114 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
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
Deterministic walks in random networks: an application to thesaurus graphs
In a landscape composed of N randomly distributed sites in Euclidean space, a
walker (``tourist'') goes to the nearest one that has not been visited in the
last \tau steps. This procedure leads to trajectories composed of a transient
part and a final cyclic attractor of period p. The tourist walk presents
universal aspects with respect to \tau and can be done in a wide range of
networks that can be viewed as ordinal neighborhood graphs. As an example, we
show that graphs defined by thesaurus dictionaries share some of the
statistical properties of low dimensional (d=2) Euclidean graphs and are easily
distinguished from random graphs. This approach furnishes complementary
information to the usual clustering coefficient and mean minimum separation
length.Comment: 12 pages, 5 figures, revised version submited to Physica A, corrected
references to figure
Mean field and corrections for the Euclidean Minimum Matching problem
Consider the length of the minimum matching of N points in
d-dimensional Euclidean space. Using numerical simulations and the finite size
scaling law , we obtain
precise estimates of for . We then consider
the approximation where distance correlations are neglected. This model is
solvable and gives at an excellent ``random link'' approximation to
. Incorporation of three-link correlations further improves
the accuracy, leading to a relative error of 0.4% at d=2 and 3. Finally, the
large d behavior of this expansion in link correlations is discussed.Comment: source and one figure. Submitted to PR
Exploratory Behavior, Trap Models and Glass Transitions
A random walk is performed on a disordered landscape composed of sites
randomly and uniformly distributed inside a -dimensional hypercube. The
walker hops from one site to another with probability proportional to , where is the inverse of a formal temperature and
is an arbitrary cost function which depends on the hop distance .
Analytic results indicate that, if and , there
exists a glass transition at . Below
, the average trapping time diverges and the system falls into an
out-of-equilibrium regime with aging phenomena. A L\'evy flight scenario and
applications to exploratory behavior are considered.Comment: 4 pages, 1 figure, new versio
Structure Factor and Electronic Structure of Compressed Liquid Rubidium
We have applied the quantal hypernetted-chain equations in combination with
the Rosenfeld bridge-functional to calculate the atomic and the electronic
structure of compressed liquid-rubidium under high pressure (0.2, 2.5, 3.9, and
6.1 GPa); the calculated structure factors are in good agreement with
experimental results measured by Tsuji et al. along the melting curve. We found
that the Rb-pseudoatom remains under these high pressures almost unchanged with
respect to the pseudoatom at room pressure; thus, the effective ion-ion
interaction is practically the same for all pressure-values. We observe that
all structure factors calculated for this pressure-variation coincide almost
into a single curve if wavenumbers are scaled in units of the Wigner-Seitz
radius although no corresponding scaling feature is observed in the
effective ion-ion interaction.This scaling property of the structure factors
signifies that the compression in liquid-rubidium is uniform with increasing
pressure; in absolute Q-values this means that the first peak-position ()
of the structure factor increases proportionally to ( being the
specific volume per ion), as was experimentally observed by Tsuji et al.Comment: 18 pages, 11 figure
Adaptive cluster expansion for the inverse Ising problem: convergence, algorithm and tests
We present a procedure to solve the inverse Ising problem, that is to find
the interactions between a set of binary variables from the measure of their
equilibrium correlations. The method consists in constructing and selecting
specific clusters of variables, based on their contributions to the
cross-entropy of the Ising model. Small contributions are discarded to avoid
overfitting and to make the computation tractable. The properties of the
cluster expansion and its performances on synthetic data are studied. To make
the implementation easier we give the pseudo-code of the algorithm.Comment: Paper submitted to Journal of Statistical Physic
Structure and thermodynamics of multi-component/multi-Yukawa mixtures
New small angle scattering experiments reveal new peaks in colloidal systems
(S.H. Chen et al) in the structure function S(k), in a region that was
inaccessible with older instruments. We propose here general closure of the
Ornstein Zernike equation, that is the sum of an arbitrary number of yukawas,
and that that will go well beyond the MSA . For this closure we get for the
Laplace transform of the pair correlation function . This function is easily
transformed into S(k) by replacing the Laplace variable by the Fourier
wariable. Although the method is general and valid for polydisperse systems, an
explicit continued fraction solution is found for the monodisperse case.Comment: 16 page
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