2,132 research outputs found
Scaling and Universality in Continuous Length Combinatorial Optimization
We consider combinatorial optimization problems defined over random
ensembles, and study how solution cost increases when the optimal solution
undergoes a small perturbation delta. For the minimum spanning tree, the
increase in cost scales as delta^2; for the mean-field and Euclidean minimum
matching and traveling salesman problems in dimension d>=2, the increase scales
as delta^3; this is observed in Monte Carlo simulations in d=2,3,4 and in
theoretical analysis of a mean-field model. We speculate that the scaling
exponent could serve to classify combinatorial optimization problems into a
small number of distinct categories, similar to universality classes in
statistical physics.Comment: 5 pages; 3 figure
Reduction of Two-Dimensional Dilute Ising Spin Glasses
The recently proposed reduction method is applied to the Edwards-Anderson
model on bond-diluted square lattices. This allows, in combination with a
graph-theoretical matching algorithm, to calculate numerically exact ground
states of large systems. Low-temperature domain-wall excitations are studied to
determine the stiffness exponent y_2. A value of y_2=-0.281(3) is found,
consistent with previous results obtained on undiluted lattices. This
comparison demonstrates the validity of the reduction method for bond-diluted
spin systems and provides strong support for similar studies proclaiming
accurate results for stiffness exponents in dimensions d=3,...,7.Comment: 7 pages, RevTex4, 6 ps-figures included, for related information, see
http://www.physics.emory.edu/faculty/boettcher
Near-Minimal Spanning Trees: a Scaling Exponent in Probability Models
We study the relation between the minimal spanning tree (MST) on many random
points and the "near-minimal" tree which is optimal subject to the constraint
that a proportion of its edges must be different from those of the
MST. Heuristics suggest that, regardless of details of the probability model,
the ratio of lengths should scale as . We prove this
scaling result in the model of the lattice with random edge-lengths and in the
Euclidean model.Comment: 24 pages, 3 figure
Disorder induced rounding of the phase transition in the large q-state Potts model
The phase transition in the q-state Potts model with homogeneous
ferromagnetic couplings is strongly first order for large q, while is rounded
in the presence of quenched disorder. Here we study this phenomenon on
different two-dimensional lattices by using the fact that the partition
function of the model is dominated by a single diagram of the high-temperature
expansion, which is calculated by an efficient combinatorial optimization
algorithm. For a given finite sample with discrete randomness the free energy
is a pice-wise linear function of the temperature, which is rounded after
averaging, however the discontinuity of the internal energy at the transition
point (i.e. the latent heat) stays finite even in the thermodynamic limit. For
a continuous disorder, instead, the latent heat vanishes. At the phase
transition point the dominant diagram percolates and the total magnetic moment
is related to the size of the percolating cluster. Its fractal dimension is
found d_f=(5+\sqrt{5})/4 and it is independent of the type of the lattice and
the form of disorder. We argue that the critical behavior is exclusively
determined by disorder and the corresponding fixed point is the isotropic
version of the so called infinite randomness fixed point, which is realized in
random quantum spin chains. From this mapping we conjecture the values of the
critical exponents as \beta=2-d_f, \beta_s=1/2 and \nu=1.Comment: 12 pages, 12 figures, version as publishe
Near optimal configurations in mean field disordered systems
We present a general technique to compute how the energy of a configuration
varies as a function of its overlap with the ground state in the case of
optimization problems. Our approach is based on a generalization of the cavity
method to a system interacting with its ground state. With this technique we
study the random matching problem as well as the mean field diluted spin glass.
As a byproduct of this approach we calculate the de Almeida-Thouless transition
line of the spin glass on a fixed connectivity random graph.Comment: 13 pages, 7 figure
The Random-bond Potts model in the large-q limit
We study the critical behavior of the q-state Potts model with random
ferromagnetic couplings. Working with the cluster representation the partition
sum of the model in the large-q limit is dominated by a single graph, the
fractal properties of which are related to the critical singularities of the
random Potts model. The optimization problem of finding the dominant graph, is
studied on the square lattice by simulated annealing and by a combinatorial
algorithm. Critical exponents of the magnetization and the correlation length
are estimated and conformal predictions are compared with numerical results.Comment: 7 pages, 6 figure
Phase coexistence and finite-size scaling in random combinatorial problems
We study an exactly solvable version of the famous random Boolean
satisfiability problem, the so called random XOR-SAT problem. Rare events are
shown to affect the combinatorial ``phase diagram'' leading to a coexistence of
solvable and unsolvable instances of the combinatorial problem in a certain
region of the parameters characterizing the model. Such instances differ by a
non-extensive quantity in the ground state energy of the associated diluted
spin-glass model. We also show that the critical exponent , controlling
the size of the critical window where the probability of having solutions
vanishes, depends on the model parameters, shedding light on the link between
random hyper-graph topology and universality classes. In the case of random
satisfiability, a similar behavior was conjectured to be connected to the onset
of computational intractability.Comment: 10 pages, 5 figures, to appear in J. Phys. A. v2: link to the XOR-SAT
probelm adde
Zero-temperature behavior of the random-anisotropy model in the strong-anisotropy limit
We consider the random-anisotropy model on the square and on the cubic
lattice in the strong-anisotropy limit. We compute exact ground-state
configurations, and we use them to determine the stiffness exponent at zero
temperature; we find and respectively
in two and three dimensions. These results show that the low-temperature phase
of the model is the same as that of the usual Ising spin-glass model. We also
show that no magnetic order occurs in two dimensions, since the expectation
value of the magnetization is zero and spatial correlation functions decay
exponentially. In three dimensions our data strongly support the absence of
spontaneous magnetization in the infinite-volume limit
- …