234 research outputs found
Scaling universalities of kth-nearest neighbor distances on closed manifolds
Take N sites distributed randomly and uniformly on a smooth closed surface.
We express the expected distance from an arbitrary point on the
surface to its kth-nearest neighboring site, in terms of the function A(l)
giving the area of a disc of radius l about that point. We then find two
universalities. First, for a flat surface, where A(l)=\pi l^2, the k-dependence
and the N-dependence separate in . All kth-nearest neighbor distances
thus have the same scaling law in N. Second, for a curved surface, the average
\int d\mu over the surface is a topological invariant at leading and
subleading order in a large N expansion. The 1/N scaling series then depends,
up through O(1/N), only on the surface's topology and not on its precise shape.
We discuss the case of higher dimensions (d>2), and also interpret our results
using Regge calculus.Comment: 14 pages, 2 figures; submitted to Advances in Applied Mathematic
Implicit Density Functional Theory
A fermion ground state energy functional is set up in terms of particle
density, relative pair density, and kinetic energy tensor density. It satisfies
a minimum principle if constrained by a complete set of compatibility
conditions. A partial set, which thereby results in a lower bound energy under
minimization, is obtained from the solution of model systems, as well as a
small number of exact sum rules. Prototypical application is made to several
one-dimensional spinless non-interacting models. The effectiveness of "atomic"
constraints on model "molecules" is observed, as well as the structure of
systems with only finitely many bound states.Comment: 9 pages, 4 figure
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
The Peculiar Phase Structure of Random Graph Bisection
The mincut graph bisection problem involves partitioning the n vertices of a
graph into disjoint subsets, each containing exactly n/2 vertices, while
minimizing the number of "cut" edges with an endpoint in each subset. When
considered over sparse random graphs, the phase structure of the graph
bisection problem displays certain familiar properties, but also some
surprises. It is known that when the mean degree is below the critical value of
2 log 2, the cutsize is zero with high probability. We study how the minimum
cutsize increases with mean degree above this critical threshold, finding a new
analytical upper bound that improves considerably upon previous bounds.
Combined with recent results on expander graphs, our bound suggests the unusual
scenario that random graph bisection is replica symmetric up to and beyond the
critical threshold, with a replica symmetry breaking transition possibly taking
place above the threshold. An intriguing algorithmic consequence is that
although the problem is NP-hard, we can find near-optimal cutsizes (whose ratio
to the optimal value approaches 1 asymptotically) in polynomial time for
typical instances near the phase transition.Comment: substantially revised section 2, changed figures 3, 4 and 6, made
minor stylistic changes and added reference
The random link approximation for the Euclidean traveling salesman problem
The traveling salesman problem (TSP) consists of finding the length of the
shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where
the cities are distributed randomly and independently in a d-dimensional unit
hypercube. Working with periodic boundary conditions and inspired by a
remarkable universality in the kth nearest neighbor distribution, we find for
the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with
beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive
analytical predictions for these quantities using the random link
approximation, where the lengths between cities are taken as independent random
variables. From the ``cavity'' equations developed by Krauth, Mezard and
Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3,
numerical results show that the random link approximation is a good one, with a
discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we
argue that the approximation is exact up to O(1/d^2) and give a conjecture for
beta_E(d), in terms of a power series in 1/d, specifying both leading and
subleading coefficients.Comment: 29 pages, 6 figures; formatting and typos correcte
The density functional theory of classical fluids revisited
We reconsider the density functional theory of nonuniform classical fluids
from the point of view of convex analysis. From the observation that the
logarithm of the grand-partition function is a convex
functional of the external potential it is shown that the Kohn-Sham free
energy is a convex functional of the density . and constitute a pair of Legendre transforms and each
of these functionals can therefore be obtained as the solution of a variational
principle. The convexity ensures the unicity of the solution in both cases. The
variational principle which gives as the maximum of a
functional of is precisely that considered in the density functional
theory while the dual principle, which gives as the maximum of
a functional of seems to be a new result.Comment: 10 page
Fundamental measure theory for lattice fluids with hard core interactions
We present the extension of Rosenfeld's fundamental measure theory to lattice
models by constructing a density functional for d-dimensional mixtures of
parallel hard hypercubes on a simple hypercubic lattice. The one-dimensional
case is exactly solvable and two cases must be distinguished: all the species
with the same lebgth parity (additive mixture), and arbitrary length parity
(nonadditive mixture). At the best of our knowledge, this is the first time
that the latter case is considered. Based on the one-dimensional exact
functional form, we propose the extension to higher dimensions by generalizing
the zero-dimensional cavities method to lattice models. This assures the
functional to have correct dimensional crossovers to any lower dimension,
including the exact zero-dimensional limit. Some applications of the functional
to particular systems are also shown.Comment: 22 pages, 7 figures, needs IOPP LaTeX styles file
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