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On defining partition entropy by inequalities
Partition entropy is the numerical metric of uncertainty within
a partition of a finite set, while conditional entropy measures the degree of
difficulty in predicting a decision partition when a condition partition is
provided. Since two direct methods exist for defining conditional entropy
based on its partition entropy, the inequality postulates of monotonicity,
which conditional entropy satisfies, are actually additional constraints on
its entropy. Thus, in this paper partition entropy is defined as a function
of probability distribution, satisfying all the inequalities of not only partition
entropy itself but also its conditional counterpart. These inequality
postulates formalize the intuitive understandings of uncertainty contained
in partitions of finite sets.We study the relationships between these inequalities,
and reduce the redundancies among them. According to two different
definitions of conditional entropy from its partition entropy, the convenient
and unified checking conditions for any partition entropy are presented, respectively.
These properties generalize and illuminate the common nature
of all partition entropies
Residue formulae for vector partitions and Euler-MacLaurin sums
Given a finite set of vectors spanning a lattice and lying in a halfspace of
a real vector space, to each vector in this vector space one can associate
a polytope consisting of nonnegative linear combinations of the vectors in the
set which sum up to . This polytope is called the partition polytope of .
If is integral, this polytope contains a finite set of lattice points
corresponding to nonnegative integral linear combinations. The partition
polytope associated to an integral is a rational convex polytope, and any
rational convex polytope can be realized canonically as a partition polytope.
We consider the problem of counting the number of lattice points in partition
polytopes, or, more generally, computing sums of values of
exponential-polynomial functions on the lattice points in such polytopes. We
give explicit formulae for these quantities using a notion of multi-dimensional
residue due to Jeffrey-Kirwan. We show, in particular, that the dependence of
these quantities on is exponential-polynomial on "large neighborhoods" of
chambers. Our method relies on a theorem of separation of variables for the
generating function, or, more generally, for periodic meromorphic functions
with poles on an arrangement of affine hyperplanes.Comment: Latex, 44 pages, eepic picture file
Geometric asymptotics for spin foam lattice gauge gravity on arbitrary triangulations
We study the behavior of holonomy spin foam partition functions, a form of
lattice gauge gravity, on generic 4d-triangulations using micro local analysis.
To do so we adapt tools from the renormalization theory of quantum field theory
on curved space times. This allows us, for the first time, to study the
partition function without taking any limits on the interior of the
triangulation.
We establish that for many of the most widely used models the geometricity
constraints, which reduce the gauge theory to a geometric one, introduce strong
accidental curvature constraints. These limit the curvature around each
triangle of the triangulation to a finite set of values. We demonstrate how to
modify the partition function to avoid this problem. Finally the new methods
introduced provide a starting point for studying the regularization ambiguities
and renormalization of the partition function.Comment: 4+6 pages, 1 figur
Entanglement, Replicas, and Thetas
We compute the single-interval Renyi entropy (replica partition function) for
free fermions in 1+1d at finite temperature and finite spatial size by two
methods: (i) using the higher-genus partition function on the replica Riemann
surface, and (ii) using twist operators on the torus. We compare the two
answers for a restricted set of spin structures, leading to a non-trivial
proposed equivalence between higher-genus Siegel -functions and Jacobi
-functions. We exhibit this proposal and provide substantial evidence
for it. The resulting expressions can be elegantly written in terms of Jacobi
forms. Thereafter we argue that the correct Renyi entropy for modular-invariant
free-fermion theories, such as the Ising model and the Dirac CFT, is given by
the higher-genus computation summed over all spin structures. The result
satisfies the physical checks of modular covariance, the thermal entropy
relation, and Bose-Fermi equivalence.Comment: 34 page
Partition Function Zeros of an Ising Spin Glass
We study the pattern of zeros emerging from exact partition function
evaluations of Ising spin glasses on conventional finite lattices of varying
sizes. A large number of random bond configurations are probed in the framework
of quenched averages. This study is motivated by the relationship between
hierarchical lattice models whose partition function zeros fall on Julia sets
and chaotic renormalization flows in such models with frustration, and by the
possible connection of the latter with spin glass behaviour. In any finite
volume, the simultaneous distribution of the zeros of all partition functions
can be viewed as part of the more general problem of finding the location of
all the zeros of a certain class of random polynomials with positive integer
coefficients. Some aspects of this problem have been studied in various
branches of mathematics, and we show how polynomial mappings which are used in
graph theory to classify graphs, may help in characterizing the distribution of
zeros. We finally discuss the possible limiting set as the volume is sent to
infinity.Comment: LaTeX, 18 pages, hardcopies of 15 figures by request to
[email protected], CERN--TH-7383/94 (a note and a reference added
Ising Model Observables and Non-Backtracking Walks
This paper presents an alternative proof of the connection between the
partition function of the Ising model on a finite graph and the set of
non-backtracking walks on . The techniques used also give formulas for
spin-spin correlation functions in terms of non-backtracking walks. The main
tools used are Viennot's theory of heaps of pieces and turning numbers on
surfaces.Comment: 33 pages, 11 figures. Typos and errors corrected, exposition
improved, results unchange
Condensation of Ideal Bose Gas Confined in a Box Within a Canonical Ensemble
We set up recursion relations for the partition function and the ground-state
occupancy for a fixed number of non-interacting bosons confined in a square box
potential and determine the temperature dependence of the specific heat and the
particle number in the ground state. A proper semiclassical treatment is set up
which yields the correct small-T-behavior in contrast to an earlier theory in
Feynman's textbook on Statistical Mechanics, in which the special role of the
ground state was ignored. The results are compared with an exact quantum
mechanical treatment. Furthermore, we derive the finite-size effect of the
system.Comment: 18 pages, 8 figure
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