4,677 research outputs found

    Periodic orbits contribution to the 2-point correlation form factor for pseudo-integrable systems

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    The 2-point correlation form factor, K2(τ)K_2(\tau), for small values of τ\tau is computed analytically for typical examples of pseudo-integrable systems. This is done by explicit calculation of periodic orbit contributions in the diagonal approximation. The following cases are considered: (i) plane billiards in the form of right triangles with one angle π/n\pi/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+ϵ(n))/(3(n−2))K_2(0)=(n+\epsilon(n))/(3(n-2)) where ϵ(n)=0\epsilon(n)=0 for odd nn, ϵ(n)=2\epsilon(n)=2 for even nn not divisible by 3, and ϵ(n)=6\epsilon(n)=6 for even nn 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)=1−3αˉ+4αˉ2K_2(0)=1-3\bar{\alpha}+4\bar{\alpha}^2 where αˉ\bar{\alpha} is the fractional part of the flux through the rectangle when 0≤αˉ≤1/20\le \bar{\alpha}\le 1/2 and it is symmetric with respect to the line αˉ=1/2\bar{\alpha}=1/2 when 1/2≤αˉ≤11/2 \le \bar{\alpha}\le 1. The comparison of these results with numerical calculations of the form factor is discussed in detail. The above values of K2(0)K_2(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.Comment: 61 pages, 13 figures. Submitted to Communications in Mathematical Physics, 200

    Move ordering and communities in complex networks describing the game of go

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    We analyze the game of go from the point of view of complex networks. We construct three different directed networks of increasing complexity, defining nodes as local patterns on plaquettes of increasing sizes, and links as actual successions of these patterns in databases of real games. We discuss the peculiarities of these networks compared to other types of networks. We explore the ranking vectors and community structure of the networks and show that this approach enables to extract groups of moves with common strategic properties. We also investigate different networks built from games with players of different levels or from different phases of the game. We discuss how the study of the community structure of these networks may help to improve the computer simulations of the game. More generally, we believe such studies may help to improve the understanding of human decision process.Comment: 14 pages, 21 figure

    Distinguishing humans from computers in the game of go: a complex network approach

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    We compare complex networks built from the game of go and obtained from databases of human-played games with those obtained from computer-played games. Our investigations show that statistical features of the human-based networks and the computer-based networks differ, and that these differences can be statistically significant on a relatively small number of games using specific estimators. We show that the deterministic or stochastic nature of the computer algorithm playing the game can also be distinguished from these quantities. This can be seen as tool to implement a Turing-like test for go simulators.Comment: 7 pages, 6 figure

    Eigenfunction entropy and spectral compressibility for critical random matrix ensembles

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    Based on numerical and perturbation series arguments we conjecture that for certain critical random matrix models the information dimension of eigenfunctions D_1 and the spectral compressibility chi are related by the simple equation chi+D_1/d=1, where d is the system dimensionality.Comment: 4 pages, 3 figure

    Multifractality and intermediate statistics in quantum maps

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    We study multifractal properties of wave functions for a one-parameter family of quantum maps displaying the whole range of spectral statistics intermediate between integrable and chaotic statistics. We perform extensive numerical computations and provide analytical arguments showing that the generalized fractal dimensions are directly related to the parameter of the underlying classical map, and thus to other properties such as spectral statistics. Our results could be relevant for Anderson and quantum Hall transitions, where wave functions also show multifractality.Comment: 4 pages, 4 figure

    Anticoherence of spin states with point group symmetries

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    We investigate multiqubit permutation-symmetric states with maximal entropy of entanglement. Such states can be viewed as particular spin states, namely anticoherent spin states. Using the Majorana representation of spin states in terms of points on the unit sphere, we analyze the consequences of a point-group symmetry in their arrangement on the quantum properties of the corresponding state. We focus on the identification of anticoherent states (for which all reduced density matrices in the symmetric subspace are maximally mixed) associated with point-group symmetric sets of points. We provide three different characterizations of anticoherence, and establish a link between point symmetries, anticoherence and classes of states equivalent through stochastic local operations with classical communication (SLOCC). We then investigate in detail the case of small numbers of qubits, and construct infinite families of anticoherent states with point-group symmetry of their Majorana points, showing that anticoherent states do exist to arbitrary order.Comment: 15 pages, 5 figure

    Random matrix ensembles associated with Lax matrices

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    A method to generate new classes of random matrix ensembles is proposed. Random matrices from these ensembles are Lax matrices of classically integrable systems with a certain distribution of momenta and coordinates. The existence of an integrable structure permits to calculate the joint distribution of eigenvalues for these matrices analytically. Spectral statistics of these ensembles are quite unusual and in many cases give rigorously new examples of intermediate statistics
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