5,515 research outputs found

### Integrable random matrix ensembles

We propose new classes of random matrix ensembles whose statistical
properties are intermediate between statistics of Wigner-Dyson random matrices
and Poisson statistics. The construction is based on integrable N-body
classical systems with a random distribution of momenta and coordinates of the
particles. The Lax matrices of these systems yield random matrix ensembles
whose joint distribution of eigenvalues can be calculated analytically thanks
to integrability of the underlying system. Formulas for spacing distributions
and level compressibility are obtained for various instances of such ensembles.Comment: 32 pages, 8 figure

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

The 2-point correlation form factor, $K_2(\tau)$, for small values of $\tau$
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 $\pi/n$ 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
$K_2(0)=(n+\epsilon(n))/(3(n-2))$ where $\epsilon(n)=0$ for odd $n$,
$\epsilon(n)=2$ for even $n$ not divisible by 3, and $\epsilon(n)=6$ 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, $K_2(0)=1-3\bar{\alpha}+4\bar{\alpha}^2$ where $\bar{\alpha}$ is the
fractional part of the flux through the rectangle when $0\le \bar{\alpha}\le
1/2$ and it is symmetric with respect to the line $\bar{\alpha}=1/2$ when $1/2
\le \bar{\alpha}\le 1$. The comparison of these results with numerical
calculations of the form factor is discussed in detail. The above values of
$K_2(0)$ 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

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

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

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

### Multifractal dimensions for all moments for certain critical random matrix ensembles in the strong multifractality regime

We construct perturbation series for the q-th moment of eigenfunctions of
various critical random matrix ensembles in the strong multifractality regime
close to localization. Contrary to previous investigations, our results are
valid in the region q<1/2. Our findings allow to verify, at first leading
orders in the strong multifractality limit, the symmetry relation for anomalous
fractal dimensions Delta(q)=Delta(1-q), recently conjectured for critical
models where an analogue of the metal-insulator transition takes place. It is
known that this relation is verified at leading order in the weak
multifractality regime. Our results thus indicate that this symmetry holds in
both limits of small and large coupling constant. For general values of the
coupling constant we present careful numerical verifications of this symmetry
relation for different critical random matrix ensembles. We also present an
example of a system closely related to one of these critical ensembles, but
where the symmetry relation, at least numerically, is not fulfilled.Comment: 12 pages, 12 figure

### Quantum computation of multifractal exponents through the quantum wavelet transform

We study the use of the quantum wavelet transform to extract efficiently
information about the multifractal exponents for multifractal quantum states.
We show that, combined with quantum simulation algorithms, it enables to build
quantum algorithms for multifractal exponents with a polynomial gain compared
to classical simulations. Numerical results indicate that a rough estimate of
fractality could be obtained exponentially fast. Our findings are relevant e.g.
for quantum simulations of multifractal quantum maps and of the Anderson model
at the metal-insulator transition.Comment: 9 pages, 9 figure

### Multifractality and intermediate statistics in quantum maps

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

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