172 research outputs found
A generalized Kac-Ward formula
The Kac-Ward formula allows to compute the Ising partition function on a
planar graph G with straight edges from the determinant of a matrix of size 2N,
where N denotes the number of edges of G. In this paper, we extend this formula
to any finite graph: the partition function can be written as an alternating
sum of the determinants of 2^{2g} matrices of size 2N, where g is the genus of
an orientable surface in which G embeds. We give two proofs of this generalized
formula. The first one is purely combinatorial, while the second relies on the
Fisher-Kasteleyn reduction of the Ising model to the dimer model, and on
geometric techniques. As a consequence of this second proof, we also obtain the
following fact: the Kac-Ward and the Fisher-Kasteleyn methods to solve the
Ising model are one and the same.Comment: 23 pages, 8 figures; minor corrections in v2; to appear in J. Stat.
Mech. Theory Ex
Cycle bases for lattices of binary matroids with no Fano dual minor and their one-element extensions
Unsigned state models for the Jones polynomial
It is well a known and fundamental result that the Jones polynomial can be
expressed as Potts and vertex partition functions of signed plane graphs. Here
we consider constructions of the Jones polynomial as state models of unsigned
graphs and show that the Jones polynomial of any link can be expressed as a
vertex model of an unsigned embedded graph.
In the process of deriving this result, we show that for every diagram of a
link in the 3-sphere there exists a diagram of an alternating link in a
thickened surface (and an alternating virtual link) with the same Kauffman
bracket. We also recover two recent results in the literature relating the
Jones and Bollobas-Riordan polynomials and show they arise from two different
interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric
Equilibration in the time-dependent Hartree-Fock approach probed with the Wigner distribution function
Calculating the Wigner distribution function in the reaction plane, we are
able to probe the phase-space behavior in time-dependent Hartree-Fock during a
heavy-ion collision. We compare the Wigner distribution function with the
smoothed Husimi distribution function. Observables are defined to give a
quantitative measure for local and global equilibration. We present different
reaction scenarios by analyzing central and non-central and
collisions. It is shown that the initial phase-space
volumes of the fragments barely merge. The mean values of the observables are
conserved in fusion reactions and indicate a "memory effect" in time-dependent
Hartree-Fock. We observe strong dissipation but no evidence for complete
equilibration.Comment: 12 pages, 10 figure
Uniqueness of Ground States for Short-Range Spin Glasses in the Half-Plane
We consider the Edwards-Anderson Ising spin glass model on the half-plane with zero external field and a wide range of choices, including
mean zero Gaussian, for the common distribution of the collection J of i.i.d.
nearest neighbor couplings. The infinite-volume joint distribution
of couplings J and ground state pairs with periodic
(respectively, free) boundary conditions in the horizontal (respectively,
vertical) coordinate is shown to exist without need for subsequence limits. Our
main result is that for almost every J, the conditional distribution
is supported on a single ground state pair.Comment: 20 pages, 3 figure
Transportation energy conservation data book edition i 5
This document contains statistical information on the major transportation modes, their respective energy consumption patterns, and other pertinent factors influencing performance in the transportation sector. Data relating to past, present, and projected energy use and conservation in the transportation sector are presented under seven chapter headings. These focus on (1) modal transportation characteristics, (2) energy characteristics of the transportation sector, (3) energy conservation alternatives involving the transportation sector, (4) government impacts on the transportation sector, (5) the supply of energy to the transportation sector, (6) characteristics of transportation demand, and (7) miscellaneous reference materials such as energy conversion factors and geographical maps. References are included for each set of data presented, and a more general bibliography is included at the end of the book. In addition, a glossary of key terms and a subject index is provided for the user. A second edition of this document is scheduled for publication in September 1977.
Document type: Repor
Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs
We define 2-decompositions of ribbon graphs, which generalise 2-sums and
tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial
of such a 2-decomposition, and derive the classical Brylawski formula for the
Tutte polynomial of a tensor product as a (very) special case. This study was
initially motivated from knot theory, and we include an application of our
formulae to mutation in knot diagrams.Comment: Version 2 has minor changes. To appear in Annals of Combinatoric
Anomalous Dynamics of Forced Translocation
We consider the passage of long polymers of length N through a hole in a
membrane. If the process is slow, it is in principle possible to focus on the
dynamics of the number of monomers s on one side of the membrane, assuming that
the two segments are in equilibrium. The dynamics of s(t) in such a limit would
be diffusive, with a mean translocation time scaling as N^2 in the absence of a
force, and proportional to N when a force is applied. We demonstrate that the
assumption of equilibrium must break down for sufficiently long polymers (more
easily when forced), and provide lower bounds for the translocation time by
comparison to unimpeded motion of the polymer. These lower bounds exceed the
time scales calculated on the basis of equilibrium, and point to anomalous
(sub-diffusive) character of translocation dynamics. This is explicitly
verified by numerical simulations of the unforced translocation of a
self-avoiding polymer. Forced translocation times are shown to strongly depend
on the method by which the force is applied. In particular, pulling the polymer
by the end leads to much longer times than when a chemical potential difference
is applied across the membrane. The bounds in these cases grow as N^2 and
N^{1+\nu}, respectively, where \nu is the exponent that relates the scaling of
the radius of gyration to N. Our simulations demonstrate that the actual
translocation times scale in the same manner as the bounds, although influenced
by strong finite size effects which persist even for the longest polymers that
we considered (N=512).Comment: 13 pages, RevTeX4, 16 eps figure
The critical Ising model via Kac-Ward matrices
The Kac-Ward formula allows to compute the Ising partition function on any
finite graph G from the determinant of 2^{2g} matrices, where g is the genus of
a surface in which G embeds. We show that in the case of isoradially embedded
graphs with critical weights, these determinants have quite remarkable
properties. First of all, they satisfy some generalized Kramers-Wannier
duality: there is an explicit equality relating the determinants associated to
a graph and to its dual graph. Also, they are proportional to the determinants
of the discrete critical Laplacians on the graph G, exactly when the genus g is
zero or one. Finally, they share several formal properties with the Ray-Singer
\bar\partial-torsions of the Riemann surface in which G embeds.Comment: 30 pages, 10 figures; added section 4.4 in version
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