79 research outputs found
Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity
The computational complexity of internal diffusion-limited aggregation (DLA)
is examined from both a theoretical and a practical point of view. We show that
for two or more dimensions, the problem of predicting the cluster from a given
set of paths is complete for the complexity class CC, the subset of P
characterized by circuits composed of comparator gates. CC-completeness is
believed to imply that, in the worst case, growing a cluster of size n requires
polynomial time in n even on a parallel computer.
A parallel relaxation algorithm is presented that uses the fact that clusters
are nearly spherical to guess the cluster from a given set of paths, and then
corrects defects in the guessed cluster through a non-local annihilation
process. The parallel running time of the relaxation algorithm for
two-dimensional internal DLA is studied by simulating it on a serial computer.
The numerical results are compatible with a running time that is either
polylogarithmic in n or a small power of n. Thus the computational resources
needed to grow large clusters are significantly less on average than the
worst-case analysis would suggest.
For a parallel machine with k processors, we show that random clusters in d
dimensions can be generated in O((n/k + log k) n^{2/d}) steps. This is a
significant speedup over explicit sequential simulation, which takes
O(n^{1+2/d}) time on average.
Finally, we show that in one dimension internal DLA can be predicted in O(log
n) parallel time, and so is in the complexity class NC
Understanding the temperature and the chemical potential using computer simulations
Several Monte Carlo algorithms and applications that are useful for
understanding the concepts of temperature and chemical potential are discussed.
We then introduce a generalization of the demon algorithm that measures the
chemical potential and is suitable for simulating systems with variable
particle number.Comment: 23 pages including 6 figure
Evidence against a mean field description of short-range spin glasses revealed through thermal boundary conditions
A theoretical description of the low-temperature phase of short-range spin
glasses has remained elusive for decades. In particular, it is unclear if
theories that assert a single pair of pure states, or theories that are based
infinitely many pure states-such as replica symmetry breaking-best describe
realistic short-range systems. To resolve this controversy, the
three-dimensional Edwards-Anderson Ising spin glass in thermal boundary
conditions is studied numerically using population annealing Monte Carlo. In
thermal boundary conditions all eight combinations of periodic vs antiperiodic
boundary conditions in the three spatial directions appear in the ensemble with
their respective Boltzmann weights, thus minimizing finite-size corrections due
to domain walls. From the relative weighting of the eight boundary conditions
for each disorder instance a sample stiffness is defined, and its typical value
is shown to grow with system size according to a stiffness exponent. An
extrapolation to the large-system-size limit is in agreement with a description
that supports the droplet picture and other theories that assert a single pair
of pure states. The results are, however, incompatible with the mean-field
replica symmetry breaking picture, thus highlighting the need to go beyond
mean-field descriptions to accurately describe short-range spin-glass systems.Comment: 13 pages, 11 figures, 3 table
Population annealing: Theory and application in spin glasses
Population annealing is an efficient sequential Monte Carlo algorithm for
simulating equilibrium states of systems with rough free energy landscapes. The
theory of population annealing is presented, and systematic and statistical
errors are discussed. The behavior of the algorithm is studied in the context
of large-scale simulations of the three-dimensional Ising spin glass and the
performance of the algorithm is compared to parallel tempering. It is found
that the two algorithms are similar in efficiency though with different
strengths and weaknesses.Comment: 16 pages, 10 figures, 4 table
Ground states and thermal states of the random field Ising model
The random field Ising model is studied numerically at both zero and positive
temperature. Ground states are mapped out in a region of random and external
field strength. Thermal states and thermodynamic properties are obtained for
all temperatures using the the Wang-Landau algorithm. The specific heat and
susceptibility typically display sharp peaks in the critical region for large
systems and strong disorder. These sharp peaks result from large domains
flipping. For a given realization of disorder, ground states and thermal states
near the critical line are found to be strongly correlated--a concrete
manifestation of the zero temperature fixed point scenario.Comment: 5 pages, 5 figures; new material added in this versio
