1,444 research outputs found
Ensemble dependence in the Random transverse-field Ising chain
In a disordered system one can either consider a microcanonical ensemble,
where there is a precise constraint on the random variables, or a canonical
ensemble where the variables are chosen according to a distribution without
constraints. We address the question as to whether critical exponents in these
two cases can differ through a detailed study of the random transverse-field
Ising chain. We find that the exponents are the same in both ensembles, though
some critical amplitudes vanish in the microcanonical ensemble for correlations
which span the whole system and are particularly sensitive to the constraint.
This can \textit{appear} as a different exponent. We expect that this apparent
dependence of exponents on ensemble is related to the integrability of the
model, and would not occur in non-integrable models.Comment: 8 pages, 12 figure
The Cost of Stability in Coalitional Games
A key question in cooperative game theory is that of coalitional stability,
usually captured by the notion of the \emph{core}--the set of outcomes such
that no subgroup of players has an incentive to deviate. However, some
coalitional games have empty cores, and any outcome in such a game is unstable.
In this paper, we investigate the possibility of stabilizing a coalitional
game by using external payments. We consider a scenario where an external
party, which is interested in having the players work together, offers a
supplemental payment to the grand coalition (or, more generally, a particular
coalition structure). This payment is conditional on players not deviating from
their coalition(s). The sum of this payment plus the actual gains of the
coalition(s) may then be divided among the agents so as to promote stability.
We define the \emph{cost of stability (CoS)} as the minimal external payment
that stabilizes the game.
We provide general bounds on the cost of stability in several classes of
games, and explore its algorithmic properties. To develop a better intuition
for the concepts we introduce, we provide a detailed algorithmic study of the
cost of stability in weighted voting games, a simple but expressive class of
games which can model decision-making in political bodies, and cooperation in
multiagent settings. Finally, we extend our model and results to games with
coalition structures.Comment: 20 pages; will be presented at SAGT'0
Percolation in random environment
We consider bond percolation on the square lattice with perfectly correlated
random probabilities. According to scaling considerations, mapping to a random
walk problem and the results of Monte Carlo simulations the critical behavior
of the system with varying degree of disorder is governed by new, random fixed
points with anisotropic scaling properties. For weaker disorder both the
magnetization and the anisotropy exponents are non-universal, whereas for
strong enough disorder the system scales into an {\it infinite randomness fixed
point} in which the critical exponents are exactly known.Comment: 8 pages, 7 figure
Griffiths-McCoy Singularities in the Random Transverse-Field Ising Spin Chain
We consider the paramagnetic phase of the random transverse-field Ising spin
chain and study the dynamical properties by numerical methods and scaling
considerations. We extend our previous work [Phys. Rev. B 57, 11404 (1998)] to
new quantities, such as the non-linear susceptibility, higher excitations and
the energy-density autocorrelation function. We show that in the Griffiths
phase all the above quantities exhibit power-law singularities and the
corresponding critical exponents, which vary with the distance from the
critical point, can be related to the dynamical exponent z, the latter being
the positive root of [(J/h)^{1/z}]_av=1. Particularly, whereas the average spin
autocorrelation function in imaginary time decays as [G]_av(t)~t^{-1/z}, the
average energy-density autocorrelations decay with another exponent as
[G^e]_av(t)~t^{-2-1/z}.Comment: 8 pages RevTeX, 8 eps-figures include
Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions
We study XY and dimerized XX spin-1/2 chains with random exchange couplings
by analytical and numerical methods and scaling considerations. We extend
previous investigations to dynamical properties, to surface quantities and
operator profiles, and give a detailed analysis of the Griffiths phase. We
present a phenomenological scaling theory of average quantities based on the
scaling properties of rare regions, in which the distribution of the couplings
follows a surviving random walk character. Using this theory we have obtained
the complete set of critical decay exponents of the random XY and XX models,
both in the volume and at the surface. The scaling results are confronted with
numerical calculations based on a mapping to free fermions, which then lead to
an exact correspondence with directed walks. The numerically calculated
critical operator profiles on large finite systems (L<=512) are found to follow
conformal predictions with the decay exponents of the phenomenological scaling
theory. Dynamical correlations in the critical state are in average
logarithmically slow and their distribution show multi-scaling character. In
the Griffiths phase, which is an extended part of the off-critical region
average autocorrelations have a power-law form with a non-universal decay
exponent, which is analytically calculated. We note on extensions of our work
to the random antiferromagnetic XXZ chain and to higher dimensions.Comment: 19 pages RevTeX, eps-figures include
Numerical studies of the two- and three-dimensional gauge glass at low temperature
We present results from Monte Carlo simulations of the two- and
three-dimensional gauge glass at low temperature using the parallel tempering
Monte Carlo method. Our results in two dimensions strongly support the
transition being at T_c=0. A finite-size scaling analysis, which works well
only for the larger sizes and lower temperatures, gives the stiffness exponent
theta = -0.39 +/- 0.03. In three dimensions we find theta = 0.27 +/- 0.01,
compatible with recent results from domain wall renormalization group studies.Comment: 7 pages, 10 figures, submitted to PR
Monte Carlo simulations of the four-dimensional XY spin glass at low temperatures
We report results for simulations of the four-dimensional XY spin glass using
the parallel tempering Monte Carlo method at low temperatures for moderate
sizes. Our results are qualitatively consistent with earlier work on the
three-dimensional gauge glass as well as three- and four-dimensional
Edwards-Anderson Ising spin glass. An extrapolation of our results would
indicate that large-scale excitations cost only a finite amount of energy in
the thermodynamic limit. The surface of these excitations may be fractal,
although we cannot rule out a scenario compatible with replica symmetry
breaking in which the surface of low-energy large-scale excitations is space
filling.Comment: 6 pages, 8 figure
Properties of the random field Ising model in a transverse magnetic field
We consider the effect of a random longitudinal field on the Ising model in a
transverse magnetic field. For spatial dimension , there is at low
strength of randomness and transverse field, a phase with true long range order
which is destroyed at higher values of the randomness or transverse field. The
properties of the quantum phase transition at zero temperature are controlled
by a fixed point with no quantum fluctuations. This fixed point also controls
the classical finite temperature phase transition in this model. Many critical
properties of the quantum transition are therefore identical to those of the
classical transition. In particular, we argue that the dynamical scaling is
activated, i.e, the logarithm of the diverging time scale rises as a power of
the diverging length scale
Nature of the Spin-glass State in the Three-dimensional Gauge Glass
We present results from simulations of the gauge glass model in three
dimensions using the parallel tempering Monte Carlo technique. Critical
fluctuations should not affect the data since we equilibrate down to low
temperatures, for moderate sizes. Our results are qualitatively consistent with
earlier work on the three and four dimensional Edwards-Anderson Ising spin
glass. We find that large scale excitations cost only a finite amount of energy
in the thermodynamic limit, and that those excitations have a surface whose
fractal dimension is less than the space dimension, consistent with a scenario
proposed by Krzakala and Martin, and Palassini and Young.Comment: 5 pages, 7 figure
Some extremal functions in Fourier analysis, III
We obtain the best approximation in , by entire functions of
exponential type, for a class of even functions that includes
, where , and , where . We also give periodic versions of these results where the
approximating functions are trigonometric polynomials of bounded degree.Comment: 26 pages. Submitte
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