7,561 research outputs found
Spin-spin Correlation in Some Excited States of Transverse Ising Model
We consider the transverse Ising model in one dimension with
nearest-neighbour interaction and calculate exactly the longitudinal spin-spin
correlation for a class of excited states. These states are known to play an
important role in the perturbative treatment of one-dimensional transverse
Ising model with frustrated second-neighbour interaction. To calculate the
correlation, we follow the earlier procedure of Wu, use Szego's theorem and
also use Fisher-Hartwig conjecture. The result is that the correlation decays
algebraically with distance () as and is oscillatory or
non-oscillatory depending on the magnitude of the transverse field.Comment: 5 pages, 1 figur
The 1999 Heineman Prize Address- Integrable models in statistical mechanics: The hidden field with unsolved problems
In the past 30 years there have been extensive discoveries in the theory of
integrable statistical mechanical models including the discovery of non-linear
differential equations for Ising model correlation functions, the theory of
random impurities, level crossing transitions in the chiral Potts model and the
use of Rogers-Ramanujan identities to generalize our concepts of Bose/Fermi
statistics. Each of these advances has led to the further discovery of major
unsolved problems of great mathematical and physical interest. I will here
discuss the mathematical advances, the physical insights and extraordinary lack
of visibility of this field of physics.Comment: Text of the 1999 Heineman Prize address given March 24 at the
Centenial Meeting of the American Physical Society in Atlanta 20 pages in
latex, references added and typos correcte
Negative virial coefficients and the dominance of loose packed diagrams for D-dimensional hard spheres
We study the virial coefficients B_k of hard spheres in D dimensions by means
of Monte-Carlo integration. We find that B_5 is positive in all dimensions but
that B_6 is negative for all D >= 6. For 7<=k<=17 we compute sets of Ree-Hoover
diagrams and find that either for large D or large k the dominant diagrams are
"loose packed". We use these results to study the radius of convergence and the
validity of the many approximations used for the equations of state for hard
spheres.Comment: 26 pages, 69 figures. Some typos corrected. Final version, to appear
in the Journal of Statistical Physic
The energy density of an Ising half plane lattice
We compute the energy density at arbitrary temperature of the half plane
Ising lattice with a boundary magnetic field at a distance rows from
the boundary and compare limiting cases of the exact expression with recent
calculations at done by means of discrete complex analysis methods.Comment: 12 pages, 1 figur
The anisotropic Ising correlations as elliptic integrals: duality and differential equations
We present the reduction of the correlation functions of the Ising model on
the anisotropic square lattice to complete elliptic integrals of the first,
second and third kind, the extension of Kramers-Wannier duality to anisotropic
correlation functions, and the linear differential equations for these
anisotropic correlations. More precisely, we show that the anisotropic
correlation functions are homogeneous polynomials of the complete elliptic
integrals of the first, second and third kind. We give the exact dual
transformation matching the correlation functions and the dual correlation
functions. We show that the linear differential operators annihilating the
general two-point correlation functions are factorised in a very simple way, in
operators of decreasing orders.Comment: 22 page
From Steiner Formulas for Cones to Concentration of Intrinsic Volumes
The intrinsic volumes of a convex cone are geometric functionals that return
basic structural information about the cone. Recent research has demonstrated
that conic intrinsic volumes are valuable for understanding the behavior of
random convex optimization problems. This paper develops a systematic technique
for studying conic intrinsic volumes using methods from probability. At the
heart of this approach is a general Steiner formula for cones. This result
converts questions about the intrinsic volumes into questions about the
projection of a Gaussian random vector onto the cone, which can then be
resolved using tools from Gaussian analysis. The approach leads to new
identities and bounds for the intrinsic volumes of a cone, including a
near-optimal concentration inequality.Comment: This version corrects errors in Propositions 3.3 and 3.4 and in Lemma
8.3 that appear in the published versio
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