoretical study of irregular tetrahedral microwave networks consisting of coaxial cables annular waveguides connected by t joints the spectra of the networks were measured in the frequency range 0 0001 16 ghz in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity d3sld the comparison of our experimental and theoretical results shows that microwave networks can simulate quantum graphs with time reversal symmetry in particular we use the spectra of the microwave networks to study the periodic orbits of the simulated quantum graphs we also present experimental study of directional microwave networks consisting of coaxial cables and faraday isolators for which the time reversal symmetry is broken in this case our experimental results indicate that spectral statistics of directional microwave networks deviate from predictions of gaussian orthogonal ensembles in random matrix theory approaching especially for small eigenfrequency spacing s results for gaussian unitary ensembles experimental results are supported by the theoretical analysis of directional graphs

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

periodic orbit action correlations in the baker map periodic orbit action correlations are studied for the piecewise linear area preserving baker map semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum baker map suggest the existence of universal periodic orbit correlations the calculation of periodic orbit sums for the baker map can be performed with the help of a perron frobenius type operator this makes it possible to study periodic orbit correlations for orbits with period up to 500 iterations of the map periodic orbit correlations are found to agree quantitatively with the predictions from random matrix theory up to a critical length determined by the semiclassical error exponentially increasing terms dominate the correlations for longer orbits which are due to the violation of unitarity in the semiclassical approximation

families of line graphs and their quantization any directed graph g with n vertices and j edges has an associated line graph l g where the j edges form the vertices of l g we show that the non zero eigenvalues of the adjacency matrices are the same for all graphs of such a family ln g we give necessary and sufficient conditions for a line graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions line graphs may therefore serve as models to study the semiclassical limit of large matrix size of a quantum dynamics on graphs with fixed classical behaviour

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

classical 1d maps quantum graphs and ensembles of unitary matrices we study a certain class of classical one dimensional piecewise linear maps for these systems we introduce an infinite family of markov partitions in equal cells the symbolic dynamics generated by these systems is described by bi stochastic doubly stochastic matrices we analyse the structure of graphs generated from the corresponding symbolic dynamics we demonstrate that the spectra of quantized graphs corresponding to the regular classical systems have locally poissonian statistics while quantized graphs derived from classically chaotic systems display statistical properties characteristic of the circular unitary ensemble even though the corresponding unitary matrices are sparse

mathematical aspects of vacuum energy on quantum graphs we use quantum graphs as a model to study various mathematical aspects of the vacuum energy such as convergence of periodic path expansions consistency among different methods trace formulae versus method of images and the possible connection with the underlying classical dynamics in our study we derive an expansion for the vacuum energy in terms of periodic paths on the graph and prove its convergence and smooth dependence on the bond lengths of the graph for an important special case of graphs with equal bond lengths we derive a simpler explicit formula with minor changes this formula also applies to graphs with rational up to a common factor bond lengths the main results are derived using the trace formula we also discuss an alternative approach using the method of images and prove that the results are consistent this may have important consequences for other systems since the method of images unlike the trace formula includes a sum over special bounce paths we succeed in showing that in our model bounce paths do not contribute to the vacuum energy finally we discuss the proposed possible link between the magnitude of the vacuum energy and the type chaotic versus integrable of the underlying classical dynamics within a random matrix model we calculate the variance of the vacuum energy over several ensembles and find evidence that the level repulsion leads to suppression of the vacuum energy