Numerical study of the random field Ising model at zero and positive temperature
In this paper the three dimensional random field Ising model is studied at
both zero temperature and positive temperature. Critical exponents are
extracted at zero temperature by finite size scaling analysis of large
discontinuities in the bond energy. The heat capacity exponent is
found to be near zero. The ground states are determined for a range of external
field and disorder strength near the zero temperature critical point and the
scaling of ground state tilings of the field-disorder plane is discussed. At
positive temperature the specific heat and the susceptibility are obtained
using the Wang-Landau algorithm. It is found that sharp peaks are present in
these physical quantities for some realizations of systems sized and
larger. These sharp peaks result from flipping large domains and correspond to
large discontinuities in ground state bond energies. Finally, zero temperature
and positive temperature spin configurations near the critical line are found
to be highly correlated suggesting a strong version of the zero temperature
fixed point hypothesis.Comment: 11 pages, 14 figure
Asymptotic behavior of the finite-size magnetization as a function of the speed of approach to criticality
The main focus of this paper is to determine whether the thermodynamic
magnetization is a physically relevant estimator of the finite-size
magnetization. This is done by comparing the asymptotic behaviors of these two
quantities along parameter sequences converging to either a second-order point
or the tricritical point in the mean-field Blume--Capel model. We show that the
thermodynamic magnetization and the finite-size magnetization are asymptotic
when the parameter governing the speed at which the sequence
approaches criticality is below a certain threshold . However, when
exceeds , the thermodynamic magnetization converges to 0
much faster than the finite-size magnetization. The asymptotic behavior of the
finite-size magnetization is proved via a moderate deviation principle when
.
To the best of our knowledge, our results are the first rigorous confirmation
of the statistical mechanical theory of finite-size scaling for a mean-field
model.Comment: Published in at http://dx.doi.org/10.1214/10-AAP679 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Ginzburg-Landau Polynomials and the Asymptotic Behavior of the Magnetization Near Critical and Tricritical Points
For the mean-field version of an important lattice-spin model due to Blume
and Capel, we prove unexpected connections among the asymptotic behavior of the
magnetization, the structure of the phase transitions, and a class of
polynomials that we call the Ginzburg-Landau polynomials. The model depends on
the parameters n, beta, and K, which represent, respectively, the number of
spins, the inverse temperature, and the interaction strength. Our main focus is
on the asymptotic behavior of the magnetization m(beta_n,K_n) for appropriate
sequences (beta_n,K_n) that converge to a second-order point or to the
tricritical point of the model and that lie inside various subsets of the
phase-coexistence region. The main result states that as (beta_n,K_n) converges
to one of these points (beta,K), m(beta_n,K_n) ~ c |beta - beta_n|^gamma --> 0.
In this formula gamma is a positive constant, and c is the unique positive,
global minimum point of a certain polynomial g that we call the Ginzburg-Landau
polynomial. This polynomial arises as a limit of appropriately scaled
free-energy functionals, the global minimum points of which define the
phase-transition structure of the model. For each sequence (beta_n,K_n) under
study, the structure of the global minimum points of the associated
Ginzburg-Landau polynomial mirrors the structure of the global minimum points
of the free-energy functional in the region through which (beta_n,K_n) passes
and thus reflects the phase-transition structure of the model in that region.
The properties of the Ginzburg-Landau polynomials make rigorous the predictions
of the Ginzburg-Landau phenomenology of critical phenomena, and the asymptotic
formula for m(beta_n,K_n) makes rigorous the heuristic scaling theory of the
tricritical point.Comment: 70 pages, 8 figure
Critical fluctuations of noisy period-doubling maps
We extend the theory of quasipotentials in dynamical systems by calculating,
within a broad class of period-doubling maps, an exact potential for the
critical fluctuations of pitchfork bifurcations in the weak noise limit. These
far-from-equilibrium fluctuations are described by finite-size mean field
theory, placing their static properties in the same universality class as the
Ising model on a complete graph. We demonstrate that the effective system size
of noisy period-doubling bifurcations exhibits universal scaling behavior along
period-doubling routes to chaos.Comment: 11 pages, 5 figure
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